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Answer all the questions provided on the PDF, providing an explanation when asked. Provide proof and formulas used for each question. You MUST be knowledgeable in Corporate Finance to complete this.

Econ 372/572 & Fintech 522,
Asset Pricing & Risk Management
Homework 4
**Due 5pm Duke time on Thursday April 19**
** Scan or compile your homework answer sheet into a single PDF file and
upload it to the course Sakai site **
Instructions on how to submit your completed homework on Sakai can be found at:
https://sakai-duke.screenstepslive.com/s/sakai_support/m/71158/l/1091053-how-do-studentssubmit-an-assignment
Fixed Income
1. A zero coupon bond with 2.5 years to maturity has a yield to maturity of 25% per annum. A 3year maturity annual-pay coupon bond has a face value of \$1000 and a 25% coupon rate. The
coupon bond also has a yield to maturity of 25%.
a) Does the longer maturity bond have a larger interest rate sensitivity? Why or why not?
b) Calculate for each bond the percentage price change associated with a change of yield to
maturity from 25% to 26%.
Options
2. Construct profit diagrams or profit tables on expiration to show what position in IBM puts, calls
and/or underlying stock best expresses the investor’s objectives described below. Assume IBM
currently sells for \$150 and for the profit diagrams/tables use share prices between \$100 and
\$200 (in \$10 increments). Also assume that “at the money” puts and calls cost \$15 each. (As
usual, the profit calculations ignore dividends and interest.)
a) An investor wants upside potential if IBM increases but wants (net) losses no greater than
\$15 if prices decline.
b) An investor wants to capture profits if IBM declines in price but wants a guaranteed limited
loss if prices increase.
c) An investor wants to profit if IBM’s upcoming earnings announcement is either
3. Stock XYZ is trading at \$100 and pays no dividends. The interest rate is 5% per year continuously
compounded. A one-year European call option on XYZ has a strike \$104. The volatility of XYZ is
20%.
a) What is the intrinsic value of the call?
b) What is the adjusted intrinsic value of the call?
c) Find the price of the European call, using the Black-Scholes model. What is the price of an
American call option with the same characteristics?
d) How many European calls do you need to buy/sell to hedge one share of XYZ?
4. Excel Question. Use the Black and Scholes file posted on Sakai. We want to explore the effect of
changing the time to expiration T from 1 to 2 years on the value of an out-of-the-money put
option. Throughout this exercise we keep the stock price at SÃ¢â€šâ‚¬=70, the strike price at X=40, the
dividend rate at ÃŽÂ´=0, and the stock price volatility at ÃÆ’=35%. For the interest rate r, consider
various values (from very low to very high).
a) Show how the effect on the put price for T=1Ã¢â€ â€™2 depends on the interest rate and explain
intuitively why this happens.
b) Confirm that the call price is always increasing in the time to expiration T.
Fixed-Income Securities:
Forward Rates & Yield Curve
Prof. Aino Levonmaa
1
Overview
Ã¯â€šÂ§ Forward rate: an interest rate on a future loan that is fixed
today
Ã¯Æ’Ëœ Inferred from the term structure of interest rates
y02
0
y01
1
f12
2
Ã¯â€šÂ§ Forward rate (f12) vs. spot rate (y01, y02)
Ã¯â€šÂ§ Yield curve (the term structure of interest rates)
Ã¢â‚¬â€œ the relationship between yield and maturity
Ã¯â€šÂ§ Theories for explaining the term structure of interest rates
Examples of Forward Rates
Ã¯â€šÂ§ A forward rate is an interest rate on a future loan
that is fixed today.
Ã¯â€šÂ§ The forward rate for 1-year lending starting t years
from now is denoted ft.
Ã¯â€šÂ§ A company will receive a payment next year and must
make a payment two years from now. The company
is worried about the re-investment risk related to the
incoming payment. Can the company lock in a lending
rate, starting one year from now?
Ã¯â€šÂ§ A firm foresees the need for short-term funds one
year from now but is worried about the interest rate
rising. Can they Ã¢â‚¬Å“lock inÃ¢â‚¬Â a rate for a one-year loan,
starting one year from now?
3
Engineering Forward Rates
Ã¯â€šÂ§ Suppose that the yields and prices of 1- and 2-year
zero coupon bonds are
P1 Ã¯â‚¬Â½
100
1 Ã¯â‚¬Â« y1
Ã¯Æ’â€º
100
P2 Ã¯â‚¬Â½
(1 Ã¯â‚¬Â« y2 ) 2
y1 Ã¯â‚¬Â½
Ã¯Æ’â€º
100
Ã¯â‚¬Â­1
P1
Ã¯Æ’Â¦ 100 Ã¯Æ’Â¶
Ã¯Æ’Â·Ã¯Æ’Â·
y2 Ã¯â‚¬Â½ Ã¯Æ’Â§Ã¯Æ’Â§
Ã¯Æ’Â¨ P2 Ã¯Æ’Â¸
1
2
Ã¯â‚¬Â­1
Ã¯â€šÂ§ Then, under the law of one price, what is the interest
rate specified today on a contract to buy a 1 year ZCB
1 year from now, i.e. what is the 1-year forward
rate?
