Description
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Lecture 6.2
Using Derivatives to Hedge
FIÃ¢â‚¬â„¢s Exposures to Financial Risks
Ã¯â€šÂ§ Hedging
– Definition: Using one transaction (position) to offset the effect of another transaction
(position) to reduce risk of loss
– Micro vs. Macro Hedge:
o Micro: hedge a particular asset or liability against a given risk (e.g. hedge a bond
portfolio against rise/fall in interest rate)
o Macro: hedge entire balance sheet against a given risk (e.g. hedge duration gap
against interest rate risk)
o Macro hedge gives better result due to portfolio effect; individual risks net out
each other
– On vs. Offbalance sheet risk management:
o Gap analysis uses direct refinancing on the balance sheet to manage interest rate
risk
o Macro hedge uses derivatives off the balance sheet to manage/reduce various onB/S risks
– Commonality of hedging instruments:
o They are derivatives; the value of a derivative is derived from the value of an
underlying variable such as stock price, interest rate, exchange rate, or any
security/property price.
o Each calls for delivery of an asset at a future date, based on the terms agreed on
today; the deliverable asset can be gold, cash, stock, bond, currency, or property.
o They involve exchange of some variable feature for some fixed feature at a future
time.
Ã¯â€šÂ§ Futures
– Definition: Agreement between a seller and a buyer that calls for delivery of an asset in
exchange for fixed delivery price at some future date.
– Nature of futures contracts:
o Later on, the seller has the obligation to make delivery of the asset, while the
buyer has the obligation to take delivery at todayÃ¢â‚¬â„¢s agreed delivery price;
o The seller will gain if the value of the deliverable asset falls; (if the deliverable is
bond, then gain if rate rises);
o The buyer will gain if the value of the deliverable asset rises; (if the deliverable is
bond, then gain if interest rate falls);
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o The fixed delivery price of a futures contract reflects the market price/value of the
deliverable asset in the future;
o The duration of a bond futures contract depends on and is equal to the duration of
the deliverable asset (i.e. bond).
– Macro hedging with bond futures (the deliverable is bond)
OnB/S, an FIÃ¢â‚¬â„¢s exposure to interest rate risk:
Ã¯Ââ€žE Ã¯â‚¬Â½ Ã¯â‚¬Â DG Ã¯â€šÂ´ A
Ã¯Ââ€žR
1Ã¯â‚¬Â« R
— (1)
OffB/S, an FIÃ¢â‚¬â„¢s exposure to interest rate risk:
Ã¯Ââ€žF Ã¯â‚¬Â½ Ã¯â‚¬Â DF Ã¯â€šÂ´ F Ã¯â€šÂ´
Ã¯Ââ€žR
1Ã¯â‚¬Â« R
— (2)
where,
F Ã¯â‚¬Â½ N F Ã¯â€šÂ´ PF is the futures position
— (3)
NF: number of futures contracts making up the futures position F
PF: futures price of each futures contract (i.e. the agreed fixed delivery price for
delivering the underlying asset Ã¢â‚¬â€œ the bond)
DF: duration of the deliverable asset used in the futures contract (here bond)
To hedge perfectly the onB/S duration gap against interest rate risk (i.e. to realise
gain from offB/S futures position that will exactly offset loss from onB/S position), we
need:
Ã¯Ââ€žE Ã¯â‚¬Â½ Ã¯Ââ€žF
(perfect hedge against ÃŽâ€R)
Using equations (1) to (3) to solve for NF:
NF Ã¯â‚¬Â½
DG Ã¯â€šÂ´ A
DF Ã¯â€šÂ´ PF
Number of bond futures contracts to be
bought (if DG < 0) or sold (if DG > 0)
– Hedging strategy using futures:
DG > 0
(or Dollar Gap < 0)
If R rises, then loss on B/S;
DG < 0
(or Dollar Gap > 0)
If R falls, then loss on B/S;
But if R rises, then gain off B/S for
seller of futures.
So sell futures against rising rates
But if R falls, then gain off B/S for
buyer of futures.
So buy futures against falling rates
i.e. shorthedge
i.e. longhedge
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– Example (using futures to hedge):
Bank One has:
A = $64 million; DA = 4.75 years.
L = $58 million; DL = 3.28 years.
A futures contract is currently priced at 102.6 on Treasury bonds with a duration of 8
years. Should the bank take a long or short hedge with the Treasury bond futures
contract? How many contracts would the bank need to conduct a perfect hedge?
Ã¯â€šÂ§ Swap
– Definition: Contract that obliges two parties to exchange financial obligations (e.g.
interest payments on some notional amount) that have different features, e.g. variable
interest rate vs. fixed interest rate
– Example (interest rate swap)
FI A: fixedrate assets & floating rate liabilities
DG > 0 (liability sensitive)
If interest rate rises, then it suffers loss (of course, if rate falls, then it gains)
FI B: floatingrate assets & fixedrate liabilities
DG < 0 (asset sensitive)
If interest rate falls, then it suffers loss (of course, if rate rises, then it gains)
These two FIs have complementary needs, so they can arrange a swap, thereby
reducing their exposures to interest rate risk. How?
In this swap, A pays to B fixedrate interests, and B pays to A variablerate interests on
a notional amount (e.g. $2 millions). Why should B make variablerate payments and A
make fixedrate payments?
o FI B has comparative advantage in paying variable rate. Note that B has floating
rate assets; if rates go up, these floating rate assets will earn more interest
income), so B can afford to pay variable rates. However, if rates go down, B loses
as its floating rate assets will earn less, so B needs protection against falling rates,
which can be offered by a fixedrate payer like A. Of course, in entering a swap, B
forgoes potential gains from rising rates, but it obtains protection against falling
rates.
o The intuition is reversed with FI A.
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 Hedging strategy using swap:
DG > 0
(or Dollar Gap < 0)
If R rises, then fixedrate payer will
receive more interests than it will pay in
the swap, thus realizing gains offB/S
DG < 0
(or Dollar Gap > 0)
If R falls, then variablerate payer will
receive more interests than it will pay in
the swap, thus realizing gains offB/S
fixedrate payer or
(variablerate receiver)
variablerate payer or
(fixedrate receiver)
Note: – By convention, fixedrate payer is called swap buyer, and variablerate
payer is called swap seller.
– Notional amount of swap in macro hedge:
To determine the notional amount of the swap, notice that an interest rate swap is
effectively a pair of bonds with face value NS — one with fixed rate and the other with
variable rate, and each has its own duration, DFix and DVar. When interest rates
( Ã¯Ââ€žR /(1 Ã¯â‚¬Â« R) ) move, the change in the value of the swap (ÃŽâ€S) equals the change in the
value of the pair of bonds, which will depend on the relative interest sensitivity of the
two bonds, (DFix Ã¢â‚¬â€œ DVar). In general,
Ã¯Ââ€žR
Ã¯Ââ€žS Ã¯â‚¬Â½ Ã¯â‚¬Â( DFix Ã¯â‚¬Â DVar ) Ã¯â€šÂ´ N S Ã¯â€šÂ´
1Ã¯â‚¬Â« R
where, NS is the Notional amount of the swap.
The optimal notional amount of swap that will allow gain on offB/S swap to
exactly offset loss in equity onB/S so as to achieve perfect hedge Ã¯Ââ€žS Ã¯â‚¬Â½ Ã¯Ââ€žE implies:
Ã¯â‚¬Â ( DFix Ã¯â‚¬Â DVar ) Ã¯â€šÂ´ N S Ã¯â€šÂ´
Ã¯Ââ€žR
Ã¯Ââ€žR
Ã¯â‚¬Â½ Ã¯â‚¬Â DG Ã¯â€šÂ´ A Ã¯â€šÂ´
1Ã¯â‚¬Â« R
1Ã¯â‚¬Â« R
Solving for NS,
NS Ã¯â‚¬Â½
DG Ã¯â€šÂ´ A
DFix Ã¯â‚¬Â DVar
Notional amount of swap to buy (if DG > 0)
or to sell (if DG < 0)
 Example (using swap to hedge):
An FI has DA = 4.5, DL = 3, L/A = 0.8, and A = $52 million. Duration of the fixed side of
a swap is 7 years, while the variable side is repriced annually, so DFix = 7 and DVar =1.
Ã¯Æ’Â
Ã¯Æ’Â
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Ã¯â€šÂ§ Option
 Definition: Contract that grants option buyer (owner) the right (not obligation as with
futures) to buy or sell an underlying asset (the deliverable asset as with futures) at fixed
price during specified time period to hedge against loss (or bet on profits for a
speculator).
 Macro hedging interest rate risk with option
(The underlying asset is bond). Similar intuition to that with futures where the deliverable
asset is a bond.
 Hedging strategy using options:
DG > 0
(or Dollar Gap < 0)
DG < 0
(or Dollar Gap > 0)
Buy Put / Sell Call options on bond to
offset with premium (profit) loss from
rising interest rate
Buy Call / Sell Put options on bond to
offset with premium (profit) loss from
falling interest rate
o Rationale:
(To see why, first be clear about the nature of the exposure, i.e., whether FIÃ¢â‚¬â„¢s onB/S is
exposed to rising or falling interest rate?)
DG > 0 and if R rises
DG < 0 and if R falls
Then loss onB/S
FI sells call and receives premium; price of
the underlying asset (bond) falls; buyer
(FIÃ¢â‚¬â„¢s counter party in the option
transaction) of the call will not call and buy
the underlying at the (higher) strike price
(since the bond can be bought at lower
market price). So the seller (the FI) will not
deliver bond and can use the premium to
offset loss onB/S.
FI buys put and pays premium; price of the
underlying asset (bond) falls; buyer (the FI)
of the put will put and sell the underlying at
the (higher) strike price (since the bondÃ¢â‚¬â„¢s
market price is lower). So the buyer (the
FI) will deliver bond and can make profit
to offset premium paid and loss onB/S
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 Size of offB/S option position to hedge onB/S exposure
OnB/S , FIÃ¢â‚¬â„¢s exposure to interest rate risk is:
L
Ã¯Ââ€žR
Ã¯Ââ€žR
Ã¯Ââ€žE Ã¯â‚¬Â½ Ã¯â‚¬Â[ D A Ã¯â‚¬Â DL ] Ã¯â€šÂ´ A Ã¯â€šÂ´
Ã¯â‚¬Â½ Ã¯â‚¬Â DG Ã¯â€šÂ´ A Ã¯â€šÂ´
1Ã¯â‚¬Â« R
A
1Ã¯â‚¬Â« R
OffB/S, FIÃ¢â‚¬â„¢s position in put option on bond is:
P Ã¯â‚¬Â½ Np Ã¯â€šÂ´ p
 (4)
where, P: put option position (in $) taken by the FI;
p: price of a put option contract;
Np: number of put options making up the FIÃ¢â‚¬â„¢s option position.
In terms of changes: Ã¯Ââ€žP Ã¯â‚¬Â½ N p Ã¯â€šÂ´ Ã¯Ââ€žp
 (5)
From option pricing theory, change in option price is:
D
Ã¯Ââ€žp Ã¯â‚¬Â½ (Ã¯â‚¬ÂÃ¯ÂÂ¤ ) Ã¯â€šÂ´ (Ã¯â‚¬Â B ) Ã¯â€šÂ´ B Ã¯â€šÂ´ Ã¯Ââ€žR
1 Ã¯â‚¬Â« R)
where,
 (6)
ÃŽÂ´: delta of an option; it indicates the optionÃ¢â‚¬â„¢s price change given a $1
change in the price of the deliverable bond underlying the option;
DB: duration of the bond underlying the option;
B: market price of the bond;
R: interest rate.
Ã¯Ââ€žR Ã¯Æ’Â¹
Ã¯Æ’Â©
Using (5) and (6), we have Ã¯Ââ€žP Ã¯â‚¬Â½ N p Ã¯â€šÂ´ Ã¯Æ’ÂªÃ¯ÂÂ¤ Ã¯â€šÂ´ DB Ã¯â€šÂ´ B Ã¯â€šÂ´
1 Ã¯â‚¬Â« R Ã¯Æ’ÂºÃ¯Æ’Â»
Ã¯Æ’Â«
To achieve perfect hedge, we need: Ã¯Ââ€žE Ã¯â‚¬Â½ Ã¯Ââ€žP
i.e.
Ã¯â‚¬Â DG A
Ã¯Ââ€žR
Ã¯Ââ€žR
Ã¯â‚¬Â½ Ã¯â‚¬Â N p Ã¯â€šÂ´ Ã¯ÂÂ¤ Ã¯â€šÂ´ DB Ã¯â€šÂ´ B Ã¯â€šÂ´
1Ã¯â‚¬Â« R
1Ã¯â‚¬Â« R
Solving for Np:
Np Ã¯â‚¬Â½
DG Ã¯â€šÂ´ A
Ã¯ÂÂ¤ Ã¯â€šÂ´ DB Ã¯â€šÂ´ B
Number of put options to buy (if DG > 0).
If using call options to hedge:
Nc Ã¯â‚¬Â½
DG Ã¯â€šÂ´ A
Ã¯â‚¬Â Ã¯ÂÂ¤ Ã¯â€šÂ´ DB Ã¯â€šÂ´ B
Number of call options to sell (if DG > 0).
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– Example (using options to hedge):
Corporate Bank has $84 million of assets with duration of 9.2 years, and liabilities
worth $72 million with duration of 7 years. The bank is contemplating a macro hedge
with bond options to preserve its equity in the event of change in interest rates.
The call and put options have a delta (Ã¯ÂÂ¤) of 0.6 and Ã¢â‚¬â€œ0.4, respectively. The bonds
underlying the put options have a market value of $104,258 and the duration of these
bonds is 8.25 years.
a. What type of option should Corporate Bank use for the macro hedge?
b. How many options should be purchased?
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Financial Institutions
Identifying How a Derivative is Used
The RSA & RSL of a bank for its repricing buckets read as following:
0.5year
1year
3year
Beyond
RSA
58,638
84,586
82,300
Ã¢â‚¬Â¦
RSL
65,000
52,300
85,800
Ã¢â‚¬Â¦
futures
Ã¢â‚¬Â¦
Swap (notional)/Futures
Ã¢â‚¬Â¦
Postderivative
periodic dollar gap
a) For the 0.5year bucket:
Without swap, the priorswap dollar gap would be:
__________________________________
With the swap, the postswap dollar gap is – $1,362.
This is equivalent to adding $5,000 RSA in this bucket. The bank has used the
swap of $5,000 notional to move its 0.5year dollar gap closer to zero (less negative). So,
the use of the swap represents a
period (but
against
interest rate over this time
).
b) For the 1year bucket:
Without swap, the priorswap dollar gap would be:
__________________________________
With the swap, the postswap dollar gap is $40,286.