4
Engineering Forward Rates
Ã¯â€šÂ§ The cash flows of the three instruments are as follows:
0
1
1-year zero
-P1
100
2-year zero
-P2
Forward contract
2
100
-Pf
100
Ã¯â€šÂ§ To find the price Pf and the forward rate from time 1 to
time 2
Ã¯â€šÂ§ Consider a strategy of:
Ã¢â‚¬â€œ Buying 1 unit of the 2 year zero and
Ã¢â‚¬â€œ Short selling P2/P1 units of the 1 year zero.
5
Engineering Forward Rates
Ã¯â€šÂ§ Cash flows to the strategy are:
t=0
t=1
t=2
Sell short
ZCB(1)
+(P2/P1)*P1=
P2
-(P2/P1)*100
0
ZCB(2)
-P2
0
100
Net CFs
0
-(P2/P1)*100
100
Ã¯â€šÂ§ Then, under no arbitrage, the price of the ZCB one year
from now must be:
Ã°Ââ€˜Â·Ã°ÂÅ¸Â
Ã°Ââ€˜Â·Ã°Ââ€™â€¡ = Ã°ÂÅ¸ÂÃ°ÂÅ¸Å½Ã°ÂÅ¸Å½
Ã°Ââ€˜Â·Ã°ÂÅ¸Â
What about the corresponding forward rate?
6
Engineering Forward Rates
Ã¯â€šÂ§ The forward rate is:
100
f1, 2
100
P1
(1 Ã¯â‚¬Â« y2 ) 2
(1 Ã¯â‚¬Â« y1 )
Ã¯â‚¬Â½
Ã¯â‚¬Â­1 Ã¯â‚¬Â½ Ã¯â‚¬Â­1 Ã¯â‚¬Â½
Ã¯â‚¬Â­1 Ã¯â‚¬Â½
Ã¯â‚¬Â­1
100
Pf
P2
1 Ã¯â‚¬Â« y1
2
(1 Ã¯â‚¬Â« y2 )
f1, 2
(1 Ã¯â‚¬Â« y2 ) 2
Ã¯â‚¬Â½
Ã¯â‚¬Â­1
1 Ã¯â‚¬Â« y1
Ã¯â€šÂ§ This result can be generalized to calculate the
forward rate at any maturity:
(1 Ã¯â‚¬Â« yn Ã¯â‚¬Â«1 ) n Ã¯â‚¬Â«1
f n ,n Ã¯â‚¬Â«1 Ã¯â‚¬Â½
Ã¯â‚¬Â­1
n
(1 Ã¯â‚¬Â« yn )
7
Engineering Forward Rates:
Example
Ã¯â€šÂ§ Suppose that
Ã¢â‚¬â€œ A 1-year zero has a YTM of 2%
Ã¢â‚¬â€œ A 2-year zero has a YTM of 3%
1) What are the prices of these bonds (assume FV =
100)?
2) How can you trade these bonds to replicate a loan
between year 1 and year 2 ? (Ã¢â‚¬Å“syntheticÃ¢â‚¬Â loan.)
3) What is the implicit interest rate in the second year?
Ã¢â‚¬â€œ This interest rate is the forward rate, f.
8
Forward Rates
Ã¯â€šÂ§ The forward rate is determined by no-arbitrage
(1 Ã¯â‚¬Â« yn Ã¯â‚¬Â«1 ) n Ã¯â‚¬Â«1
f n ,n Ã¯â‚¬Â«1 Ã¯â‚¬Â½
Ã¯â‚¬Â­1
n
(1 Ã¯â‚¬Â« yn )
Ã¯â€šÂ§ Example with n=2 (and rewriting)
(1 Ã¯â‚¬Â« y2 ) Ã¯â‚¬Â½ Ã¯Ââ€ºÃ¯â‚¬Â¨1 Ã¯â‚¬Â« y1 Ã¯â‚¬Â©Ã¯â‚¬Â¨1 Ã¯â‚¬Â« f1, 2 Ã¯â‚¬Â©Ã¯ÂÂ
1/ 2
Ã¯â€šÂ§ Forward rate = interest rate that would need to
prevail in second year to make the long- and shortterm investments equally attractive.
9
Forward contracts and Futures
Ã¯â€šÂ§ Forward rates are also traded directly:
Ã¢â‚¬â€œ FRAÃ¢â‚¬â„¢s : Forward Rate Agreements
Ã¢â‚¬â€œ Interest Rate Futures
Ã¯â€šÂ§ We will come back to this in our class on Futures
10
The Yield Curve or the
Term-Structure of Interest
Ã¯â€šÂ§ The collection of YTM of zero-coupon bonds has many
names:
Ã¢â‚¬â€œ The term structure of zero-coupon bond yields
Ã¢â‚¬â€œ The term structure of (spot) interest rates
Ã¢â‚¬â€œ The yield curve
Ã¯â€šÂ§ Typical shapes of the term structure of interest:
Ã¢â‚¬â€œ Upward sloping (most common)
Ã¢â‚¬â€œ Downward sloping
Ã¢â‚¬â€œ Flat
Ã¢â‚¬â€œ Hump-shaped
11
The Yield Curve
Ã¯â€šÂ§ The yield curve can take many shapes; for example flat, upward
sloping, inverted, or humped.
15
The yield curve Jan 2020 vs. Jan 2021
13
1. The Expectations Hypothesis
Ã¯â€šÂ§ Assumptions:
Ã¢â‚¬â€œ No transaction costs.
Ã¢â‚¬â€œ No default.
Ã¢â‚¬â€œ Investors maximize profits; they are riskneutral.
Ã¢â‚¬â€œ All bonds are zero-coupon bonds.
Ã¯â€šÂ§ Implication of risk-neutrality: Investors choose
the maturity of their bonds to maximize holding
period returns.
14
Expectation Hypothesis
Ã¯â€šÂ§ Consider the choice between:
Ã¢â‚¬â€œ Buying 1-year zero coupon bond and holding it to maturity
Ã¢â‚¬â€œ Buying a 2-year zero coupon bond and then selling it after 1 year
when it will be a 1-year zero coupon bond.
Ã¯â€šÂ§ What are the expected holding period returns for these two
Ã¢â‚¬Å“one yearÃ¢â‚¬Â strategies?
Ã¯â€šÂ§ Alternatively, consider the choice between:
Ã¢â‚¬â€œ Buying 2-year zero coupon bond and holding it to maturity
Ã¢â‚¬â€œ Buying a 1-year zero coupon bond and then rolling the proceeds
into a second 1-year zero coupon bond at time 1.
Ã¯â€šÂ§ What are the expected holding period returns for these two
Ã¢â‚¬Å“two yearÃ¢â‚¬Â strategies?
15
E[HPR] One-year Strategies
Ã¯â€šÂ§ Expected HPR on the 1-year bond (known with certainty)
V1
100
E[ HPR ] Ã¯â‚¬Â½ HPR Ã¯â‚¬Â½
Ã¯â‚¬Â­1 Ã¯â‚¬Â½
Ã¯â‚¬Â­ 1 Ã¯â‚¬Â½ y 0,1
V0
P0,1
Ã¯â€šÂ§ Expected HPR on buying a 2-year zero coupon bond at t=0 and
selling it at t=1
100
(1 Ã¯â‚¬Â« E[ y1, 2 ])
(1 Ã¯â‚¬Â« y0, 2 ) 2
E[V1 ]
E[ P1, 2]
E[ HPR] Ã¯â‚¬Â½
Ã¯â‚¬Â­1 Ã¯â‚¬Â½
Ã¯â‚¬Â­1 Ã¯â€šÂ»
Ã¯â‚¬Â­1 Ã¯â‚¬Â½
Ã¯â‚¬Â­1
100
V0
P0, 2
1 Ã¯â‚¬Â« E[ y1, 2 ]
2
(1 Ã¯â‚¬Â« y0, 2 )
16
E[HPR] One-year Strategies
Ã¯â€šÂ§ Under the expectations hypothesis, this HPR has to be equal to
the return from holding the 1-year bonds:
(1 Ã¯â‚¬Â« y0, 2 ) 2
1 Ã¯â‚¬Â« E[ y1, 2 ]
Ã¯â‚¬Â­ 1 Ã¯â‚¬Â½ y0,1 Ã¯Æ’â€º (1 Ã¯â‚¬Â« y0, 2 ) 2 Ã¯â‚¬Â½ (1 Ã¯â‚¬Â« y0,1 )(1 Ã¯â‚¬Â« E[ y1, 2 ])
Ã¯â€šÂ§ Or equivalently:
(1 Ã¯â‚¬Â« y0, 2 ) Ã¯â‚¬Â½ [(1 Ã¯â‚¬Â« y0,1 )(1 Ã¯â‚¬Â« E[ y1, 2 ])]
1
2
Ã¯â€šÂ§ The 2-year rate is the geometric average of the expected 1year rates
17
E[HPR] Two-year Strategies
bond and rolling the proceeds over into a second 1-year bond
at maturity (in 1 year)
Ã¯â€šÂ§ Buy and hold 2-year bond:
Ã¯Æ’Â¦ V2
E[ HPR ] Ã¯â‚¬Â½ Ã¯Æ’Â§Ã¯Æ’Â§
Ã¯Æ’Â¨ V0
Ã¯Æ’Â¶
Ã¯Æ’Â·Ã¯Æ’Â·
Ã¯Æ’Â¸
1
2
Ã¯Æ’Â¦ 100 Ã¯Æ’Â¶
Ã¯Æ’Â·
Ã¯â‚¬Â­ 1 Ã¯â‚¬Â½ Ã¯Æ’Â§Ã¯Æ’Â§
Ã¯Æ’Â·
P
0
,
2
Ã¯Æ’Â¨
Ã¯Æ’Â¸
1
2
Ã¯Æ’Â¦
Ã¯Æ’Â§
100
Ã¯â‚¬Â­1 Ã¯â‚¬Â½ Ã¯Æ’Â§
Ã¯Æ’Â§Ã¯Æ’Â§ 100
2
(
1
Ã¯â‚¬Â«
y
)
0, 2
Ã¯Æ’Â¨
Ã¯Æ’Â¶
Ã¯Æ’Â·
Ã¯Æ’Â·
Ã¯Æ’Â·Ã¯Æ’Â·
Ã¯Æ’Â¸
1
2
Ã¯â‚¬Â­ 1 Ã¯â‚¬Â½ y 0, 2
18
E[HPR] Two-year Strategies
Ã¯â€šÂ§ Rolling over the 1-year bond generates:
Ã¯Æ’Â¦
Ã¯Æ’Â¦100
Ã¯Æ’Â¶Ã¯Æ’Â¶
1
Ã¯Æ’Â§ 100Ã¯Æ’Â§
Ã¯Æ’Â·Ã¯Æ’Â·
E
[
P
]
Ã¯Æ’Â¦ E[V2 ] Ã¯Æ’Â¶ 2