This is equivalent to adding $8,000 RSA in this bucket. The bank has used the
swap of $8,000 notional to move its 1year gap more away from zero (more positive). So,
the use of the swap represents
period.
on
interest rate over this time
Financial Institutions
c) For the 3year bucket:
Without futures, the priorfutures dollar gap would be:
________________________________
With the futures, the postfutures dollar gap is $2,450.
This means that the bank has used the futures to reverse its 3year gap (from
negative to positive). So, the use of the futures represents
interest rate over this time period (now
).
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Lecture 7.1
Market Risk & Value at Risk (VaR)
Ã¯â€šÂ§
What is VaR?
o Recall in Statistics – Quantile, L( p) :
1 p
p
L( p)
Quantile is the quantity or value that is associated with a particular
probability. For example, with normal distribution, the quantiles of the popular
probabilities of 50%, 1%, 5%, and 10% are, respectively:
L(50%) Ã¯â‚¬Â½
L(1%) Ã¯â‚¬Â½
L(5%) Ã¯â‚¬Â½
L(10%) Ã¯â‚¬Â½ E Ã¯â‚¬Â 1.2816Ã¯ÂÂ³
o VaR – Definition:
1 p (confidence level)
p
$
VaR
VaR = pth quantile
ValueatRisk is the critical loss (relative to the original or expected value)
that can be exceeded over a certain time period (h) with a given probability (p).
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Ã¯â€šÂ§
Relative VaR & Absolute VaR:
p
Absolute VaR
$
V0 EV
Relative VaR
Ã¯â€šÂ§
What determine VaR?
1. How are VaR and confidence level related?
The higher the confidence level, 1 p, the larger the (negative) VaR will be.
2. How are VaR and volatility related?
At the same confidence level, 1 p, the larger the volatility, Ã¯ÂÂ³ , the larger the
(negative) VaR:
The larger the volatility, Ã¯ÂÂ³ , the wider the range of the possible values, and the
larger the extreme values (both large positive and larger negative), so the larger the VaR
for the same confidence level, 1 – p.
3. How are VaR and time related?
o The independently and identically distributed (i.i.d.) changes assumptions:
– changes of value are uncorrelated over successive time intervals (the Efficient
Market Hypothesis)
– changes have the same distribution over time (a technical assumption)
– Volatility grows with the square root of time, t .
e.g., volatility of loss over 2 periods is:
Ã¯ÂÂ³ 1~ 2 Ã¯â‚¬Â½ Var ( L1~ 2 ) Ã¯â‚¬Â½ 2Ã¯ÂÂ³ 2 Ã¯â‚¬Â½ 2 Ã¯â€šÂ´ Ã¯ÂÂ³
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This last property (square root of time) of VaR is extremely useful when we
calculate VaR for different time horizons.
o Example — converting VaR between different time periods:
If we know the daily VaR, how to find the annual VaR?
VaRp,1year = year/day Ã¯â€šÂ´ VaRp,1day
or
VaRp,1year =
250 Ã¯â€šÂ´ VaRp,1day
Since there are 250 trading days in a year.
4. How are VaR and position related?
The larger the position in dollar amount, the larger the potential loss, and larger
the VaR.
Ã¯Æ’Ëœ To summarize, (negative) VaR increases with:
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¯â€šÂ§
confidence level, (1 p)
volatility, ( Ã¯ÂÂ³ )
time horizon, (T)
position, ($)
Calculate VaR Ã¢â‚¬â€œ Example
Suppose we want to measure the VaR of a $100 million equity position which
expects to grow by 0.98% over a time horizon of 10 days, at a confidence level of 99%.
The annual volatility of rate of return on the equity Ã¯ÂÂ³ 1Ã¯â‚¬Â year Ã¯â‚¬Â½ 15% , which is s. d. of the
distribution of percentage change in price over 1 year.
a. Marktomarket of the current position: $100 million;
b. Set the confidence level: 99%, which corresponds to 2.3263 times s.d. with a
normal distribution;
c. Measure 10day volatility, Ã¯ÂÂ³ 10Ã¯â‚¬Âday (as only Ã¯ÂÂ³ 1Ã¯â‚¬Â year Ã¯â‚¬Â½ 15% is given), using rule
of Ã¢â‚¬Å“square root of timeÃ¢â‚¬Â:
Ã¯ÂÂ³ 10Ã¯â‚¬Âday Ã¯â‚¬Â½
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distribution of rate of return in 10 days
Ã¯ÂÂ³ 10Ã¯â‚¬Â d
1%
Ã¯ÂÂ³ 10Ã¯â‚¬Â d
2.3263 Ã¯â€šÂ´3%
r (%)
0.98%
3%
3%
rVaR =
Absolute VaR of position10day = V0Ãƒâ€”(1 + rVaR) =
distribution of position in 10 days
1%
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$
$
94
$
100 $100.98
Interpretations:
– with an expected 0.98% growth rate over the next 10 days, we have 1%
probability to end up with less than $94M (i.e. lose more than $6M);
– with an expected 0.98% growth rate over the next 10 days, we have 99%
confidence level to end up with more than $94M (i.e. lose less than $6M).
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Lecture 7.2
Applications of VaR Ã¢â‚¬â€œ Risk Capital & RAROC
Ã¯â€šÂ§
Risk Capital
So, risk capital is the minimum level of equity capital needed today to weather
potential losses that occur at horizon T with probability (1 p).
$
+ 8M
+ 7M
Distribution of
values at horizon
gain
+ 6M
+ 5M
current
position
(V0)
risk capital
(EVT)
growth
expected value (most likely position)
expected profit
zero loss
potential loss
0
– 1M
loss
– 2M
VaRp,T
– 3M
p
– 4M
time
t=0
t=T
Note: The VaRp,T here is the pth quantile, which is always 1.6449 (if p = 5%) or 2.3263 (if p = 1%)
standard deviations away from the central expected value of a normal distribution.
Risk capital is the gap between the pth quantile (i.e. the VaRp,T) from the
distribution of a position at horizon T and the expected value of the distribution, with
the gap being discounted to reflect its present value:
risk capital = PV of potential loss = PV of (VaRp,T Ã¢â‚¬â€œ EVT)
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Ã¯â€šÂ§
RAROC
o Definition:
RAROC, for RiskAdjusted Return On Capital, is a ratio of expected profit from
taking a position (in trading, investing, lending, etc.) over potential loss arising from
taking the position. It reflects how much capital will be put at risk (of being lost) in
order to earn this profit.
RAROC =
annual expected profit of a position
EV Ã¢â‚¬â€œ V0
=
risk capital due to the position
VaR Ã¢â‚¬â€œ V0
Note:
– It is profit as a percent of the risk capital that an FIÃ¢â‚¬â„¢s shareholders need to set
aside and may potentially lose with some probability.
– RAROC is conventionally an annual rate of return. So, the VaRÃ¢â‚¬â„¢s time interval
should be one year:
VaRp,1year =
250 Ã¯â€šÂ´ VaRp,1day
– In comparison with RAROC which is a VaRbased measure of profitability,
here is the traditional measure of profitability:
Traditional measure of profitability =
Ã¯â€šÂ§
EV Ã¢â‚¬â€œ V0
V0
Example — Calculating RAROC:
A Canadian FI has a bond portfolio consisting of $100 million long position in
Canadian bonds and $100 million short position in foreign bonds (so current position
is zero!). The expected profit of the portfolio over the next two weeks (i.e. 10 trading
days) is $4,000. Suppose daily percentage changes in bond price (not yield) are
normally distributed, with zero mean, standard deviations (ÃÆ’1day) for the Canadian
and foreign bonds being 0.1897% and 0.3162%, respectively, and correlation between
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the two bond markets being 0.67. The probability of loss is chosen at 1% (i.e. a
confidence level of 99%).
Long position: buying/holding of assets, with intent to sell them at higher prices.
Short position: shortselling of assets, with intent to buy them back at lower prices.
— For the Canadian bond long position:
VaRlong (1%, 1day) =
VaRlong (1%, 10days) =
— For the foreign bond short position:
VaR short (1%, 1day) =
VaR short (1%, 10days) =
(Note: the negative sign for short position)
— Capital at risk associated with the portfolio over the 10day period:
VaR Total (1%, 10days)
= (VaRC. Bond 2 + VaRF. Bond 2 + 2 Corr Ã¯â€šÂ´ VaRC. Bond Ãƒâ€”VaRF. Bond )1/2
=
=
— RAROC of the portfolio:
RAROC10days =
RAROC1year =
Now, this annual RAROC can be used to decide if the two positions should be
taken. For example, if the benchmark (i.e. required annual rate of return on equity) is
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7%, then this portfolio with an annual RAROC of 5.94% is not worth investing and
should be divested if already held.
This example illustrates how VaR can resolve the puzzle of calculating rates of
return on many investments that require no upfront capital, such as here with equal
long and short positions, and other futures and swap positions. Apparently, these
activities involve no initial capital, but in reality, they put an FIÃ¢â‚¬â„¢s capital at risk if
losses occur later.
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Lecture 7.3
Credit VaR & Risk Capital
— Using actual data
A bank is lending a 5year, fixed 6% annual interest rate, $100 million face value
loan to a corporate borrower. The borrowing firm has a credit rating of BBB.
From historical data, a BBBclass borrower has 86.93% probability to remain a
BBBborrower, but also has some probabilities that its creditworthiness may improve or
deteriorate over the next year. The following table summarizes these transition
probabilities and associated loan values.
Rating
AAA
AA
A
BBB
BB
B
CCC
Default
Probability
0.02%
0.33
5.95
86.93
5.30
1.17
0.12
0.18
Loan Value
(include.
interest)
Probabilityweighted
Value
$109.37
109.19
108.66
107.55
102.02
98.10
83.64
51.13
$0.02
0.36
6.47
93.49
5.41
1.15
0.10
0.09
Mean = $107.09
Source: Standard &PoorÃ¢â‚¬â„¢s CreditWeek
Difference
(Value Ã¢â‚¬â€œ
Mean)
Probabilityweighted
Difference
Squared
$2.28
2.10
1.57
0.46
5.07
8.99
23.45
55.96
0.0010
0.0146
0.1467
0.1839
1.3624
0.9456
0.6599
5.6367
Variance = 8.95
Ã¯ÂÂ³ Ã¯â‚¬Â½ $2.99
The probability distribution of loan values is plotted below. We see that the
distribution is not normal Ã¢â‚¬â€œ it has limited maximum and minimum values, and is
negatively skewed to the left (not symmetric, with longtail downside risk).
Probability
51.13
107.09 109.37
1
Value of loan
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Based on this actual distribution, the bank can estimate its Ã¢â‚¬Å“worstcase scenariosÃ¢â‚¬Â
with certain confidence levels, for example, the 1% VaR over a 1year risk horizon.
From the table, we see that there is a 1.47% (= 0.18% + 0.12% + 1.17%)
probability that the loan will fall below $98.10 million, implying an actual 1.47% VaR of
$107.09 Ã¢â‚¬â€œ $98.10 = $8.99 million. The exact 1% VaR can be obtained by employing
linear interpolation. Specifically, since 1.47% corresponds to $98.10, and 0.3% (= 0.18%
+ 0.12%) corresponds to $83.64, then:
which suggests an actual VaR1% (relative to mean):
VaR1% = $107.09 Ã¢â‚¬â€œ $92.29 = $14.80 million
Today, the market value of this BBBgrade loan is $103.02 million. Given the
loanÃ¢â‚¬â„¢s expected value of $107.09 next year, the marketÃ¢â‚¬â„¢s expected rate of return on the
loan, ER, is thus:
ERBBB Ã¯â‚¬Â½
EVBBB
Ã¯â‚¬Â1 Ã¯â‚¬Â½
PBBB
2
Ã‚Â© Jie Dai
0
PBBB
($103.02)
$6
$6
$6
$6
$106
1
2
3
4
5
EV
($107.09)
What capital structure should the bank use for financing this loan? Capital
structure is stated in present value (PV). Indeed, eq. (7.4) in lecture note 7.2 defines risk
capital as:
risk capital = PV of potential loss = PV of (VaRp,T Ã¢â‚¬â€œ EVT)
When future values are lognormally distributed, present value can be calculated
using BlackScholes formula. However, when future values follow some unconventional
distribution, such as the case here with a risky loan, no general formula can be used for
calculating the present value, and what discount rate should be used becomes an issue. A
sensible way to obtain present value is to use expected rate of return over the relevant
time horizon as the discount rate.
Thus, the risk capital is calculated as the following:
Risk capital = E =
EVT Ã¯â‚¬Â absolute VaR T
=
1 Ã¯â‚¬Â« ERBBB
This risk capital implies that to absorb Ã¢â‚¬Å“worstÃ¢â‚¬Â loan losses Ã¢â‚¬â€œ defined as the 1%
relative VaR of loan value distribution over a risk horizon of 1 year, the bank needs to
hold today at least $14.24 million of equity (risk capital) Ã¢â‚¬â€œ for engaging in this lending
activity.
The capital structure for financing this risky loan is thus:
Capital structure =
D V Ã¯â‚¬ÂE
Ã¯â‚¬Â½
Ã¯â‚¬Â½
V
V
A lending FI employing a higher D/V ratio is said to be undercapitalized.
3
Financial Institutions
Lecture 8.2
Examples in Economics of Information
Example 1 (Adverse Selection)
Suppose: Used car market with three levels of quality: qG > qM > qB, where
value (qG) = $10,000, value (qM) = $5,000, and value (qB) = $0.
Riskneutral valuation: price = expected value. So, owner will not sell his car at
a price lower than its value, and buyer will not pay more than the carÃ¢â‚¬â„¢s value.
Asymmetric information: owner knows his carÃ¢â‚¬â„¢s quality qi, (i = G, M, B),
whereas buyer knows only proportions/probabilities of Pr(qG)= 0.4, Pr(qM)=0.2,
and Pr(qB)=0.4
Question: What will happen in such a market?
— SOLUTION TO COME –Example 2 (Signaling)
Suppose: Different qualities have different probabilities of engine failure after one year:
Pr(qG fails) = 0.1, Pr(qM fails) = 0.5, and Pr(qB fails) = 1.0
Question: What to do to reestablish the market?
— SOLUTION TO COME –Notes: WM = $10,000 is twice the value of qM! Does this make sense?