1, 2 Ã¯Æ’Â¸ Ã¯Æ’Â·
Ã¯Æ’Â¨
Ã¯Æ’Â·Ã¯Æ’Â· Ã¯â‚¬Â­ 1 Ã¯â€šÂ» Ã¯Æ’Â§
E[ HPR] Ã¯â€šÂ» Ã¯Æ’Â§Ã¯Æ’Â§
Ã¯Æ’Â§
Ã¯Æ’Â·
P0,1
Ã¯Æ’Â¨ V0 Ã¯Æ’Â¸
Ã¯Æ’Â§
Ã¯Æ’Â·
Ã¯Æ’Â¨
Ã¯Æ’Â¸
1
2
Ã¯Æ’Â¦
Ã¯Æ’Â¶
Ã¯Æ’Â§ 100Ã¯â‚¬Â¨1 Ã¯â‚¬Â« E[ y1, 2 ]Ã¯â‚¬Â© Ã¯Æ’Â·
Ã¯â‚¬Â­1 Ã¯â€šÂ» Ã¯Æ’Â§
Ã¯Æ’Â·
100
Ã¯Æ’Â§
Ã¯Æ’Â·
(
1
Ã¯â‚¬Â«
y
)
0
,
1
Ã¯Æ’Â¨
Ã¯Æ’Â¸
1
2
Ã¯â‚¬Â­1
Ã¯â‚¬Â½ Ã¯Ââ€º(1 Ã¯â‚¬Â« y0,1 )Ã¯â‚¬Â¨1 Ã¯â‚¬Â« E[ y1, 2 ]Ã¯â‚¬Â©Ã¯ÂÂ 2 Ã¯â‚¬Â­ 1
1
19
Main Insights
Ã¯â€šÂ§ The general result, under the expectations hypothesis and its
risk neutrality assumption is that the yield on a long-term
bond is equal to the geometric average of expected short
term yields.
1
2
Ã¯â€šÂ§ All expected HPR are equalized in equilibrium.
Ã¯â€šÂ§ Typical shape of yield curve is flat.
Ã¯â€šÂ§ Short-term interest rates are more volatile than long-term
interest rates.
Ã¯â€šÂ§ What does an upward-sloping yield curve, i.e. y(2) > y(1), imply
about the expected future short-term interest rate?
20
Expectations Hypothesis: Example
Ã¯â€šÂ§ Suppose:
yt(1) = y01 = 2%
Et[yt+1(1)]= Et(y12) = 4%
Ã¯â€šÂ§ What should yt(2) (=y02) be under the expectations hypothesis?
Ã¯â€šÂ§ Investigate if short-term (1 year) investors are indifferent
between:
1) buying & holding a 1-year ZCB
2) buying a 2-year ZCB and selling it after 1 year
Ã¯â€šÂ§ Investigate if long-term (2 year) investors are indifferent
between:
1) buying & holding a 2-year ZCB
2) buying a 1-year ZCB and rolling the proceeds into
another 1-year ZCB at time 1
21
Ã¯â€šÂ§ Recall the definition of the forward rate:
1
2
Ã¯â€šÂ§ Compare with prediction of EH theory
1
2
Ã¯â€šÂ§ Therefore, under the EH theory, the forward rate
equals the expected future 1-year interest rate :
f t (1) Ã¯â‚¬Â½ Et [ yt Ã¯â‚¬Â«1 (1)]
22
2. Liquidity Preference Theory
Ã¯â€šÂ§ Problem with the EH theory: yield curve is flat on average while
in the data it is upward sloping 90% of the time.
Ã¯â€šÂ§ Source of problem: risk-neutrality assumption in EH theory.
Ã¯â€šÂ§ short rate next period is not known, it is risky!
Ã¯â€šÂ§ When investors are risk averse, they care not only about the
expected short rate but also about its volatility.
Ã¯â€šÂ§ Investors in long-term bonds want to be compensated
Ã¢â‚¬â€œ For Ã¢â‚¬Å“tying upÃ¢â‚¬Â money for a long time.
Ã¢â‚¬â€œ For facing price risk if they need to sell before maturity.
Ã¯â€šÂ§ Conversely, issuers of bonds are willing to pay a higher interest
rate on long-term bonds because
Ã¢â‚¬â€œ They can lock in an interest rate for many years
23
Liquidity Preference Theory
Ã¯â€šÂ§ The associated risk premium is denoted the
1
2
Ã¯â‚¬Â« LP
Ã¯â€šÂ§ Based on this theory,
Ã¢â‚¬â€œ What is the typical shape of the yield curve?
Ã¢â‚¬â€œ Is the forward rate still equal to the expected
future short rate?
24
2. Liquidity Preference Theory: Examples
LP (+)
E (+)
YC (+)
Yield
Yield
YC (+)
LP (+)
E (-)
Maturity
YC (-)
1y
1y
10y
Yield
Yield
1y
10y
Maturity
LP(+)
YC (0)
E (-)
LP (+)
10y
1y
Maturity
Maturity
expected future changes in short rate; YC is the total yield spread.
E (-)
10y
25
3. Segmented Markets Theory
Ã¯â€šÂ§ Some investors trade short-term bonds:
Ã¢â‚¬â€œ Short-term interest rates are determined by
supply and demand among these investors.
Ã¯â€šÂ§ Other investors trade long-term bonds:
Ã¢â‚¬â€œ Long-term interest rates are determined by supply
and demand among these investors.
Ã¯â€šÂ§ Each maturity sector is then a Ã¢â‚¬Ëœsegmented
marketÃ¢â‚¬â„¢ and the yield in that market is
determined independently.
26
4. Preferred Habitat Theory
Ã¯â€šÂ§ Borrowers and lenders have strong preferences
for particular maturities, but yields are not
determined independently.
Ã¯â€šÂ§ If expected returns are large enough, investors
will deviate from their preferred maturities.
Ã¯Æ’Ëœ Investors will accept additional risk in return for
27
Fixed Income Securities:
Interest Rate Management
Professor Aino Levonmaa
1
Outline
Ã¯â€šÂ§ Interest rate sensitivity of a bond price
Ã¢â‚¬â€œ What happens to prices of fixed income
securities as interest rates change?
Ã¯â€šÂ§ Duration
Ã¯â€šÂ§ Convexity
Ã¯â€šÂ§ Immunization: duration matching
2
Interest-Rate Sensitivity
Ã¢â‚¬Â¢ First order effect: Bond prices and interest
rates are negatively related.
Ã¢â‚¬Â¢ Maturity matters: Prices of long-term
bonds are more sensitive to interest-rate
changes than short-term bonds.
Ã¢â‚¬Â¢ Convexity: An increase in a bondÃ¢â‚¬â„¢s YTM
results in a smaller price decline than the
price gain associated with a decrease of
equal magnitude in the YTM.
3
Example: Interest-Rate Sensitivity
Price
90
89
88
P = 100 / (1+y)^T
5%
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
6%
7%
YTM
Price
1-Year ZCB
Price
2-Year ZCB
5%
95.24
90.70
6%
94.34
89.00
7%
93.46
87.34
YTM
Negative convex relation YTM and price:
Ã¢â‚¬â€œ Price change 2-year ZCB 6 Ã¢â€ â€™ 5%:
Ã¢â‚¬â€œ Price change 2-year ZCB 6 Ã¢â€ â€™ 7%:
Maturity Matters:
Ã¢â‚¬â€œ % change 1y 6 Ã¢â€ â€™ 7%:
Ã¢â‚¬â€œ % change 2y 6 Ã¢â€ â€™ 7%:
90.70-89.0 = 1.70
87.34-89.0 = -1.66
(93.46-94.34)/94.34 = -.93%
(87.34-89.00)/89.00 = -1.87%
4
Price Sensitivity of BondsÃ¢â‚¬Â¦
Ã¢â‚¬Â¢ Ã¢â‚¬Â¦ as interest rates change
Ã¢â‚¬Â¢ Maturity clearly plays a role
Ã¢â‚¬Â¢ This class introduce the precise concept that
matters: DURATION
5
Duration: The Concept
Ã¢â‚¬Â¢ Duration (or Macaulay duration)
Ã¢â‚¬â€œ Is defined as Ã¢â‚¬Å“the averageÃ¢â‚¬Â time you have to wait for
Ã¢â‚¬â€œ Is used to determine how sensitive the price of a bond
is to changes in the yield
(maturity matters Ã¢â€ â€™ duration matters)
Ã¢â‚¬Â¢ How long on average do you have to wait for your
payments in the following two situations?
-P
A:
B:
1000
t=0
t=1
t=2
t=3
t=4
-P
80
80
80
1080
t=0
t=1
t=2
t=3
t=4
6
Duration: The Derivation
The price sensitivity is linked to Ã¢â‚¬Å“the average timeÃ¢â‚¬Â you
have to wait for your payments, provided the weights
are the contributions to price of the bond:
CF1
CF2
CFT
PÃ¯â‚¬Â½
Ã¯â‚¬Â«
Ã¯â‚¬Â« … Ã¯â‚¬Â«
1
2
dP
CF1
CF2
CFT
Ã¯â‚¬Â½ Ã¯â‚¬Â­1
Ã¯â‚¬Â­2
Ã¯â‚¬Â­ … Ã¯â‚¬Â­ T
2
3
dy
dP 1 Ã¯â‚¬Â« y
CF1
CF2
CFT
Ã¯â‚¬Â­
Ã¯â‚¬Â½1
Ã¯â‚¬Â«2
Ã¯â‚¬Â« … Ã¯â‚¬Â« T
1
2
T
dy P
7
Duration: Summary
Ã¢â‚¬Â¢ The duration (D) of a bond is defined as minus the
elasticity of its price (P) with respect to (1 plus) its YTM (y):
dP 1 Ã¯â‚¬Â« y T
Cashflow (t )
DÃ¯â‚¬Â½Ã¯â‚¬Â­
Ã¯â‚¬Â½ Ã¯Æ’Â¥ wt t , where wt Ã¯â‚¬Â½
t
dy P
(
1
Ã¯â‚¬Â«
y
)
P
1
Ã¢â‚¬Â¢ We see that the duration D is equal to the average of the
cash-flow times, weighted by their contribution to the
present value of the bond!