1) WM will only be paid with probability 0.5, so we should look at expected
liability which is $10,000Ãƒâ€”0.5 = $5,000. But still this is quite high relative to qMÃ¢â‚¬â„¢s value
of $5,000.
2) This is due to the need to install a credible mechanism (the IC conditions) to
overcome the information problem (adverse selection).
3) The IC conditions here ensure perfect revelation, which is a strong requirement.
Generally, partial revelation imposes less strong condition.
4) Finally, the price PM is actually buying two things: immediate value of car
($5,000) and future value of warranty ($5,000 = 0.5Ãƒâ€”10,000)! (The net value received by
seller of qM equals exactly the value of the car — $5,000 = immediate price minus future
warranty).
(The same is true for qG, PG = $11,250, net value = 11,250 Ã¢â‚¬â€œ 0.1WG = $10,000)
Now, with the warranty plan {WG, WM, 0}, qG and qM are traded at PG and PM,
respectively, and qB is withdrawn from the market Ã¯â€šÂ® market reestablished.
Here, the warranties offered serve as signals of qualities. They are observable and
credible with credibility being guaranteed by the IC conditions.
The very activities of offering warranty plan {WG, WM, 0} constitute signaling.
What allows for such signaling/revelation of private information is the IC conditions.
Lecture 9.1
Credit Contracting
Example 3 (Asymmetric information & Capital)
Suppose: With a pool of firms, each firm can invest $150,000 (at t = 0) in a project that
will yield uncertain payoff one period hence (at t = 1). There are two types of
firms: L and H (low and highrisk projects).
firm L:
0.8
$300,000
if project succeeds
0.2
$0
if project fails
0.5
$600,000
if project succeeds
0.5
$0
if project fails
$150,000
firm H:
$150,000
Note: H is of high risk because it yields higher payoff ($600,000) with lower probability
(0.5), compared to LÃ¢â‚¬â„¢s payoff ($300,000) and probability (0.8).
The firms want to borrow the funds for the investments from a bank. While each
firm knows which type its project is, the bank does not know whether a particular firm
has a Lrisk or Hrisk project (even after conducting a careful credit analysis). Interest
payments on loans are tax deductible and firm corporate tax rate is 30%.
Question: If you are the banker, how would you make the lending decision?
Solution:
You need to craft a mechanism that will elicit the private information. That is, you
design loans in such a way that different types of borrowers will use different loans.
Specifically, you can offer the borrowers two loan choices:
loan (h): borrow the entire $150,000, and repay PH (P is the price of the loan);
loan (l): put up $E in equity capital, borrow $150,000 Ã¢â‚¬â€œ E, and repay PL.
1
First, you assume/infer that typeH borrower Ã¯â€šÂ¾choose
Ã¯â€šÂ¾Ã¯â€šÂ¾Ã¯â€šÂ® loan (h);
typeL borrower Ã¯â€šÂ¾choose
Ã¯â€šÂ¾Ã¯â€šÂ¾Ã¯â€šÂ® loan (l).
Second, you set the loan prices (repayment obligations) PH and PL to break even:
bankÃ¢â‚¬â„¢s payoff with loan (h):
0.5
PH
0.5
$0
bankÃ¢â‚¬â„¢s payoff with loan (l):
0.8
PL
0.2
$0
$(150,000 Ã¢â‚¬â€œ E)
$150,000
For loan (h): PH Ãƒâ€” (0.5) + $0 Ãƒâ€” (0.5) = $150,000
Ã¯Æ’Â¨(BEh)
Ã¯â€šÂ® PH = $300,000 or an interest rate rH = 100%
For loan (l): PL Ãƒâ€” (0.8) + $0 Ãƒâ€” (0.2) = $150,000 Ã¢â‚¬â€œ E
Ã¯Æ’Â¨(BEl)
Ã¯â€šÂ® PL = 150,000 Ã¯â‚¬Â E
0.8
Third, to assure that your assumption/inference about the association between loan choice
and borrower type is correct, it requires that borrower H be incentive compatible:
2
Solving the above condition (ICH), we get E Ã¢â€°Â¥ $70,000.
So the loan price (repayment obligation) for typeL borrower is:
PL =
150,000 Ã¯â‚¬Â E 150,000 Ã¯â‚¬Â 70,000
Ã¢â€°Â¤
= $100,000.
0.8
0.8
Since PL = (150,000 Ã¢â‚¬â€œ E)(1+rL), with E Ã¢â€°Â¥ 70,000 and PL Ã¢â€°Â¤ 100,000, rL = 25%
(lower default, lower rate).
(When E = 70,000, borrower H is indifferent between truthfully selfidentifying
and misrepresenting; we assume that borrower H will truthfully identify himself).
Finally, we need borrower L to be incentive compatible as well:
You can check that condition (ICL) is met. So borrower L will also tell the truth.
As you (the uninformed lender) rationally anticipate the association between loan
choice and borrower type, on one side, and your borrower (the informed firm) optimally
choose the loan that is best to him, on the other side, this constitutes a Nash equilibrium
characterized by:
3
choose / inf er
Highrisk borrower Ã¯â€šÂ¬Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ® highleverage loan (h) that
repays PH, (rH = 100%)
puts up no capital,
borrows the entire funds needed.
choose / inf er
Low risk borrower Ã¯â€šÂ¬Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ® lowleverage loan (l) that
repays PL, (rL = 25%)
puts up at least capital E,
borrows only part of the funds needed.
Discussions:
1) Here, capital serves as a signal of project type. The Ltype borrower signals his low
risk by funding part of the required investment with equity capital, E. For this, he is
rewarded with a lower interest rate (rL = 25%).
2) While putting up equity capital certainly offers a cushion to protect creditors and
reduces default probability of a firm, as we have seen in VAR analysis, the analysis here
provides another role to equity capital: not only functioning as cushion, capital also helps
resolve asymmetric information problems, which involves far more intuition than the
cushion explanation.
3) Interest rate is positively related to borrowerÃ¢â‚¬â„¢s leverage (or negatively related to
borrowerÃ¢â‚¬â„¢s equity position), since more leveraged (or less capitalized) borrowers are
more risky and choose loan (h), and are charged higher interest rates (rH = 100%).
4
Lecture 9.2
Credit contracting
Most loans are secured with collateral. But using collateral is costly:
1) borrower may undertake actions that undermine the value of the collateral to
bank. So ongoing monitoring of the collateral is required, which is costly.
2) transferring control of assets from borrower to bank involves legal costs.
3) liquidating collateral by striping it from the other assets of the firm is costly
(assets are often worth less piecemeal to the bank than they are to the borrower as
component of a productive whole).
4) from the borrowerÃ¢â‚¬â„¢s standpoint, use of collateral makes subsequent borrowing
more expensive, since fewer assets are available to other creditors.
Despite these costs, then why is collateral still widely used? (It helps resolve
information problems and agency problems).
Example 4 (Asymmetric information & Collateral)
Suppose: There are two types of firm; each can invest $30 in a project that yields the
following cash flows one period hence:
firm A:
0.7
$100
if project succeeds
0.3
$0
if project fails
0.5
$200
if project succeeds
0.5
$0
if project fails
$30
firm B:
$30
The firms (typeA or typeB) seek to borrow the needed funds from a bank. The
two types of firm look equally risky to the bank, even after all credit analyses are done.
However, the bank suspects some firms to be riskier than others, although it cannot tell
which. The cash flows from the projects are unrelated to states of economy (i.e. risk is
diversifiable), thus the projects betas are zero and discount rate for time value of money
is the riskless interest rate which is 5%. The bank needs to earn 5% as the normal rate of
1
return when lending. Also, any collateral worth $1 to the borrower firm is worth only 80
cents to the lending bank, as repossessing and liquidating collateral is costly to the bank.
Question: How to use interest rate along with collateral such that each borrower will be
induced to truthfully reveal its privately known type of risk?
Solution: (The logic is similar to that used in explaining the signaling role of equity
capital). The bank can design two loan contracts distinguished by absence/presence of
collateral requirement, and expects that different borrowers (different risks) will opt for
different loan contracts.
Bank uses two instruments in a loan contract: interest rate, ri, along with
collateral, C, and bank assumes:
inf er / choose
Ã¯â€šÂ¬Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ® less risky borrower A
(rC, C):
collateralized loan (c)
(rN, N):
inf er / choose
noncollateralized loan (n) Ã¯â€šÂ¬Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ¾
Ã¯â€šÂ¾Ã¯â€šÂ® more risky borrower B
— With loan (c):
bankÃ¢â‚¬â„¢s payoff structure:
0.7
PC = 30(1+ rC)
if success
0.3
0.8C
if failure
$30
To determine loan price PC for loan (c) which would allow the bank to earn the
normal rate of return of 5%, bank breaks even:
PC Ã¯â€šÂ´ 0.7 Ã¯â‚¬Â« 0.8C Ã¯â€šÂ´ 0.3
Ã¯â‚¬Â½ 30
1 Ã¯â‚¬Â« 5% Ã¯â‚¬Â« 5%
Ã¯Æ’Â¨ (BE – c)
— With loan (n):
bankÃ¢â‚¬â„¢s payoff structure:
0.5
PN = 30(1+ rN)
if success
0.5
$0 (no collateral)
if failure
$30
2
To determine loan price PN for loan (n) which would allow the bank to earn the
normal rate of return of 5%, bank breaks even:
PN Ã¯â€šÂ´ 0.5 Ã¯â‚¬Â« 0 Ã¯â€šÂ´ 0.5
Ã¯â‚¬Â½ 30
1 Ã¯â‚¬Â« 5 % Ã¯â‚¬Â« 5%
Ã¯Æ’Â¨ (BE – n)
So, repayment obligation PN = $66, or an interest rate rN = 120%.
— For borrower B to be incentive compatible:
The condition (ICB) simplifies to PC + C Ã¢â€°Â¥ 66. Combining it with (BE – c), we
can solve for the value of PC and C:
collateral:
C Ã¢â€°Â¥
repayment obligation:
PC Ã¢â€°Â¥
or an interest rate:
rC Ã¢â€°Â¥
Thus, if the bank requires a collateral worth no less than $28.70 and an interest
rate no lower than 24.33% in loan (c), then borrower B will indeed opt for noncollateralized loan (n), which indicates to the bank that he is of the more risky type
borrower B.
3
— For borrower A to be also incentive compatible, i.e., to choose loan (c) rather than (n):
Intuition:
Why is that the less risky borrower A chooses loan (c) with collateral $C but low
interest rate rC, while the more risky borrower B chooses loan (n) without collateral but
high interest rate rN?
1) lower risk means that chance of repaying interest is greater, so lower interest rate is
more appealing;
2) lower risk means that chance of defaulting and losing collateral to bank is smaller, so
offering collateral is less onerous. Thus, A will choose loan (c). For similar reasons, B
will choose loan (n).
Conclusion:
Collateral here serves as a signal of borrowerÃ¢â‚¬â„¢s risk; it resolves asymmetric
information by inducing borrowers to reveal their privately known risk types.
4
Lecture 10.1
Credit Contracting
Example 5 (Moral hazard & Covenants)
Suppose: A firm has borrowed (at t = 1) a bank loan that will be due one period hence (at
t = 1). The repayment obligation (principal & interest) of the loan is $100,000. The firm
currently (at t = 0) can invest $30,000 in a risky venture (which is not specified under the
loan) using retained earnings. If the investment is not taken, then the firmÃ¢â‚¬â„¢s shareholders
get a dividend of $100,000 (at t = 0). The total value of the firm at t = 1 depends on both
the decision of the firm at t = 0 and the state of the economy at t = 1:
Total value
at t=1
Economy at
t=1
Decision at t=0
Action 1: Stay with
current project
Ã¯â€šÂ®
Action 2: Switch to
riskier project
Ã¯â€šÂ®
Boom
with probability
0.5
Bust
with probability
0.5
$110,000
$60,000
$200,000
$5,000
No investment
taken; and $100,000
dividend paid
With investment
taken; and $70,000
dividend paid
That is, the firm has a production function: CFt=1 = f (decisiont=0; economyt=1). Ignore
taxes and discounting.
Note: Action 2 (taking the investment project) actually increases the firmÃ¢â‚¬â„¢s risk profile Ã¯ÂÂ³ :
if success, then greater cash flow; but if failure, then utter bankruptcy, though the
probabilities of success or failure are the same.
Time line of events (a typical story of switching borrowed funds to riskier project):
loan obtained
investment considered


value realized

Question: What should the firm do (Action 1 or 2) with respect of its investment decision
at t = 0?
1
Solution:
Compare the NPV associated with decision of investing vs. not investing.
Scenario i) objective: maximize S/HÃ¢â‚¬â„¢s wealth (bankÃ¢â‚¬â„¢s objective is ignored)
If not invest (Action 1): then S/Hs get a $100,000 dividend now, plus $10,000
(total value of $110,000 Ã¢â‚¬â€œ loan repayment of $100,000) if economy is in the boom state;
and S/Hs get nothing in the bust state of economy (limited liability implies that the bank
will seize $60,000 and the S/Hs get nothing). Thus, the expected payoff of this decision
to the S/Hs is:
$100,000 Ãƒâ€” (1.0) + $10,000 Ãƒâ€” (0.5) + $0 Ãƒâ€” (0.5) = $105,000
div.
now
Pr(Boom)
Pr(Bust)
If invest (Action 2): then S/Hs get $70,000 div. now, plus $100,000 in the boom
state (total value of $200,000 Ã¢â‚¬â€œ loan repayment of $100,000), and nothing in the bust
state. Thus, the expected payoff of this decision to the S/Hs is:
$70,000 Ãƒâ€” (1.0) + $100,000 Ãƒâ€” (0.5) + $0 Ãƒâ€” (0.5) = $120,000
div.
now
As $105,000not
Pr(Boom)
invest
Pr(Bust)
< $120,000invest, so the firm will invest (i.e., will take
Action 2).
Scenario ii) objective: maximize total value of firm (closer to bankÃ¢â‚¬â„¢s objective)
If not invest (Action 1): the expected value of the firm:
$100,000 + $110,000 Ãƒâ€”(0.5) + $60,000 Ãƒâ€”(0.5) = $185,000
If invest (Action 2): the expected value of the firm:
$70,000 + $200,000 Ãƒâ€”(0.5) + $5,000 Ãƒâ€”(0.5) = $172,500
As $185,000not invest > $172,500invest, so the firm will not invest (i.e., will take
Action 1).
2
In Scenario ii), we see that total firm value is lower with the investment than
without, which means that the investment project has a negative NPV and so should be
rejected.
However, in Scenario i), the project is accepted. The dramatically opposite actions
are due to the divergent objectives of S/Hs (the borrower) and bank (the lender).