Ã¢â‚¬Â¢ The relative price-response to a yield change is therefore:
Ã¯Ââ€žP
D
Ã¯Ââ‚¬Ã¯â‚¬Â­
Ã¯Ââ€žy
P
1
Ã¯â‚¬Â«
y
Ã¯ÂÂ»
Modified
Duration
8
Duration: Example
Ã¢â‚¬Â¢ What is the duration of a 4-year coupon bond
with a face value of \$1,000, a coupon rate of
8% and a YTM of 10%?
80
80
80
1080
PÃ¯â‚¬Â½
Ã¯â‚¬Â«
Ã¯â‚¬Â«
Ã¯â‚¬Â«
Ã¯â‚¬Â½ 936.6
1
2
3
4
80
80
w1 Ã¯â‚¬Â½
Ã¯â‚¬Â½
0
.
0777
,
w
2 Ã¯â‚¬Â½
Ã¯â‚¬Â½ 0.0706
1
2
936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©
936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©
80
1080
w3 Ã¯â‚¬Â½
Ã¯â‚¬Â½ 0.0642 , w4 Ã¯â‚¬Â½
Ã¯â‚¬Â½ 0.7876
3
4
936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©
936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©
D Ã¯â‚¬Â½ w1 Ã¯â€šÂ´1 Ã¯â‚¬Â« w2 Ã¯â€šÂ´ 2 Ã¯â‚¬Â« w3 Ã¯â€šÂ´ 3 Ã¯â‚¬Â« w4 Ã¯â€šÂ´ 4 Ã¯â‚¬Â½ 3.56
9
Duration: Example
Ã¢â‚¬Â¢ If the YTM changes from y = 10% to 10.1%,
what would be the (relative) change in price?
Ã¯Ââ€žP
D
3.56
Ã¯Ââ‚¬Ã¯â‚¬Â­
Ã¯Ââ€žy Ã¯â‚¬Â½ Ã¯â‚¬Â­
0.001 Ã¯â‚¬Â½ Ã¯â‚¬Â­0.32%
P
1
Ã¯â‚¬Â«y
1
Ã¯â‚¬Â«Ã¯â‚¬Â²
.10
Ã¯â‚¬Â±
Ã¯â‚¬Â³
Ã¯ÂÂ»
Modified
Duration
Modified
Duration
Ã¢â‚¬Â¢ Does the duration formula imply that longterm bonds are more risky (bigger price
changes) than short-term bonds?
Yes for same change; BUT: Long-term interest rates less
volatile
10
Duration: Useful Facts
Ã¢â‚¬Â¢ The duration of a portfolio is the weighted average of the
durations of the constituents: D p Ã¯â‚¬Â½ Ã¯Æ’Â¥ Ã¯ÂÂ· i Di
i
Ã¢â‚¬Â¢ What is the duration of zero-coupon bond?
Ã¢â‚¬Â¢ What must be true for the duration of a coupon bond?
Ã¢â‚¬Â¢ What happens to the duration of a coupon bond if (all else
equal) the coupon rate increases?
Ã¢â‚¬Â¢ What happens to the duration of the bond if (all else
equal) the YTM increases?
Ã¢â‚¬Â¢ The duration of a perpetuity is: (1+y)/y
11
Going back to our previous 8% coupon bond.
When YTM increases to 11%
P becomes 906.927 (decreased)
W1 = 0.0797 (Ã¢â€ â€˜)
W2 = 0.0716 (Ã¢â€ â€˜)
W3 = 0.0645 (Ã¢â€ â€˜)
W4 = 0.07844 (Ã¢â€ â€œ)
Ã¯Æ’Â¨ New D = 3.554 (Ã¢â€ â€œ)
12
Portfolio Duration
Ã¢â‚¬Â¢ How to compute the duration of a bond portfolio?
Ã¢â‚¬Â¢ -> Simply the weighted average duration of the bonds in the
portfolio.
Ã¢â‚¬Â¢ The weight for each bond duration is calculated as the market
value of the bond divided by the total market value of the bond
Bond Market Value (\$m) Weight Mod.
portfolio.
Duration
A
10
0.1
4
B
40
0.4
7
C
30
0.3
6
D
20
0.2
2
Ã¢â‚¬Â¢ Then the portfolio duration is given by 0.1*4+0.4*7+0.3*6+0.2*2 =
5.4
Ã¢â‚¬Â¢ If the yields affecting all four bonds change by 100bps, the
portfolioÃ¢â‚¬â„¢s value will change by approximately 5.4%.