The firm, acting in the interest of its S/Hs, has incentive to undertake projects
(take actions) that benefit the S/Hs, at the expense of creditors. To see this,
Without the project, the expected payoff to the lending bank is:
With the project, the expected payoff to the lending bank is:
Thus, by investing in the risky project, the borrower firm reduces the wealth of
lending bank by: $80,000 – $52,500 = $27,500, and gains for its shareholders by:
$120,000 – $105,000 = $15,000
Ã¯Æ’Â¨
moral hazard (after loan is granted)
3
Recall:
– promised/expected vs. actual return;
– borrowerÃ¢â‚¬â„¢s business risk vs. credit risk premium;
– credit risk deterioration/improvement and its impact on lender welfare.
=> If loan rate kL is set for project with risk Ã¯ÂÂ³ L , yet the loan is used for riskier project
with risk Ã¯ÂÂ³ H (> Ã¯ÂÂ³ L ), then for the bank, lending at kL (< kH) is lending at loss.
If the lending bank has no control over the borrower firmÃ¢â‚¬â„¢s actions (here the
investment decision), then the bank stands to suffer loss (credit risk deterioration) as
consequence of moral hazard.
To control/minimize such incident of credit risk deterioration, banks usually build
into loan contracts legal clauses called covenants.
 Loan covenants: special clauses designed to protect lender and prohibit borrower from
taking actions that could adversely affect the probability of repayment.
Covenants allow the lender to have a say in borrower firmsÃ¢â‚¬â„¢ decisions.
4 types of covenants:
1) Affirmative covenants: obligations imposed on borrower.
e.g.:  periodic furnishing of financial statements;
 minimum level of working capital;
 approval of any management replacement upon resignation/death .
2) Restrictive clauses: limits imposed on borrowerÃ¢â‚¬â„¢s actions.
e.g.:  paying/amount of dividends (inclination to divert wealth to S/Hs to
expropriate creditors);
 acquiring fixed assets (replacing cash with assets that produce risky
cash flows  asset substitutions).
3) Negative covenants: prohibitions regarding borrower firmÃ¢â‚¬â„¢s actions that alter its risk
profile.
e.g.:  pledging assets to other lenders;
 mergers, consolidations, sales of assets, providing loans and guarantees
to others.
4) Default provision: make the entire loan immediately due and payable.
e.g.:  acceleration clause (upon failure or violation regarding some specified
events, bank calls the loan immediately that enables bank to take timely
actions to protect from further credit risk exposure).
4
Lecture 10.2
Credit contracting
Example 6 (Moral hazard & Collateral)
Suppose: A firm needs to borrow $100 from a bank to finance a project that will pay off
next period. The firm can choose between two projects: S (relatively safe) and R (more
risky). The bank knows this but cannot observe/control the firmÃ¢â‚¬â„¢s choice of project.
0.9
S:
$
$
300
100
R:
0.1
$
$
0.6
$
400
0.4
$
0
100
0
Everybody is risk neutral and discount rate is 10%. Any collateral worth $1 to the
borrower firm will be worth only 90 cents to the bank.
Question: How should the bank use interest rate along with collateral to design its loan
contracts so that the borrower will choose the safer project?
Solution: The bank can design two different loans:
secured (collateralized) loan (s), and
unsecured (noncollateralized) loan (u).
The purpose of the designing is to induce the borrower to take the action (here
choose the project) that will lead to the highest possible welfare for both parties.
We proceed in 4 steps.
i) No collateral  unsecured loan (u):
1) Suppose the bank believes that the borrower will choose project S and sets the interest
rate on loan (u) ruS to break even:
100(1 Ã¯â‚¬Â« ruS ) Ã¯â€šÂ´ 0.9 Ã¯â‚¬Â« 0 Ã¯â€šÂ´ 0.1
Ã¯â‚¬Â½ 100
1 Ã¯â‚¬Â« 10%
 (BEÃ¢â‚¬â„¢ Ã¢â‚¬â€œ u)
i.e., ruS = 22 %
1
We need to check whether the borrower will indeed choose project S when
offered with such ruS . To this, we compare borrowerÃ¢â‚¬â„¢s earnings from choosing S with
those from choosing R:
the profit with project S: [300  (100Ãƒâ€”1.22)]Ãƒâ€” 0.9 = $160.2
the profit with project R: [400  (100Ãƒâ€”1.22)]Ãƒâ€” 0.6 = $166.8
Thus, offering an unsecured loan (u) with an interest rate of 22% cannot be a
Nash equilibrium in the sense that bank (incorrectly) believes project S but borrower
(instead) chooses project R.
2) Suppose now the bank believes that the borrower chooses project R and sets the
interest rate ruR accordingly to break even:
100(1 Ã¯â‚¬Â« ruR ) Ã¯â€šÂ´ 0.6 Ã¯â‚¬Â« 0 Ã¯â€šÂ´ 0.4
Ã¯â‚¬Â½ 100
1 Ã¯â‚¬Â« 10%
 (BE Ã¢â‚¬â€œ u)
i.e. ruR = 83%.
Now offered with this rate, borrowerÃ¢â‚¬â„¢s earnings from
choosing S: [300  (100Ãƒâ€”1.83)] Ãƒâ€” 0.9 = $105
choosing R: [400  (100Ãƒâ€”1.83)] Ãƒâ€” 0.6 = $130
 (IC Ã¢â‚¬â€œ R)
Thus, offering an interest rate of 83% is a Nash equilibrium in the sense that the
bank (correctly) believes project R to be chosen by the borrower and the borrower
(indeed) chooses R.
Ã¯Æ’Â¨ Nash equilibrium u:
{bank breaks even; firm earns $130}
Next, examine whether by using collateral, the bank continues to break even, but
the firm can improve its earnings.
2
ii) With collateral  secured loan (s):
3) Suppose the bank believes that the borrower will choose project S and sets the interest
rate on loan (s), rsS , accordingly to break even:
100(1 Ã¯â‚¬Â« rsS ) Ã¯â€šÂ´ 0.9 Ã¯â‚¬Â« 0.9 Ã¯â€šÂ´ C Ã¯â€šÂ´ 0.1
Ã¯â‚¬Â½ 100
1 Ã¯â‚¬Â« 10%

(BE Ã¢â‚¬â€œ s)
where, 0.9Ãƒâ€”C reflects the fact that each dollar of collateral is worth only 90 cents to the
bank.
In determining the collateral, C, the bank needs to make the borrower prefer
choosing project S to choosing project R:
[300100Ãƒâ€”(1+ rsS )]Ãƒâ€”0.9  CÃƒâ€”0.1 Ã¯â€šÂ³ [400100Ãƒâ€” (1+ rsS )]Ãƒâ€”0.6  CÃƒâ€”0.4
 (IC Ã¢â‚¬â€œ S)
Set the (IC Ã¢â‚¬â€œ S) to equality, so the borrower is indifferent between choosing S and
R (then assume S is chosen). Solving for rsS and C from (BE Ã¢â‚¬â€œ s) and (IC Ã¢â‚¬â€œ S):
rsS = 20.21% and
C = $20.20.
Plugging these two values into (IC Ã¢â‚¬â€œ S), we then find the borrowerÃ¢â‚¬â„¢s earnings:
[300100Ãƒâ€”(1.2021)]Ãƒâ€”0.9 Ã¢â‚¬â€œ 20.2Ãƒâ€”0.1 = $160 and
[400100Ãƒâ€”(1.2021)]Ãƒâ€”0.6 Ã¢â‚¬â€œ 20.2Ãƒâ€”0.4 = $160
This is a Nash equilibrium in the sense that bank (correctly) believes project S to
be chosen by the borrower and the borrower (indeed) chooses S.
Ã¯Æ’Â¨ Nash equilibrium s: {bank breaks even; firm earns $160}
4) Now compare the two Nash equilibriums, s and u:
In both equilibriums, the bankÃ¢â‚¬â„¢s beliefs about what firm will do are confirmed by
firmÃ¢â‚¬â„¢s optimal actions Ã¢â‚¬â€œ project choices, and the bank breaks even.
3
However, the borrower firm is better off in equilibrium s where collateral is used,
since it earns $160, compared to $130 in equilibrium u where no collateral is used.
Conclusion:
Use of collateral induces borrower to take desirable action (here choose safer
project) that better aligns the objectives of borrower and lender, which leads to
improvement in total welfare of the two transacting parties.
Note:
In both equilibriums, the bank merely breaks even. So, why should the bank care
about whether borrower would be better off or not?
4
Solutions to
Assignment 2
Question 1
a) This year, the firm has retained 50% of its net income of $200, ending up with a
retained earnings of $400. Thus, last yearÃ¢â‚¬â„¢s retained earnings was $300.
b) The book value of equity is:
Contributed capital + Retained earnings = 260 + 400 = $660
The market value of equity is 120% of book value or 1.2 Ãƒâ€” 660 = $792.
c) AltmanÃ¢â‚¬â„¢s ZScore model is:
Z = 1.2 X1 + 1.4 X2 + 3.3 X3 + 0.6 X4 + 1.0 X5
Where
X1 = net working capital / total assets
= 320/1675 = 0.191
X2 = retained earnings / total assets
= 400/1675 = 0.239
X3 = EBIT / total assets
= 400/1675 = 0.239
X4 = market value of equity / book value of total debt
= 792/1015 = 0.780
X5 = sales / total assets
= 2110/1675 = 1.260
Z = 3.079
The Zscore is 3.079 which is larger than the critical value of 1.81. Hence, Nova Chang is not
in danger of bankruptcy within the next year  assuming AltmanÃ¢â‚¬â„¢s model is applicable here.
d) From
pÃ¯â‚¬Â½
1
1
Ã¯â‚¬Â½
Ã¯â‚¬Â½ 95.60%
Ã¯â‚¬ÂZ
1Ã¯â‚¬Â« e
1 Ã¯â‚¬Â« e Ã¯â‚¬Â 3.079
With repayment probability p = 95.60% and recovery = 0.40L(1+k), the interest rate
based on riskneutral credit pricing is calculated using:
LÃ¯â‚¬Â½
kÃ¯â‚¬Â½
p Ã¯â€šÂ´ L(1 Ã¯â‚¬Â« k ) Ã¯â‚¬Â« (1 Ã¯â‚¬Â p) Ã¯â€šÂ´ recovery
1 Ã¯â‚¬Â« rf
1 Ã¯â‚¬Â« rf
[ p Ã¯â‚¬Â« 0.4(1 Ã¯â‚¬Â p)]
Ã¯â‚¬Â1 Ã¯â‚¬Â½
1 Ã¯â‚¬Â« 2%
Ã¯â‚¬Â 1 Ã¯â‚¬Â½ 4.766%
0.956 Ã¯â‚¬Â« 0.4 Ã¯â€šÂ´ (1 Ã¯â‚¬Â 0.956)
1
Question 2
The loanÃ¢â‚¬â„¢s payoff structure would be:
LÃƒâ€”(1+ k)
Ã¯Æ’Â
timely full repayment
0.9
L
0.6
0.1
0.8L
0.3
0.5L
0.1
LGD
0
5%
Ã¯Æ’Â cost of funds
Credit pricing:
0.9 Ã¯â€šÂ´ L(1 Ã¯â‚¬Â« k ) Ã¯â‚¬Â« 0.1 Ã¯â€šÂ´ [0.6 Ã¯â€šÂ´ 0.8L Ã¯â‚¬Â« 0.3 Ã¯â€šÂ´ 0.5L Ã¯â‚¬Â« 0.1 Ã¯â€šÂ´ 0]
Ã¯â‚¬Â½L
1 Ã¯â‚¬Â« 0.05
Solving for k:
k=
1 Ã¯â‚¬Â« 0.05 1
Ã¯â‚¬Â Ã¯â€šÂ´ [0.6 Ã¯â€šÂ´ 0.8 Ã¯â‚¬Â« 0.3 Ã¯â€šÂ´ 0.5 Ã¯â‚¬Â« 0] Ã¯â‚¬Â 1 = 9.67%
0.9
9
Question 3
a) To calculate current market value of the loan:
dÃ¯â‚¬Â½
DT Ã¯â‚¬Â r f T
7,582,000
e
Ã¯â‚¬Â½
Ã¯â‚¬Â½ 0.771441
V0
9,256,000 Ã¯â€šÂ´ e0.03Ã¯â€šÂ´2
h1 Ã¯â‚¬Â½ Ã¯â‚¬Â
Ã¯Ââ€º0.5Ã¯ÂÂ³
Ã¯ÂÂ
Ã¯Ââ€º
Ã¯ÂÂ
T Ã¯â‚¬Â ln d
0.5(0.26)2 Ã¯â€šÂ´ 2 Ã¯â‚¬Â ln(0.771441)
Ã¯â‚¬Â½Ã¯â‚¬Â
Ã¯â‚¬Â½ Ã¯â‚¬Â0.889581
Ã¯ÂÂ³V T
0.26 Ã¯â€šÂ´ 2
Ã¯â‚¬Â¨
2
V
Ã¯â‚¬Â© Ã¯â‚¬Â¨
Ã¯â‚¬Â©
h2 Ã¯â‚¬Â½ Ã¯â‚¬Â h1 Ã¯â‚¬Â« Ã¯ÂÂ³ V T Ã¯â‚¬Â½ Ã¯â‚¬Â Ã¯â‚¬Â 0.889581Ã¯â‚¬Â« 0.26 2 Ã¯â‚¬Â½ 0.521886
From Excel:
N(h1) = N(0.889581) = 0.186845
N(h2) = N( 0.521886) = 0.699125
2
Ã¯Æ’Â© N(h1 )
Ã¯Æ’Â¹
Ã¯Æ’Âª d Ã¯â‚¬Â« N(h2 )Ã¯Æ’Âº
Ã¯Æ’Â«
Ã¯Æ’Â»
0
.
186845
Ã¯Æ’Â©
Ã¯Æ’Â¹
Ã¯â‚¬Â½ $7,582,000 Ã¯â€šÂ´ e Ã¯â‚¬Â 0.03Ã¯â€šÂ´ 2 Ã¯Æ’Âª
Ã¯â‚¬Â« 0.699125Ã¯Æ’Âº Ã¯â‚¬Â½ $6,721,514
Ã¯Æ’Â« 0.771441
Ã¯Æ’Â»
D0 Ã¯â‚¬Â½ DT e
Ã¯â‚¬Ârf T
b) The interest rate to be charged on the loan.
kÃ¯â‚¬Â½Ã¯â‚¬Â
1 Ã¯Æ’Â¦1
lnÃ¯Æ’Â§ N(h1 ) Ã¯â‚¬Â« N(h2 ) Ã¯Æ’Â¶Ã¯Æ’Â· Ã¯â‚¬Â« rf
T Ã¯Æ’Â¨d
Ã¯Æ’Â¸
1 Ã¯Æ’Â¦ 0.0.186845
Ã¯Æ’Â¶
Ã¯â‚¬Â½ Ã¯â‚¬Â lnÃ¯Æ’Â§
Ã¯â‚¬Â« 0.699125Ã¯Æ’Â· Ã¯â‚¬Â« 3% Ã¯â‚¬Â½ 6.02%
2 Ã¯Æ’Â¨ 0.771441
Ã¯Æ’Â¸
Question 4
a)
What is the bankÃ¢â‚¬â„¢s duration gap?
Assets:
270 day Treasury bills
$500m
2 year consumer loans
Fixed rate, 12% p.a. annually $275m
7 year commercial loans
$350m
Fixed rate, 9% p.a. annually
10 year fixed rate mortgages
$675m
Fixed rate, 6.5% p.a. annually
10 year floating rate mortgages $125m
LIBOR+50bp, monthly roll date
Durations:
Duration = 270/360 = 0.75 year
Liabilities:
1 year Certificates of Deposit $550m
Demand Deposits
$750m
2 year IntÃ¢â‚¬â„¢l Bonds
$175m
Fixed rate, 7.5% p.a. annually
Durations:
Duration = 1 year
Duration = 0 year
Duration = 1.93 years (see below)
Overnight borrowing
Duration = 1/360 = 0.00278 year
$350m
Duration of 2 year consumer loans =
Duration of 2 year IntÃ¢â‚¬â„¢l Bonds =
Duration = 1.89 years (see below)
Duration = 5.486 years (given)
Duration = 7.656 years (given)
Duration = 1/12 = 0.083 year
120(1) / 1.12 Ã¯â‚¬Â« 1120(2) / 1.122
=1.89 years
120 / 1.12 Ã¯â‚¬Â« 1120 / 1.122
75(1) / 1.075 Ã¯â‚¬Â« 1075(2) / 1.0752
= 1.93 years
75 / 1.075 Ã¯â‚¬Â« 1075 / 1.0752
3
DA=
500 Ã¯â€šÂ´ 0.75 Ã¯â‚¬Â« 275 Ã¯â€šÂ´ 1.89 Ã¯â‚¬Â« 350 Ã¯â€šÂ´ 5.486 Ã¯â‚¬Â« 675 Ã¯â€šÂ´ 7.656 Ã¯â‚¬Â« 125 Ã¯â€šÂ´ 0.083
= 4.15 years
1925
DL =
550 Ã¯â€šÂ´ 1 Ã¯â‚¬Â« 750 Ã¯â€šÂ´ 0 Ã¯â‚¬Â« 175 Ã¯â€šÂ´ 1.93 Ã¯â‚¬Â« 350 Ã¯â€šÂ´ 0.00278
= 0.487 year
1825
DG = DA DLÃƒâ€”(L/A) = 4.15 Ã¢â‚¬â€œ 0.487Ãƒâ€”(1825/1925) = + 3.69 years
b)
Exposed to interest rate increases because a positive duration gap implies that
equity values decline when interest rates rise.
c)
Ã¯Ââ€žE = (Duration gap)Ã¯â€šÂ´AÃ¯â€šÂ´Ã¯Ââ€žR/(1+R) = 3.69Ã¯â€šÂ´1925Ã¯â€šÂ´0.25% =  $17,758,125
d)
i) Fixedrate payer.
Since the bank has a positive duration gap, which means if rates rise, then it
suffers loss on balance sheet. To reduce the loss, your bank can pay fixed rate
and receive variable rate in the swap so that if rates rise, your bank will
receive more interest payments from the counterparty than it will pay, thus
making profits offbalance sheet that can offset (thus hedge) the loss on
balance sheet.
ii) Notional:
NS Ã¯â‚¬Â½
e)
DG Ã¯â€šÂ´ A
3.69 Ã¯â€šÂ´ 1,925,000,000
Ã¯â‚¬Â½
= $710,325,000
DFix Ã¯â‚¬Â DVar
12 Ã¯â‚¬Â 2
Sell futures to construct a short hedge.
If rates rise, the bank loses onbalance sheet. If the bank sells futures today, it
will sell bonds later at fixed price. But when rates rise then, bond price will be
lower. Selling bonds at fixed price means the bank can make profits with the
futures. These offbalance sheet profits offset the loss on balance sheet,
achieving thereby hedging purpose.
f)
Buy puts to construct a short hedge.
Question 5
a) Ã¯Ââ€žE Ã¯â€šÂ»  (Duration gap) Ã¯â€šÂ´ A Ã¯â€šÂ´ Ã¯Ââ€žR/(1+R) =  8.189 Ã¯â€šÂ´ 950,000 Ã¯â€šÂ´ 0.01 =  $77,796.
The value of equity would fall by $77,796.
b) The FI can sell bond futures (i.e. take a short hedge) as it has a positive duration gap.
4
c) Ã¯Ââ€žF Ã¯â€šÂ»  7 Ã¯â€šÂ´ 96,000 Ã¯â€šÂ´ 0.01 =  $6,720 per futures contract. Since this macro hedge is a
short hedge, the negative sign means there will actually be a gain of $6,720 per
contract !
d) The 8 futures contract used in the hedge will produce $6,720 Ãƒâ€” 8 = $53,760 profit.
Thus, the net impact is: Ã¯Ââ€žE + Ã¯Ââ€žF =  77,796 + 53,760 =  $24,036.
e) To hedge Ã¢â‚¬Å“perfectlyÃ¢â‚¬Â the on balancesheet loss of $77,796, the profit on the futures
position should equal to $77,796. The total number of futures contracts sold should be
NF Ã¯â‚¬Â½
DG Ã¯â€šÂ´ A 8.189 Ã¯â€šÂ´ 950,000
Ã¯â‚¬Â½
Ã¯â€šÂ» 12 total futures contracts
DF Ã¯â€šÂ´ PF
7 Ã¯â€šÂ´ 96,000
NF Ã¯â‚¬Â½
24,036
Ã¯â€šÂ» 4 more future to be sold, on top of the 8 futures already sold.
6,720
or,
Question 6
a) Absent the puts, the original gap would be 48,548 Ã¢â‚¬â€œ 79,000 =  30,452. After using the
puts, however, the actual gap is 50,500, which is more away from zero (more negative)
than before. Thus, the puts have not been used for hedging. More specifically, the puts on
bond have been sold for speculating on falling interest rates.
b) Absent the futures, the original gap would be 234,000 Ã¢â‚¬â€œ 145,450 = 88,550. After using
the futures, however, the actual gap is 18,550, which is closer to zero than before (less
positive). Thus, the swap has been used for hedging (but underhedged). More
specifically, the bond futures has been bought for hedging against falling interest rates.
c) Absent the swap, the original gap would be 105,655 Ã¢â‚¬â€œ 149,055=  43,400. After using
the swap, however, the actual gap is 16,600, which became positive. Thus, the swap has
been used for hedging (but overhedged). More specifically, the swap has been bought for
speculating on rising interest rates.
5
Assignment 2
Zhang Zepeng
A00429736
Note :
1) The Due time is by 6:00pm, Monday, November 1 (the solutions will be posted
immediately after).
2) You must Type all your assignment work using a WORD or similar software; then
convert it to PDF format before submitting.
3) Submit all your assignment work as a Single file to the folder Assignment 2 in
Brightspace of this course (only ONE submission is allowed there)
Question 1
Use AltmanÃ¢â‚¬â„¢s ZScore model to determine whether to grant a loan to Nova Chang Inc.
The borrowing firm has provided you its CURRENT yearÃ¢â‚¬â„¢s financial statements as follows:
Balance Sheet (in thousands $)
Liabilities
Cash
75 Notes payable
55
Accounts receivable
300 Debt (long term)
960
Plant and equipment 1,000 Contributed capital
260
Land
300
Retained earnings
400
Total assets
1,675
1,675
Assets
Income Statement (in thousands $)
Sales
2,110
Cost of goods sold
1,560
Depreciation
150
Interest expense
67
Taxes
133
Net income
200
Notes: Nova Chang pays 50% dividend, and its Equity is currently
selling at 120% of its book value.
a) What was the borrowing firmÃ¢â‚¬â„¢s retained earnings of LAST year?
The borrowing firmÃ¢â‚¬â„¢s retained earning of LAST year= The borrowing firmÃ¢â‚¬â„¢s retained earning
of current year earnings retained in this year
= Retained earnings balance on the current yearÃ¢â‚¬â„¢s balance sheet Net income of current year*(1dividend payout ratio)
= 400200*(150%)
1
= 300
The borrowing firmÃ¢â‚¬â„¢s earning of LAST year is 300.
b) What is the market value of its equity?
The market value of its equity= Book Value*(Contributed Capital+ Retained Earnings)
= 120%*(260+400)
= $792(in thousands $)
c) Calculate Nova ChangÃ¢â‚¬â„¢s Zscore. Should its loan application be approved?
Z is AltmanÃ¢â‚¬â„¢s Zscore
X1: Net Working Capital /Total Assets
= (Current Assets Current Liabilities)/Total assets
= (Cash + receivables Notes Payable)/Total Assets
= ($75+$300$55)/$1,675
= 0.191
X2: Retained earnings/ Total assets= $400/$1,675=0.238
X3: EBIT/Total assets
= (Sales Cost of goods soldDepreciation )/Total assets
= ($2,110$1,560$150)/$1,675= 0.24
X4: Market Value of equity/ Book value of total liabilities
= $792/($55+$960)=0.78
X5: Sales/ Total Assets
= $2,110/$1,675
= 1.26
Z score= 1.2X1+1.4X2+3.3X3+0.6X4+ 1.0X5
= 1.2*0.191+1.4*0.238+3.3*0.24+0.6*0.78+1.0*1.26
=0.2292+0.3332+0.792+0.468+1.26
= 3.08
Z Score is 3.08Ã¢â€°Â¥1.81. This means that the company is well run and reputable, and that the
company will not experience any economic crisis or even bankruptcy in the short term.
Therefore, the company's loan application is able to be approved.
d) If you ever decide to grant the loan, what interest rate would you charge on the loan?
Assume: The recovery is 40% of the repayment obligation, and discount rate is 2%.
P= 1/(1+eÃ¢â€¹â‚¬3.08)= 0.956
L= [L*(1+k)*0.956+L*(1+k)*0.4*(10.956)]/(1+r)
[L*(0.956+0.956k)+L*(1+k)*0.4*0.044]/(1+r)=L
k= 4.77%
2
Question 2
Working as a credit officer, you are examining a loan application by a corporate customer
of your bank. Based on your collected information on the repayments of similar borrowers, you
estimate that this borrower has a fullrepayment probability of 90%, default probability of 10%,
and the losses given default (LGD) distribution of:
Recovery:
Probability:
80% of principal
60%
Half of principal
30%
Total loss
10%
If your cost of funds is 5%, what interest rate are you going to charge on the loan?
[0.9L(1+k)+0.1*(0.6*0.8L+0.3*0.5L+0.1*0)]/(1+r)=L
[0.9L(1+k)+0.1*(0.48L+0.15L+0)]/(1+r)=L
[0.9L+0.9Lk+0.48L+0.15L+0] /(1+r)=L
K=9.67%
Question 3
A firm is borrowing a twoyear zerocoupon business loan from your bank with a repayment
obligation of $7.582 million. You estimate that the market value of the firmÃ¢â‚¬â„¢s total assets is
$9.256 million. Riskfree interest rate is 3%, and the standard deviation of the rate of change in
the underlying assets of the borrowing firm is 26%. Using the options framework, determine the
following:
a) The current market value of the loan.
d= $7.582/$9.256*eÃ¢Ë†Â§(0.03*2)=0.771441
The Current market Value of loan= 9.2562.5345
=6.72
3
=$7million
b) The interest rate to be charged on the loan.
Note: Keep 6 decimals during all your calculations, but round your final answer to last
dollar for debt value and 4 decimals for interest rate on the loan.
4
Question 4
Use the following to answer questions a) Ã¢â‚¬â€œ f).
Bank of Big Bucks
Assets:
270 day Treasury bills
$500m
2 year consumer loans
Fixed rate, 12% p.a. annually $275m
7 year commercial loans
$350m
Fixed rate, 9% p.a. annually
10 year fixed rate mortgages
$675m
Fixed rate, 6.5% p.a. annually
10 year floating rate mortgages $125m
LIBOR+50bp, monthly roll date
($ million)
Liabilities and Net Worth:
1 year Certificates of Deposit
Demand Deposits
2 year IntÃ¢â‚¬â„¢l Bonds
Fixed rate, 7.5% p.a. annually
Overnight borrowing
$350m
Equity
$100m
$550m
$750m
$175m
Notes:
The 1 year Certificates of Deposit pay 1.95% p.a. annually and will be rolled over at
maturity. The 7 year commercial loans have a duration of 5.486 years. The fixed rate
mortgages have a duration of 7.656 years. All values are market values and are trading at
par.
Assumption: 30 days per month; 90 days per quarter; 360 days per year.
What is the bankÃ¢â‚¬â„¢s duration gap?
a. 0.49 years
b. +4.24 years
c. Ã¢â‚¬â€œ0.94 years
d. 2.81 years
e. +3.69 years
Total Equity=$550m+$750m+$175m+$350m+$100m= $1,925
Total Assets= $500m+$275m+$350m
DA=
$500/$1,925*0.75+$275/$1,925*1.89+$350/$1,925*5.486+$675/$1,925*7.656+$125/$1,92
5*(1/12)
= 0.194805+0.27+0.99745+2.684571+0.00541
=4.152
DL= $550/$1,825*1+$175/$1,825*1.93+$350/$1,825*(1/360)
a)
= 0.30137+0.1851+0.0005327
=0.487
DG= RSAREL
=4.150.487*$1,825/$1,925
=3.69years
b)
What is the bankÃ¢â‚¬â„¢s interest rate risk exposure (i.e. exposed to rising or falling
rates)?
This is because rising interest rates tend to expose the risks that exist in banks.
c)
What is the onbalancesheet impact on the bank if all interest rates increase 25
basis points? (i.e. suppose Ã¯Ââ€žR/(1+R) is equal to an increase of 25 basis points.)