Duration: How good is the
approximation? (y=8%, T=30)
14
Limitations of Duration
Ã¢â‚¬Â¢ Duration is an approximation for the change in bond prices due to
the change in yields.
Ã¢â‚¬Â¢ Another key limitation: duration is based on the bond pricing
equation:
Ã¢â‚¬Â¢ It assumes that all cash flows are discounted at the same interest
rate, then duration assumes that yields for all maturities change
by an equal amount.
Ã¢â‚¬â€œ This means we assume a parallel shift in the yield curve.
Ã¢â‚¬Â¢ Improve the approximation by also using convexity
measure
Convexity
Ã¢â‚¬Â¢ The sensitivity of price with respect to yield is
approximated by a linear function when using duration.
Ã¢â‚¬Â¢ The relation is really non-linear, specifically, itÃ¢â‚¬â„¢s convex.
Ã¢â‚¬Â¢ The convexity of a bond is the curvature of its price-yield
relationship:
d 2P 1 T
(t 2 Ã¯â‚¬Â« t )
Cashflow (t )
Convexity Ã¯â‚¬Â½ 2 Ã¯â‚¬Â½ Ã¯Æ’Â¥ wt
,
where
w
Ã¯â‚¬Â½
t
dy P 1
(1 Ã¯â‚¬Â« y)2
(1 Ã¯â‚¬Â« y )t P
Ã¢â‚¬Â¢ The relative price response to a yield change can be
better approximated using convexity:
Ã¯Ââ€žP
D
1
Ã¯Ââ‚¬Ã¯â‚¬Â­
Ã¯Ââ€žy Ã¯â‚¬Â« Convexity Ã¯â€šÂ´ (Ã¯Ââ€žy ) 2
P
1Ã¯â‚¬Â« y
2
16
DIY: Example Convexity
Ã¢â‚¬Â¢ Same 4-year coupon bond with F = \$1,000,
coupon rate of 8% and YTM of 10%?
Ã¢â‚¬Â¢ Recall earlier calculation: w1=.0777,
w2=.0706, w3=.0642, w4=.7876, D = 3.5617.
Ã¢â‚¬Â¢ Calculate convexity = 14.133 =
=
1+12
Ã°Ââ€˜Â¤
1+0.1 2 1
+
2+22
3+32
4+42
Ã°Ââ€˜Â¤ +
Ã°Ââ€˜Â¤ +
Ã°Ââ€˜Â¤
1+0.1 2 2 1+0.1 2 3 1+0.1 2 4
Ã¢â‚¬Â¢ What happens to the bond price if YTM
increases to 11%?
17
Convexity: Summary
Ã¢â‚¬Â¢ When yields decline, the price increase in the
bond is underestimated by the simple duration
formula. A convexity term corrects the problem.
Ã¢â‚¬Â¢ The more convex a bond, the greater the expected
price increase for a given decrease in yield and the
smaller the expected price decrease.
Ã¢â‚¬Â¢ If interest rates are volatile, this is an attractive
asymmetry.
18
Interest-Rate Management
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Investors and financial institutions are subject to interest-rate
risk, for instance:
Ã¢â‚¬â€œ Homeowner: mortgage payments (ARM)
Ã¢â‚¬â€œ Bank: short-term deposits and long-term loans
Ã¢â‚¬â€œ Pension fund: owns bonds and must pay retirees
A change in the interest rate results in:
1. Price risk
2. Re-investment risk
Want to construct a portfolio which is insensitive to interestrate changes.
Ã¯Æ’Ëœ We want to Ã¢â‚¬Å“immunizeÃ¢â‚¬Â the portfolio of assets against ÃŽâ€y
19
Duration Matching: Immunization
Ã¢â‚¬Â¢ Duration matching means to make the duration
of assets and liabilities equal.
Ã¢â‚¬Â¢ Then, the sensitivity to interest-rate changes is:
D Assets Assets
D Liabilitie s Liabilitie s
Ã¯Ââ€žP Ã¯Ââ‚¬
P
Ã¯Ââ€žy Ã¯â‚¬Â­
P
Ã¯Ââ€žy Ã¯â‚¬Â½ 0
1Ã¯â‚¬Â« y
1Ã¯â‚¬Â« y
Ã¢â‚¬Â¢ Interest rate changes makes the values of assets
and liabilities change by the same amount: The
portfolio is immunized.
20
Markets: GMÃ¢â‚¬â„¢s Pension Fund
Ã¢â‚¬Â¢ General MotorsÃ¢â‚¬â„¢ pension fund has
Ã¢â‚¬â€œ Liabilities with duration of about 15 years
Ã¢â‚¬â€œ Assets (bonds) with duration of about 5 years
Ã¢â‚¬â€œ Problem: Duration mismatch!
Ã¢â‚¬Â¢ Price risk: When the interest rate falls:
Ã¢â‚¬â€œ The value of the bonds increases, but
Ã¢â‚¬â€œ The present value of the liabilities increases more!
Ã¢â‚¬Â¢ Reinvestment risk:
Ã¢â‚¬â€œ At the new interest rate, the assets could not be
reinvested to make the future payments.