5
Ã¯Ââ€žR/(1+R)=0.0025
Ã¢Ë†â€ E=DG*A*[Ã¯Ââ€žR/(1+R)
=(3.69)*$1,925m*0.0025
=$17,758,125
Hence, the bankÃ¢â‚¬â„¢s interest rate risk exposure is $17,758,125.
d)
Suppose you are a risk manager of the bank. Construct an appropriate swap hedge
for your bank.
i).
Specify whether your bank should be a fixed or variablerate payer in the
swap, and
In your bank ,you should Fixedrate payer.
ii).
Calculate the notional of swap, NS, for a perfect hedge. Suppose the fixed side
of the swap has a duration of 12 years, while the variable side of the swap is
floating on a biennial base.
NS= DG*A/(DFixDVar)
=3.69*$1,925m/(122)
= 3.69*1,925m/10
= $710.33million
6
How would you use futures contracts on Canada Bond to hedge your bankÃ¢â‚¬â„¢s
interest rate risk exposure? Specify whether your bank should conduct a long or
short hedge. Justify your strategy.
Banks can hedge their accounts by selling futures or shorting their bank accounts, thus they
can offset the potential losses that exist on the bank's balance sheet and avoid the risk that
comes with higher interest rates.
e)
f)
How would you use put option contracts to hedge the interest rate risk exposure of
your bank? That is, should the bank buy or sell put options on Canada Bond?
Hedging is achieved by buying put options and thus constructing a short account.
Question 5
Consider the following balance sheet of an FI:
Assets ($000)
Duration = 10 years
$950
Liabilities ($000)
Duration = 2 years
Equity
$860
$ 90
a) What is the impact on the FIÃ¢â‚¬â„¢s equity value if all interest rates increase such that
Ã¯Ââ€žR/(1+R) = 0.01?
DG= (102)*( $860, 000/$950,000)
= 8.19
Ã¯Ââ€žR/(1+R) = 0.01
Change in equity= DG*A*[Ã¯Ââ€žR/(1+R)
= (8.19)* $950,000*0.01
= $77,805
b) How can the FI use futures contracts to put on a macro hedge? (Note: You first need
to find out the duration gap)
Based on the conclusion from the above calculation, DG>0, if it wants to raise the
interest rate, it needs FI to sell futures to hedge the risk.
c) Suppose that the FI put on a macro hedge using zerocoupon bond futures that are
currently priced at 96. What is the impact on the FIÃ¢â‚¬â„¢s futures position (in terms of Ã¯Ââ€žF
per contract) if all interest rates increase such that Ã¯Ââ€žR/(1+R) = 0.01? The deliverable
zerocoupon bond has a maturity of 7 years.
Change in F(Ã¯Ââ€žF)= Ã¢Ë†â€ E
=DF*A*[Ã¯Ââ€žR/(1+R)
= 7*96000*0.01
= $6,720
7
d) After interest rate change of Ã¯Ââ€žR/(1+ R) = 0.01, what is the net impact on the FIÃ¢â‚¬â„¢s
equity, if the FIÃ¢â‚¬â„¢s hedge used 8 futures contracts as described in b) and c)?
Ã¢Ë†â€ E
=8.19*950000*0.01+7*8*100*960*0.01
= ($77,805)+$53,760
= $24,045
e) If the FI wanted to put on a perfect macro hedge, how many more or fewer such bond
futures contracts are needed?
NF=
=
Ã°ÂÂÂ·Ã°ÂÂÂº Ãƒâ€”Ã°ÂÂÂ´
Ã°ÂÂÂ·Ã°ÂÂÂ¹ Ãƒâ€”Ã°Ââ€˜Æ’Ã°ÂÂÂ¹
$950,000Ãƒâ€”8.19
7Ãƒâ€”$96,000
= $7,780,500/$672,000
=11.578 contracts
=12 contracts
They need 12 contracts bond futures .
Question 6
The interest rate exposures of a bank for its two repricing buckets and the use of
derivatives during the time intervals are shown below:
8
1 year
or less
13 years
35 years
beyond
RSA
48,548
234,000
105,655
Ã¢â‚¬Â¦
RSL
79,000
145,450
149,055
Ã¢â‚¬Â¦
Equity
Derivatives
Interest rate
Swap (notional)
Actual
Dollar Gap
95,000
Puts
Ã¢â‚¬Â¦
Futures
60,000
– 50,500
18,550
16,600
Ã¢â‚¬Â¦
a) For the repricing time interval of 1 year or less, identify how the puts on bond have
been used by specifying whether the puts are bought (or sold) for hedging against (or
speculation on) a rising (or falling) interest rate? (i.e. you need to make three correct
judgments that are consistent with each other)
The original gap= $485487$9000=$30452
The original gap was 30,452. but the actual gap is really 50,500, and by
comparison, the actual gap will be further from zero than the original gap was. So,
the speculative rate goes down and the puts in the debt paper are similarly sold.
b) For the repricing time interval of 13 years, what have the bond futures been used for?
(i.e. specify whether the bond futures have been bought or sold for hedging against or
speculation on a rising or falling interest rate)
The original gap= 23400145450=88550
The original shortfall was 88,550, while the actual shortfall was 18,550, which
means that the actual shortfall is infinitely closer to zero. If you want to make the
hedging rate go down, you can make the hedging rate go down by buying a lot of
futures.
c) For the repricing time interval of 35 years, how has the interest rate swap been used
for? (again, three judgments are needed as above)
The original gap= 105655149055=43400
The original gap is 43,400. while the actual gap is 16,600, which is a large difference
in value. If the company wants to speculate on a significant increase in interest rates,
it needs to inject more money.
9
Assignment 2
Zhang Zepeng
A00429736
Note :
1) The Due time is by 6:00pm, Monday, November 1 (the solutions will be posted
immediately after).
2) You must Type all your assignment work using a WORD or similar software; then
convert it to PDF format before submitting.
3) Submit all your assignment work as a Single file to the folder Assignment 2 in
Brightspace of this course (only ONE submission is allowed there)
Question 1
Use AltmanÃ¢â‚¬â„¢s ZScore model to determine whether to grant a loan to Nova Chang Inc.
The borrowing firm has provided you its CURRENT yearÃ¢â‚¬â„¢s financial statements as follows:
Balance Sheet (in thousands $)
Liabilities
Cash
75 Notes payable
55
Accounts receivable
300 Debt (long term)
960
Plant and equipment 1,000 Contributed capital
260
Land
300
Retained earnings
400
Total assets
1,675
1,675
Assets
Income Statement (in thousands $)
Sales
2,110
Cost of goods sold
1,560
Depreciation
150
Interest expense
67
Taxes
133
Net income
200
Notes: Nova Chang pays 50% dividend, and its Equity is currently
selling at 120% of its book value.
a) What was the borrowing firmÃ¢â‚¬â„¢s retained earnings of LAST year?
The borrowing firmÃ¢â‚¬â„¢s retained earning of LAST year= The borrowing firmÃ¢â‚¬â„¢s retained earning
of current year earnings retained in this year
= Retained earnings balance on the current yearÃ¢â‚¬â„¢s balance sheet Net income of current year*(1dividend payout ratio)
= 400200*(150%)
1
= 300
The borrowing firmÃ¢â‚¬â„¢s earning of LAST year is 300.
b) What is the market value of its equity?
The market value of its equity= Book Value*(Contributed Capital+ Retained Earnings)
= 120%*(260+400)
= $792(in thousands $)
c) Calculate Nova ChangÃ¢â‚¬â„¢s Zscore. Should its loan application be approved?
Z is AltmanÃ¢â‚¬â„¢s Zscore
X1: Net Working Capital /Total Assets
= (Current Assets Current Liabilities)/Total assets
= (Cash + receivables Notes Payable)/Total Assets
= ($75+$300$55)/$1,675
= 0.191
X2: Retained earnings/ Total assets= $400/$1,675=0.238
X3: EBIT/Total assets
= (Sales Cost of goods soldDepreciation )/Total assets
= ($2,110$1,560$150)/$1,675= 0.24
X4: Market Value of equity/ Book value of total liabilities
= $792/($55+$960)=0.78
X5: Sales/ Total Assets
= $2,110/$1,675
= 1.26
Z score= 1.2X1+1.4X2+3.3X3+0.6X4+ 1.0X5
= 1.2*0.191+1.4*0.238+3.3*0.24+0.6*0.78+1.0*1.26
=0.2292+0.3332+0.792+0.468+1.26
= 3.08
Z Score is 3.08Ã¢â€°Â¥1.81. This means that the company is well run and reputable, and that the
company will not experience any economic crisis or even bankruptcy in the short term.
Therefore, the company’s loan application is able to be approved.
d) If you ever decide to grant the loan, what interest rate would you charge on the loan?
Assume: The recovery is 40% of the repayment obligation, and discount rate is 2%.
P= 1/(1+eÃ¢â€¹â‚¬3.08)= 0.956
L= [L*(1+k)*0.956+L*(1+k)*0.4*(10.956)]/(1+r)
[L*(0.956+0.956k)+L*(1+k)*0.4*0.044]/(1+r)=L
k= 4.77%
2
Question 2
Working as a credit officer, you are examining a loan application by a corporate customer
of your bank. Based on your collected information on the repayments of similar borrowers, you
estimate that this borrower has a fullrepayment probability of 90%, default probability of 10%,
and the losses given default (LGD) distribution of:
Recovery:
Probability:
80% of principal
60%
Half of principal
30%
Total loss
10%
If your cost of funds is 5%, what interest rate are you going to charge on the loan?
[0.9L(1+k)+0.1*(0.6*0.8L+0.3*0.5L+0.1*0)]/(1+r)=L
[0.9L(1+k)+0.1*(0.48L+0.15L+0)]/(1+r)=L
[0.9L+0.9Lk+0.48L+0.15L+0] /(1+r)=L
K=9.67%
Question 3
A firm is borrowing a twoyear zerocoupon business loan from your bank with a repayment
obligation of $7.582 million. You estimate that the market value of the firmÃ¢â‚¬â„¢s total assets is
$9.256 million. Riskfree interest rate is 3%, and the standard deviation of the rate of change in
the underlying assets of the borrowing firm is 26%. Using the options framework, determine the
following:
a) The current market value of the loan.
d= $7.582/$9.256*eÃ¢Ë†Â§(0.03*2)=0.771441
The Current market Value of loan= 9.2562.5345
=6.72
3
=$7million
b) The interest rate to be charged on the loan.
Note: Keep 6 decimals during all your calculations, but round your final answer to last
dollar for debt value and 4 decimals for interest rate on the loan.
4
Question 4
Use the following to answer questions a) Ã¢â‚¬â€œ f).
Bank of Big Bucks
Assets:
270 day Treasury bills
$500m
2 year consumer loans
Fixed rate, 12% p.a. annually $275m
7 year commercial loans
$350m
Fixed rate, 9% p.a. annually
10 year fixed rate mortgages
$675m
Fixed rate, 6.5% p.a. annually
10 year floating rate mortgages $125m
LIBOR+50bp, monthly roll date
($ million)
Liabilities and Net Worth:
1 year Certificates of Deposit
Demand Deposits
2 year IntÃ¢â‚¬â„¢l Bonds
Fixed rate, 7.5% p.a. annually
Overnight borrowing
$350m
Equity
$100m
$550m
$750m
$175m
Notes:
The 1 year Certificates of Deposit pay 1.95% p.a. annually and will be rolled over at
maturity. The 7 year commercial loans have a duration of 5.486 years. The fixed rate
mortgages have a duration of 7.656 years. All values are market values and are trading at
par.
Assumption: 30 days per month; 90 days per quarter; 360 days per year.
What is the bankÃ¢â‚¬â„¢s duration gap?
a. 0.49 years
b. +4.24 years
c. Ã¢â‚¬â€œ0.94 years
d. 2.81 years
e. +3.69 years
Total Equity=$550m+$750m+$175m+$350m+$100m= $1,925
Total Assets= $500m+$275m+$350m
DA=
$500/$1,925*0.75+$275/$1,925*1.89+$350/$1,925*5.486+$675/$1,925*7.656+$125/$1,92
5*(1/12)
= 0.194805+0.27+0.99745+2.684571+0.00541
=4.152
DL= $550/$1,825*1+$175/$1,825*1.93+$350/$1,825*(1/360)
a)
= 0.30137+0.1851+0.0005327
=0.487
DG= RSAREL
=4.150.487*$1,825/$1,925
=3.69years
b)
What is the bankÃ¢â‚¬â„¢s interest rate risk exposure (i.e. exposed to rising or falling
rates)?
This is because rising interest rates tend to expose the risks that exist in banks.
c)
What is the onbalancesheet impact on the bank if all interest rates increase 25
basis points? (i.e. suppose Ã¯Ââ€žR/(1+R) is equal to an increase of 25 basis points.)
5
Ã¯Ââ€žR/(1+R)=0.0025
Ã¢Ë†â€ E=DG*A*[Ã¯Ââ€žR/(1+R)
=(3.69)*$1,925m*0.0025
=$17,758,125
Hence, the bankÃ¢â‚¬â„¢s interest rate risk exposure is $17,758,125.
d)
Suppose you are a risk manager of the bank. Construct an appropriate swap hedge
for your bank.
i).
Specify whether your bank should be a fixed or variablerate payer in the
swap, and
In your bank ,you should Fixedrate payer.
ii).
Calculate the notional of swap, NS, for a perfect hedge. Suppose the fixed side
of the swap has a duration of 12 years, while the variable side of the swap is
floating on a biennial base.
NS= DG*A/(DFixDVar)
=3.69*$1,925m/(122)
= 3.69*1,925m/10
= $710.33million
6
How would you use futures contracts on Canada Bond to hedge your bankÃ¢â‚¬â„¢s
interest rate risk exposure? Specify whether your bank should conduct a long or
short hedge. Justify your strategy.
Banks can hedge their accounts by selling futures or shorting their bank accounts, thus they
can offset the potential losses that exist on the bank’s balance sheet and avoid the risk that
comes with higher interest rates.
e)
f)
How would you use put option contracts to hedge the interest rate risk exposure of
your bank? That is, should the bank buy or sell put options on Canada Bond?
Hedging is achieved by buying put options and thus constructing a short account.
Question 5
Consider the following balance sheet of an FI:
Assets ($000)
Duration = 10 years
$950
Liabilities ($000)
Duration = 2 years
Equity
$860
$ 90
a) What is the impact on the FIÃ¢â‚¬â„¢s equity value if all interest rates increase such that
Ã¯Ââ€žR/(1+R) = 0.01?
DG= (102)*( $860, 000/$950,000)
= 8.19
Ã¯Ââ€žR/(1+R) = 0.01
Change in equity= DG*A*[Ã¯Ââ€žR/(1+R)
= (8.19)* $950,000*0.01
= $77,805
b) How can the FI use futures contracts to put on a macro hedge? (Note: You first need
to find out the duration gap)
Based on the conclusion from the above calculation, DG>0, if it wants to raise the
interest rate, it needs FI to sell futures to hedge the risk.
c) Suppose that the FI put on a macro hedge using zerocoupon bond futures that are
currently priced at 96. What is the impact on the FIÃ¢â‚¬â„¢s futures position (in terms of Ã¯Ââ€žF
per contract) if all interest rates increase such that Ã¯Ââ€žR/(1+R) = 0.01? The deliverable
zerocoupon bond has a maturity of 7 years.
Change in F(Ã¯Ââ€žF)= Ã¢Ë†â€ E
=DF*A*[Ã¯Ââ€žR/(1+R)
= 7*96000*0.01
= $6,720
7
d) After interest rate change of Ã¯Ââ€žR/(1+ R) = 0.01, what is the net impact on the FIÃ¢â‚¬â„¢s
equity, if the FIÃ¢â‚¬â„¢s hedge used 8 futures contracts as described in b) and c)?
Ã¢Ë†â€ E
=8.19*950000*0.01+7*8*100*960*0.01
= ($77,805)+$53,760
= $24,045
e) If the FI wanted to put on a perfect macro hedge, how many more or fewer such bond
futures contracts are needed?
NF=
=
Ã°ÂÂÂ·Ã°ÂÂÂº Ãƒâ€”Ã°ÂÂÂ´
Ã°ÂÂÂ·Ã°ÂÂÂ¹ Ãƒâ€”Ã°Ââ€˜Æ’Ã°ÂÂÂ¹
$950,000Ãƒâ€”8.19
7Ãƒâ€”$96,000
= $7,780,500/$672,000
=11.578 contracts
=12 contracts
They need 12 contracts bond futures .
Question 6
The interest rate exposures of a bank for its two repricing buckets and the use of
derivatives during the time intervals are shown below:
8
1 year
or less
13 years
35 years
beyond
RSA
48,548
234,000
105,655
Ã¢â‚¬Â¦
RSL
79,000
145,450
149,055
Ã¢â‚¬Â¦
Equity
Derivatives
Interest rate
Swap (notional)
Actual
Dollar Gap
95,000
Puts
Ã¢â‚¬Â¦
Futures
60,000
– 50,500
18,550
16,600
Ã¢â‚¬Â¦
a) For the repricing time interval of 1 year or less, identify how the puts on bond have
been used by specifying whether the puts are bought (or sold) for hedging against (or
speculation on) a rising (or falling) interest rate? (i.e. you need to make three correct
judgments that are consistent with each other)
The original gap= $485487$9000=$30452
The original gap was 30,452. but the actual gap is really 50,500, and by
comparison, the actual gap will be further from zero than the original gap was. So,
the speculative rate goes down and the puts in the debt paper are similarly sold.
b) For the repricing time interval of 13 years, what have the bond futures been used for?
(i.e. specify whether the bond futures have been bought or sold for hedging against or
speculation on a rising or falling interest rate)
The original gap= 23400145450=88550
The original shortfall was 88,550, while the actual shortfall was 18,550, which
means that the actual shortfall is infinitely closer to zero. If you want to make the
hedging rate go down, you can make the hedging rate go down by buying a lot of
futures.
c) For the repricing time interval of 35 years, how has the interest rate swap been used
for? (again, three judgments are needed as above)
The original gap= 105655149055=43400
The original gap is 43,400. while the actual gap is 16,600, which is a large difference
in value. If the company wants to speculate on a significant increase in interest rates,
it needs to inject more money.
9
FINA4471 A/B/C
FINA 4471 Ã¢â‚¬â€œ Financial Institutions
(2021 Fall)
Course Outline
Instructor: Dr. Jie Dai
Office: Online
Phone: (902) 4205269
Email: jie.dai@smu.ca
Course Description:
This course studies the managerial problems facing financial institutions such as
commercial banks, insurance companies, investment banks, pension and mutual funds.
The main objective is to enable students to understand how financial institutions
earn their profits by managing various types of risk. After successfully finishing this
course, the students should
(1) have a general understanding of the financial service industry;
(2) comprehend the process of financial intermediation by financial institutions,
the risks associated with such intermediation and, most importantly, their
management;
(3) be conversant with informational and agency problems encountered in
financial intermediation.
In accord with these three subobjectives, this course is divided into three
sections. In the first, we describe each type of financial service providers along with the
regulatory and economic environment within which they operate. This serves as an
introduction to insure that the class is familiar with the basic industry structure before
proceeding into more details. The second section examines the different types of risk that
all financial institutions face and discusses the techniques and products available to
mitigate these risks. And the third section of the course presents the modern treatment of
financial intermediation, focusing on issues of informational imperfection and agency
problems.
1
FINA4471 A/B/C
Recommended Material:
–
Textbook:
o Saunders A., M. Cornett, and P. McGraw (2014): Financial Institutions
Management Ã¢â‚¬â€œ A Risk Management Approach, 5th Canadian Edition,
McGraw Hill Ryerson. (referred to as SCM)
o Chance, D. and R. Brooks, (2013): Introduction to Derivatives and Risk
Management, 9th Edition, Nelson Education
o Greenbaum, S., A. Thakor, and A. Boot (2019): Contemporary Financial
Intermediation, 4th Edition, Elsevier. (referred to as GTB)
o Keat, P. and P. Young (2003), Managerial Economics, Prentice Hall, 4th
edition, Appendix B. (referred to as K&Y )
o Milgrom, P. and Roberts J. (1992), Economics, Organization &
Management, Prentice Hall, Chapters 5, 6. (referred to as M&R)
Optional Materials:
–
Articles:
o Note on Commercial Banking, Richard Ivey School of Business, The
University of Western Ontario, 2002. (referred to as The Note)
o A Guide to the Initial Public Offering Process, K. Ellis, R. Michaely, and
M. OÃ¢â‚¬â„¢Hara, 1999. (referred to as The Guide)
Course Delivery:
Live Sessions: Monday & Wednesday 10:00 Ã¢â‚¬â€œ 11:15am on Zoom (via Brightspace)
Q & A Sessions: Tuesday: 5:30 Ã¢â‚¬â€œ 8:15pm; Wednesday 1:00 Ã¢â‚¬â€œ 2:15pm on Zoom
Recorded Sessions: Each Live Session will be posted on Brightspace for 14 days
Grade Determination:
Participation & Progress
5%
Assignments (2Ãƒâ€”5%)
10%
Quiz 1
20%
Quiz 2
20%
Final Exam
45%
2
FINA4471 A/B/C
Academic Integrity:
Plagiarism and cheating are serious offences and may be punished by failure on
exam, in course, or expulsion from the University. As all the quizzes and exam will be
conducted online, academic integrity is of greater concern. For more information, refer to
the Ã¢â‚¬Å“Academic Integrity and Student ResponsibilityÃ¢â‚¬Â section in the Saint MaryÃ¢â‚¬â„¢s
University Academic Calendar, in particular, the Ã¢â‚¬Å“Examples of Academic OffencesÃ¢â‚¬Â in
Academic Regulation 18.
Instructor Copyright:
The materials provided to students in this course are subject to Canadian
copyright law. Further reproduction, dissemination, downloading, or sharing may not be
allowed unless permitted by an exception in the Copyright Act or with permission from
the copyright holder. Instructors own the rights to the content they create, and it is
intended for personal student use in this class. Posting this content on external sites or
sharing it with people outside of the class without permission may be an infringement of
copyright and face legal consequences.
General Information:
1. I will post my lecture notes & handouts on weekend before the two classes of
the upcoming week, in the Course Content of Brightspace.
2. Class participations (e.g. bringing about quality questions/comments that help
generate interesting inclass and Q&A discussions) are highly encouraged.
The participation marks will be awarded based on the frequency and quality of
such activities. Also, your progress in quizzes/exam over the semester is
another factor for determining class participation.
3. The two assignments should be done individually and are due one week after
the problem sets are handed out to you. The solutions will then be made
available to you before the quizzes.
3
FINA4471 A/B/C
4. There will be two online midterm quizzes, each lasting for 1h15m. They may
consist of multiple choices and short answer questions, but mainly calculation
type problems. These quizzes serve to assure that you keep up with the pace
of the course and get familiar with the topics covered in this course. They
shall be held on Brightspace at 10:00am, on Fridays of the 6th and 9th week of
the term, Oct. 15 and Nov. 5. (Anyone who cannot make it to these two
scheduled quizzes must notify me at least one week in advance to arrange an
alternative time that is before each scheduled date). After the scheduled dates,
no any makeup quizzes will be arranged, and the weight of a missed quiz will
be automatically shifted to the Final exam.
5. The university will schedule the date/time of the final exam that will also be
conducted online in Brightspace. The structure of the final exam will be
similar to that of the midterm quizzes. The content in the final will only relate
to materials covered since Quiz 2.
6. While there is a wide array of topics contained in a textbook, only those
covered in my lectures will appear in the exams (of course we will focus on
only the most interesting managerial issues in financial institutions). I may
provide more details as our classes go.
7. As all quizzes and exam will be openbook and online, you are responsible for
preparing your own formula sheet for each and all the quizzes and exam.
8. Important: All quizzes and exam are required to demonstrate your individual
efforts and honesty; any otherwise intended attempts and behaviors are strictly
prohibited. Offenses of any of these rules will face the due consequences.
4
FINA4471 A/B/C
Class Schedule
FINA4471A/B/C sections
(2021 Fall)
Note: The sequence of topics and quizzes may be subject to changes, depending on the classÃ¢â‚¬â„¢s progress and
circumstances like weather events. You will be given sufficient notice should changes occur.
Date
Course
Section
Lecture Topic
Reading
Sep. 8
– Course Introduction
Syllabus
Sep. 13
– Specialness of Financial Institutions
(FIs)
SCM Ch. 1, 7.
The Note,
The Guide.
Sep. 15
Sep. 20
Section
I
– Overview of the FIs
Sep. 22
– Liquidity Risk
Sep.
Sep.
Oct.
Oct.
– Interest Rate Risk
and Gap Analyses
27
29
4
6
Oct. 13
Oct. 18
Oct. 20
Oct. 25
Oct. 27
Nov. 1
– Credit Risk Analyses
Section
II
– Hedging Techniques in
FI Risk Management
Handout Materials
SCM Ch. 22, 23, 24.
– Market Risk
– Value at Risk (VaR)
Handout Materials
SCM Ch. 15.
Q&A for Quiz 2 on Friday Nov. 5
– Risk Capital
– RAROC
Nov. 22
Nov. 24
– Information Economics in Financial
Intermediation
Section
III
Handout Materials
SCM Ch. 10, 11.
Q&A for Quiz 1 on Friday Oct. 15
Nov. 3
Nov. 15
Nov. 17
Nov. 29
Dec. 1
Dec. 6
Dec. 8
Handout Materials
SCM Ch. 2, 3, 4, 5, 6.
Handout Materials
SCM Ch. 12, 18
Handout Materials
SCM Ch. 8, 9.
– Agency Theory in Financial
Intermediation
– Credit Contracting
Handout Materials
SCM Ch. 20.
Handout Materials
K&Y Appendix B.
M&R Ch. 5, 6.
Handout Materials
M&R Ch. 5, 6.
Handout Materials
GTB
Final Exam
Note:
Oct.13 and Nov.3 will be Q&A sessions (Preparations for the two quizzes on Fridays)
Oct. 11
No class
(Thanksgiving)
Nov. 8 – 14 No class
(Fall Break)
5
FINA4471 A/B/C
FINA 4471 Ã¢â‚¬â€œ Financial Institutions
(2021 Fall)
Course Outline
Instructor: Dr. Jie Dai
Office: Online
Phone: (902) 4205269
Email: jie.dai@smu.ca
Course Description:
This course studies the managerial problems facing financial institutions such as
commercial banks, insurance companies, investment banks, pension and mutual funds.
The main objective is to enable students to understand how financial institutions
earn their profits by managing various types of risk. After successfully finishing this
course, the students should
(1) have a general understanding of the financial service industry;
(2) comprehend the process of financial intermediation by financial institutions,
the risks associated with such intermediation and, most importantly, their
management;
(3) be conversant with informational and agency problems encountered in
financial intermediation.
In accord with these three subobjectives, this course is divided into three
sections. In the first, we describe each type of financial service providers along with the
regulatory and economic environment within which they operate. This serves as an
introduction to insure that the class is familiar with the basic industry structure before
proceeding into more details. The second section examines the different types of risk that
all financial institutions face and discusses the techniques and products available to
mitigate these risks. And the third section of the course presents the modern treatment of
financial intermediation, focusing on issues of informational imperfection and agency
problems.
1
FINA4471 A/B/C
Recommended Material:
–
Textbook:
o Saunders A., M. Cornett, and P. McGraw (2014): Financial Institutions
Management Ã¢â‚¬â€œ A Risk Management Approach, 5th Canadian Edition,
McGraw Hill Ryerson. (referred to as SCM)
o Chance, D. and R. Brooks, (2013): Introduction to Derivatives and Risk
Management, 9th Edition, Nelson Education
o Greenbaum, S., A. Thakor, and A. Boot (2019): Contemporary Financial
Intermediation, 4th Edition, Elsevier. (referred to as GTB)
o Keat, P. and P. Young (2003), Managerial Economics, Prentice Hall, 4th
edition, Appendix B. (referred to as K&Y )
o Milgrom, P. and Roberts J. (1992), Economics, Organization &
Management, Prentice Hall, Chapters 5, 6. (referred to as M&R)
Optional Materials:
–
Articles:
o Note on Commercial Banking, Richard Ivey School of Business, The
University of Western Ontario, 2002. (referred to as The Note)
o A Guide to the Initial Public Offering Process, K. Ellis, R. Michaely, and
M. OÃ¢â‚¬â„¢Hara, 1999. (referred to as The Guide)
Course Delivery:
Live Sessions: Monday & Wednesday 10:00 Ã¢â‚¬â€œ 11:15am on Zoom (via Brightspace)
Q & A Sessions: Tuesday: 5:30 Ã¢â‚¬â€œ 8:15pm; Wednesday 1:00 Ã¢â‚¬â€œ 2:15pm on Zoom
Recorded Sessions: Each Live Session will be posted on Brightspace for 14 days
Grade Determination:
Participation & Progress
5%
Assignments (2Ãƒâ€”5%)
10%
Quiz 1
20%
Quiz 2
20%
Final Exam
45%
2
FINA4471 A/B/C
Academic Integrity:
Plagiarism and cheating are serious offences and may be punished by failure on
exam, in course, or expulsion from the University. As all the quizzes and exam will be
conducted online, academic integrity is of greater concern. For more information, refer to
the Ã¢â‚¬Å“Academic Integrity and Student ResponsibilityÃ¢â‚¬Â section in the Saint MaryÃ¢â‚¬â„¢s
University Academic Calendar, in particular, the Ã¢â‚¬Å“Examples of Academic OffencesÃ¢â‚¬Â in
Academic Regulation 18.