21
Solution: Immunization
Ã¢â‚¬Â¢ Suppose:
Ã¢â‚¬â€œ GMÃ¢â‚¬â„¢s pension fund must pay \$100M in 15 years
Ã¢â‚¬Â¢ Assume no payment before year 15. Liability duration?
Ã¢â‚¬â€œ The current market interest rate is 6% at all maturities
Ã¢â‚¬Â¢ Assume the fund can only invest in 1-year and 30-year zero-coupon
bonds
Ã¢â‚¬â€œ How many securities should the fund buy to be immunized?
Ã¢â‚¬â€œ What are the prices of these bonds (face value is \$100)?
Ã¢â‚¬Â¢ Right after the fund buys the bonds, the interest rate rises to 7%
Ã¢â‚¬â€œ What is the new value of the fundÃ¢â‚¬â„¢s bond position?
Ã¢â‚¬â€œ What is the new present value of the fundÃ¢â‚¬â„¢s liabilities?
22
Solution: Immunization
1
2
Mat.
3
4
5
6
7
8
9
10
Portfolio
weight
PV
holding
@ y=6%
Bond
price @
y=6%
No. of
bonds
(mill.)
FV
15yrs
@
y=6%
Bond
price
@
y=7%
PV
bonds
@
y=7%
FV
15yrs
@
y=7%
41.73
41.73
1.00
100
36.25
36.25
100
L
15
A
1
0.517
21.58
94.34
0.23
51.72
93.46
21.38
58.99
A
30
0.483
20.14
17.41
1.16
48.27
13.14
15.20
41.93
Total A
1.000
41.73
100
23
36.58 100.90
Problems with Immunization
Ã¢â‚¬Â¢ Our example: immunization against once-andfor-all change
Ã¢â‚¬Â¢ Reality: rates change frequently;
immunization requires rebalancing.
Ã¢â‚¬Â¢ It is an approximation that assumes:
Ã¢â‚¬â€œ Only risk of changes in the level of interest rates;
not in the slope of the term-structure.
Ã¢â‚¬â€œ Small interest rate changes Ã¢â‚¬â€œ improve duration
matching by also matching convexity.
24
Concepts to Know
Ã¢â‚¬Â¢ Duration
Ã¢â‚¬Â¢ Convexity
Ã¢â‚¬Â¢ Immunization
25
Options:
Basics and Strategies
Professor Aino Levonmaa
1
Overview
Ã¢â‚¬Â¢ Option basics
Ã¢â‚¬â€œ Option valuation on expiration date
Ã¢â‚¬â€œ Option strategies
Ã¢â‚¬Â¢ Next two classes: Option valuation prior to
expiration date
Ã¢â‚¬â€œ No-arbitrage bounds on option prices
Ã¢â‚¬â€œ Put Call Parity
Ã¢â‚¬â€œ Black-Scholes Pricing model
2
Option Basics
TODAY:
Ã¢â‚¬Â¢ Derivatives
Ã¢â‚¬Â¢ Option characteristics
Ã¢â‚¬Â¢ Value of options at expiration
Ã¢â‚¬Â¢ Option strategies
3
Derivatives
Ã¢â‚¬Â¢ A derivative is a security with a payoff that depends on
the price of another security
Ã¢â‚¬Â¢ The other security is called the underlying (security)
Ã¢â‚¬Â¢ Examples: options, futures, swaps.
Ã¢â‚¬Â¢ Derivatives are used for
Ã¢â‚¬â€œ Risk management, hedging
Ã¢â‚¬â€œ Executive compensation
Ã¢â‚¬â€œ Portfolio insurance
Ã¢â‚¬â€œ Speculation
4
Options Characteristics
Ã¢â‚¬Â¢ Option types
Ã¢â‚¬â€œ Call option: right to Buy underlying at
Ã¢â‚¬â€œ Put option: right to Sell underlying at
Ã¢â‚¬Â¢ Exercise price / strike price (X)
Ã¢â‚¬Â¢ Can exercise at or before expiration (T)
Ã¢â‚¬â€œ European: exercise only at expiration
Ã¢â‚¬â€œ American: exercise any time at or before expiration
Ã¢â‚¬Â¢ Pay option price or premium
Ã¢â‚¬Â¢ Call option: In-the-money, out-of-the-money, at-the-money
S0 > X
Long stock
ST
ST
Long put
X-ST
0
Net
X
ST
3. Covered Call
Ã¢â‚¬Â¢ Suppose that you own a share of Microsoft
for S0=\$70. You think that at-the-money call
options trading at \$10 seem excessively
expensive, and you want to profit from this.
ST
40
Payoff Stock:
40
Payoff Short Call: 0
Payoff Total:
40
Profit Total:
50
50
50
0
50
60
60
60
0
60
70
70
70
0
70
80
80
80
-10
70
80
90
90
-20
70
80
100
100
-30
70
80
Ã¢â‚¬Â¢ Compare: Ã¢â‚¬Å“naked option writingÃ¢â‚¬Â
18
Covered Call
Stock payoff
Combined profit
on combo
Combined payoff
writing call
Short Call payoff
ST > X
ST Ã¢â€°Â¤ X
Long stock
ST
ST
Short call
-(ST-X)
0
Net
X
ST
19