Instructor Copyright:
The materials provided to students in this course are subject to Canadian
copyright law. Further reproduction, dissemination, downloading, or sharing may not be
allowed unless permitted by an exception in the Copyright Act or with permission from
the copyright holder. Instructors own the rights to the content they create, and it is
intended for personal student use in this class. Posting this content on external sites or
sharing it with people outside of the class without permission may be an infringement of
copyright and face legal consequences.
General Information:
1. I will post my lecture notes & handouts on weekend before the two classes of
the upcoming week, in the Course Content of Brightspace.
2. Class participations (e.g. bringing about quality questions/comments that help
generate interesting inclass and Q&A discussions) are highly encouraged.
The participation marks will be awarded based on the frequency and quality of
such activities. Also, your progress in quizzes/exam over the semester is
another factor for determining class participation.
3. The two assignments should be done individually and are due one week after
the problem sets are handed out to you. The solutions will then be made
available to you before the quizzes.
3
FINA4471 A/B/C
4. There will be two online midterm quizzes, each lasting for 1h15m. They may
consist of multiple choices and short answer questions, but mainly calculation
type problems. These quizzes serve to assure that you keep up with the pace
of the course and get familiar with the topics covered in this course. They
shall be held on Brightspace at 10:00am, on Fridays of the 6th and 9th week of
the term, Oct. 15 and Nov. 5. (Anyone who cannot make it to these two
scheduled quizzes must notify me at least one week in advance to arrange an
alternative time that is before each scheduled date). After the scheduled dates,
no any makeup quizzes will be arranged, and the weight of a missed quiz will
be automatically shifted to the Final exam.
5. The university will schedule the date/time of the final exam that will also be
conducted online in Brightspace. The structure of the final exam will be
similar to that of the midterm quizzes. The content in the final will only relate
to materials covered since Quiz 2.
6. While there is a wide array of topics contained in a textbook, only those
covered in my lectures will appear in the exams (of course we will focus on
only the most interesting managerial issues in financial institutions). I may
provide more details as our classes go.
7. As all quizzes and exam will be openbook and online, you are responsible for
preparing your own formula sheet for each and all the quizzes and exam.
8. Important: All quizzes and exam are required to demonstrate your individual
efforts and honesty; any otherwise intended attempts and behaviors are strictly
prohibited. Offenses of any of these rules will face the due consequences.
4
FINA4471 A/B/C
Class Schedule
FINA4471A/B/C sections
(2021 Fall)
Note: The sequence of topics and quizzes may be subject to changes, depending on the classÃ¢â‚¬â„¢s progress and
circumstances like weather events. You will be given sufficient notice should changes occur.
Date
Course
Section
Lecture Topic
Reading
Sep. 8
– Course Introduction
Syllabus
Sep. 13
– Specialness of Financial Institutions
(FIs)
SCM Ch. 1, 7.
The Note,
The Guide.
Sep. 15
Sep. 20
Section
I
– Overview of the FIs
Sep. 22
– Liquidity Risk
Sep.
Sep.
Oct.
Oct.
– Interest Rate Risk
and Gap Analyses
27
29
4
6
Oct. 13
Oct. 18
Oct. 20
Oct. 25
Oct. 27
Nov. 1
– Credit Risk Analyses
Section
II
– Hedging Techniques in
FI Risk Management
Handout Materials
SCM Ch. 22, 23, 24.
– Market Risk
– Value at Risk (VaR)
Handout Materials
SCM Ch. 15.
Q&A for Quiz 2 on Friday Nov. 5
– Risk Capital
– RAROC
Nov. 22
Nov. 24
– Information Economics in Financial
Intermediation
Section
III
Handout Materials
SCM Ch. 10, 11.
Q&A for Quiz 1 on Friday Oct. 15
Nov. 3
Nov. 15
Nov. 17
Nov. 29
Dec. 1
Dec. 6
Dec. 8
Handout Materials
SCM Ch. 2, 3, 4, 5, 6.
Handout Materials
SCM Ch. 12, 18
Handout Materials
SCM Ch. 8, 9.
– Agency Theory in Financial
Intermediation
– Credit Contracting
Handout Materials
SCM Ch. 20.
Handout Materials
K&Y Appendix B.
M&R Ch. 5, 6.
Handout Materials
M&R Ch. 5, 6.
Handout Materials
GTB
Final Exam
Note:
Oct.13 and Nov.3 will be Q&A sessions (Preparations for the two quizzes on Fridays)
Oct. 11
No class
(Thanksgiving)
Nov. 8 – 14 No class
(Fall Break)
5
B Quizzes – Financial Institutions X
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Time Limit: 2:30:00
Time Left:2:29:15
Zepeng Zhang: Attempt 1
Page 1:
For a M/C question, circle only ONE right answer:
1
2
3
—
—
Question 1 (3 points)
4
5
6
—
—
A lender knows that her potential borrowers are of three types in terms of credit risk: 24% of them are H
risk, 48% are Mrisk, and the remaining are Lrisk. The acceptable loan rates for the three types of borrowers
are, respectively, 13.2%, 9.8%, and 7.5%. While each borrower knows his own risk, the lender lacks such
information, so she charges a weightedaverage loan rate of 9.97%. At this rate, however, 4 out of 7 (i.e. 4/7)
of the Lrisk borrowers will no longer borrow, as they deem 9.97% to be too high and unacceptable.
Recognizing this, what new weightedaverage loan rate would the lender charge?
(Keep 4 decimals, e.g. if 0.12345, keep 0.1235, and submit 12.35%)
Page 2:
7
8
9
?
 00
A
10
11
12

Question 2 (1 point)
Page 3:
Which of the following statement is correct?
13
14
15
Adverse selection is the reason for asymmetric information
16
17
18
Moral hazard is equivalent to adverse selection
—
Page 4:
Agency problem and asymmetric information are similar
Asymmetric information gives rise to adverse selection
Adverse selection and moral hazard are of the same nature
Time Left:2:28:50
Zepeng Zhang: Attempt 1
Question 2 (1 point)
Which of the following statement is correct?
Adverse selection is the reason for asymmetric information
Moral hazard is equivalent to adverse selection
Agency problem and asymmetric information are similar
Asymmetric information gives rise to adverse selection
Adverse selection and moral hazard are of the same nature
Question 3 (1 point)
Which of the following statement describes best the concept of incentive compatible?
The incentives of both the agent and the principal are optimal.
The agent is willing to disclose its information.
The optimal choice of the agent to maximize its own welfare will also reveal the information or action
of the agent.
Time Left:2:28:50
Zepeng Zhang: Attempt 1
Question 2 (1 point)
Which of the following statement is correct?
Adverse selection is the reason for asymmetric information
Moral hazard is equivalent to adverse selection
Agency problem and asymmetric information are similar
Asymmetric information gives rise to adverse selection
Adverse selection and moral hazard are of the same nature
Question 3 (1 point)
Which of the following statement describes best the concept of incentive compatible?
The incentives of both the agent and the principal are optimal.
The agent is willing to disclose its information.
The optimal choice of the agent to maximize its own welfare will also reveal the information or action
of the agent.
Question 3 (1 point)
Which of the following statement describes best the concept of incentive compatible?
The incentives of both the agent and the principal are optimal.
The agent is willing to disclose its information.
The optimal choice of the agent to maximize its own welfare will also reveal the information or action
of the agent.
The choice of the principal maximizes the welfare of the agent.
Both the agent and the principal maximize their own welfare.
The action of agent is not observed by the principal but can be assured of being the right choice.
Question 4 (1 point)
Which of the following statement is correct?
A lending bank does not need to monitor how its borrowing firm will use the bank loan in the firm’s
business activities
A lending bank can trust its borrowing firm, since both parties are business partners striving for success
of the project financed with bank loan
A borrower wants to avoid high risk project when bank loan is used in financing the project
It makes no sense that a borrower may incr Main Content project’s risk when using borrowed money
The goals of a lending bank and its borrowing firm are divergent, and covenants help control
consequences from such divergence.
Question 5 (3 points)
A firm’s existing operations will realize a present value of $50,000 upon success, but $30,000 if fail. By
spending $42,000, the firm can modify its operations which will change the present value to $135,000 upon
success, but to $7,000 if fail. The success probability is 0.6 and the failure probability is 0.4, regardless of
whether the modification is implemented or not. To decide whether the firm should implement it or not, what
is the NPV of this operational modification?
(Round to the last dollar)
A/
Question 6 (1 point)
Why is asymmetric information an issue for financial markets and institutions?
Because the two parties will transact at unfair terms.
Because without information, we cannot make right decision.
Because the more informed party can take advantage of it.
Because it will lead to closure of market.
Because the less informed party will be exploited.
Question 6 (1 point)
Why is asymmetric information an issue for financial markets and institutions?
Because the two parties will transact at unfair terms.
Because without information, we cannot make right decision.
Because the more informed party can take advantage of it.
Because it will lead to closure of market.
Because the less informed party will be exploited.
For a M/C question, circle only ONE right answer:
Question 1 (3 points)
A lender knows that her potential borrowers are of three types in terms of credit risk: 24% of them are H
risk, 48% are Mrisk, and the remaining are Lrisk. The acceptable loan rates for the three types of borrowers
are, respectively, 13.2%, 9.8%, and 7.5%. While each borrower knows his own risk, the lender lacks such
information, so she charges a weightedaverage loan rate of 9.97%. At this rate, however, 4 out of 7 (i.e. 4/7)
of the Lrisk borrowers will no longer borrow, as they deem 9.97% to be too high and unacceptable.
Recognizing this, what new weightedaverage loan rate would the lender charge?
(Keep 4 decimals, e.g. if 0.12345, keep 0.1235, and submit 12.35%)
AJ
You work as a trader for a financial institution and manage a portfolio of stock whose rates of return follow a
normal distribution with expected rate of return of 6.70% and volatility of 14.50% on an annual base of 250
trading days. For the purpose of market risk measurement, you set the confidence level at 98%. The current
value of your portfolio is $680,000.
Question 7 (2 points)
What is the weekly (i.e. over 5 trading days) expected rate of return?
5
(Round to 6 decimals for your final answer, e.g. if 0.0123456, round to 0.012346 and submit 1.2346%)
A
Question 8 (3 points)
What is the weekly (i.e. over 5 trading days) volatility of your portfolio? (Round to 6 decimals for your final
answer)
A
Question 9 (1 point)
What would be the expected value of your portfolio in one week of five trading days? (Round your answer to
the last dollar)
$1,280
$912,950
$465,120
$680,883
O $798,720
$568,720
Question 10 (2 points)
Use Excel to find the quantile of the unit normal distribution that you may need for any calculations
corresponding to 98% confidence level.
Hint: To find the pth quantile of a normal distribution in Excel, use the function NORMINV(p, u, o).
0.2255
Question 10 (2 points)
Use Excel to find the quantile of the unit normal distribution that you may need for any calculations
corresponding to 98% confidence level.
Hint: To find the pth quantile of a normal distribution in Excel, use the function NORMINV(p, 4, 0).
0.2255
2.0537
1.8808
0.3915
1.8808
2.0537
1.6449
Question 11 (4 points)
What is the absolute VaR (in dollar amount) of your portfolio in one week of 5 trading days? i.e., what
critical value your portfolio can fall below with 98% confidence level?
(Round to 6 decimals for rate and round to last dollar for amount)
$762,120
$652,246
$709,520
$670,560
$549,180
$598,334
$526,500
$650,480
$765,230
Question 12 (3 points)
What is the relative VaR (in dollar amount) of your portfolio in one week? i.e., what loss your portfolio could
exceed with 98% confidence level?
(Round to last dollar; use 1000 separator, and put negative sign before dollar sign for loss when submitting
your answer)
A
There are two types of firm L and H; each can invest $140 in a project that will yield the following
payoffs one period hence:
0.82 $210
0.72 $240
L: $140
H: $140
0.18 $0
0.28 $0
The firms want to borrow the funds from a bank. While each firm knows its own type, the bank does not
a
know whether a particular firm is of type L or H, even after a credit analysis. Interest payments on loans are
tax deductible, and the firm tax rate is 34%. Assume no discounting and everyone is risk neutral.
If you are the banker and will use interest rate and equity capital as instruments in your lending, design
appropriate loan contracts by answering the following questions.
Question 13 (2 points)
You assume that firm L will finance part of the investment with some equity, E, and borrow the rest of the
funds needed from your bank using a loan (I), while firm H will use no equity, and borrow all from your
bank using a loan (h).
Use probability trees to draw the payoff structures of the two loan contracts, on a piece of paper, and insert
an image of your drawings in JPG or PDF type, using the “Insert Image” button in the menu under “+”.
(Make sure you see you image in the response box after you have inserted it).
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Question 14 (2 points)
To break even, what interest rate should you charge on loan (h)?
(Round your final answer to 4 decimals, e.g. if 0.12468, submit 12.47%)
A
Question 15 (2 points)
What break even interest rate should you charge on loan (1)?
(Round your final answer to 4 decimals, e.g. if 0.12468, submit 12.47%)
A
Question 16 (4 points)
A
Question 16 (4 points)
To assure that firm H will actually borrow with loan (h), what condition will you impose on firm H?
Write down neatly the condition on a piece of paper and insert an image of it in JPG or PDF type, using the ”
is Insert Image” button in the menu under “+”.
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Add a File
Record Audio
Record Video
Question 17 (4 points)
To make sure that firm H will actually borrow with loan (h), what minimum equity E should be in your loan
contract?
Time Left:1:06:33
Zepeng Zhang: Attempt 1
Question 17 (4 points)
To make sure that firm H will actually borrow with loan (h), what minimum equity E should be in your loan
contract?
(Keep 4 decimals in your calculations and round your final answer to the last cent)
A/
Question 18 (3 points)
With the equity you found above, how much would the firm L be better off by using the loan you designed
for the borrower L, rather than using the loan you designed for firm H? (Round your final answer to the last
cent)
Write down neatly your two li…
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