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C15.0002 Foundations of Financial Markets

Fall 2010

Sample Final Solutions

 

Multiple choice questions

 

1. Which of the following risk-free securities has the lowest Macaulay duration (assuming the yields on all the securities are identical)?

a. A 5-year, zero-coupon bond

b. A 5-year, annual pay, 6% coupon bond

c. A 5-year, semi-annual pay, 6% coupon bond (6% is the annual coupon rate)

d. A 5-year, annual pay, 4% coupon bond

Holding all else fixed, duration decreases as the coupon increases because more cash flow is received earlier. For the same coupon rate, semi-annual pay bonds have a lower duration than annual pay bonds for the same reason.

 

2. If the earnings retention (plowback) rate is positive and the ROE is less than the required return on the equity (k), then the present value of growth opportunities (PVGO) is

a. Positive

b. Negative

c. Zero

d. Not enough information to tell

For ROE < k, growth has negative value.

 

3. Holding duration constant, as the convexity of a bond increases, the percentage price change for a given decrease in yields

a. Increases

b. Decreases

c. Does not change

d. Not enough information to tell

More convex means a greater deviation from the duration (linear) approximation. For yield decreases price deviations are positive, i.e., the actual price/yield relationship is above the linear approximation.

 

4. For an at-the-money put option, if the stock price increases by $1, holding all else equal, then the put price

a. Decreases by more than $1

b. Decreases by less than $1

c. Increases by less than $1

d. Not enough information to tell

The delta (hedge ratio) of an at-the-money put option is between 0 and -1, with the exact value depending on the parameters.

 

5. 
Under the liquidity preference theory, if the term structure of interest rates (yield curve) is upward sloping, then investors expect short-term rates to

a. Increase in the future

b. Stay the same

c. Decrease in the future

d. Not enough information to tell

Under the LPT, long bonds contain a (positive) risk premium. Thus, an upward sloping yield curve is not necessarily associated with increasing expected short-term rates.

 

6. If the volatility of the stock increases, holding all other factors constant, the value of a portfolio that is long a call and short a put, both with the same exercise price and maturity,

a. Increases

b. Decreases

c. Stays the same

d. Not enough information to tell

Under put-call parity, call minus put equals stock minus bond. Since the values of neither the stock nor the bond depend (directly) on volatility, the increase in the call price must be perfectly offset by the increase in the put price.

 

7. If interest rates are positive, for futures contracts on a non-dividend paying stock,

a. Longer maturity futures have a lower price than shorter maturity futures

b. Longer maturity futures have a higher price than shorter maturity futures

c. Longer maturity futures have the same price as shorter maturity futures

d. Not enough information to tell

The relevant formula is F0=S0(1+r-d)T. For d=0, F increases as T increases.

 

8. If you buy a 10-year, zero-coupon bond with a yield of 4% (annual compounding) and sell it 5 years later at a yield of 5% (annual compounding), your annualized holding period return (HPR) is

a. 4.0%

b. 5.0%

c. 3.0%

d. 6.0%

Since you sell the bond at a higher yield than the original YTM, the sale price must be lower than it would have been had the yield remained the same. Thus, the HPR is lower than the original YTM.

 

9. A stock sells for $50 based on a discount rate of 10% and a perpetual expected growth rate of 6%. If the market suddenly realizes that growth will be 5% in perpetuity instead of 6%, starting immediately, what will be the new stock price (holding the discount rate constant)?

a. 40.38

b. 40.00

c. 39.62

d. Not enough information to tell

P=D0(1+g)/(k-g) so D0=P(k-g)/(1+g)=50(10%-6%)/(1+6%)=1.887. At the new growth rate P=D0(1+g)/(k-g)=1.887(1+5%)/(10%-5%)=39.62

 

10. A bond currently has a price of $1,000, a yield of 5% and a Macaulay duration of 6 years. If the yield changes to 5.10% what will be the new price (based on the duration approximation)?

a. 994.29

b. 942.86

c. 994.00

d. 1,006.00

ΔP/P=[-D/(1+y)]Δy=[-6/(1+5%)]0.1%=-0.57% so the new price is 1,000(1-0.57%)=994.29

 

11. If a stock has a beta of 1, an at-the-money call option (T = 1,  r = 0%) on the stock will have a beta of

a. Less than 1

b. 1

c. Greater than 1

d. Not enough information to tell

The call is a levered position in the stock and thus has magnified risk.

 

12. If the 1-year, risk-free yield is 2% (y1 = 2%), and the 1-year forward rate is 2.5% (f1 = 2.5%), then under the expectations hypothesis what is the expected holding period return over the next year on a 3-year, zero-coupon bond? Assume annual compounding.

a. 2.00%

b. 2.25%

c. 2.50%

d. Not enough information to tell

Under the expectations hypothesis, the expected HPR on all bonds is the same. The HPR on the 1-year bond is 2%, so the HPR on the 3-year bond must also be 2%. (Note: This question uses a different notation than we used in class this year. Instead of f1, it should be f2 (in our notation).)

 

13. If a stock is trading at a forward-looking PE ratio (P0/E1) of 20, has an expected retention (plowback) rate of 50% in perpetuity, and a discount rate of 10%, what is its expected perpetual growth rate?

a. 2.5%

b. 5.0%

c. 7.5%

d. Not enough information to tell

P0/E1=(1-b)/(k-g) so g=k-(1-b)/[ P0/E1]=10%-50%/20=7.5%

 

14. 
Under the liquidity preference theory, if the term structure of interest rates (yield curve) is downward sloping, then investors expect short-term rates to

a. Increase in the future

b. Stay the same

c. Decrease in the future

d. Not enough information to tell

Under the LPT there is a risk premium in long-term bonds. Therefore, the curve would be even more downward sloping absent this risk premium and future spot rates must be expected to decline.

 

15. If a stock has a price of $40 and the expected dividend next year is $2 (D1 = 2), what must be the expected price at the end of the year (P1) to generate an expected holding period return of 10%?

a. 44

b. 42

c. 40

d. Not enough information to tell

HPR=[( P1+D1)/ P0]-1 so P1=(1+HPR) P0- D1=(1+10%)40-2=42

 

16. If a put option is certain to finish in-the-money, its hedge ratio/delta is

a. H=1

b. H=-1

c. H=0

d. -1<H<0

If a put option is certain to finish in-the-money, then it will be exercised and it is equivalent to shorting 1 share of stock.

 

17. To hedge the risk of future increases in jet fuel prices, airlines should

a. Sell futures on jet fuel

b. Buy futures on jet fuel

c. Sell calls on jet fuel

d. Buy puts on jet fuel

Buying futures looks in the price of the commodity the airline is buying.

 

18. A company has pension liabilities with a value of $100 million at an interest rate/yield of 4%. If the value of these liabilities falls to $98 million when the interest rate/yield increases to 4.5%, what is the modified duration of the liabilities (based on the duration approximation)?

a. 2

b. 4

c. 20

d. 40

D*=[-Δp/p]/Δy=[-(98-100)/100]/0.5%=4

 

19. 
If two stocks have the same expected growth rates and retention (plowback) rates in perpetuity, then the stock with the higher PE ratio must have

a. A higher required return

b. A higher ROE

c. A lower required return

d. A lower ROE

Controlling for growth and plowback (thus controlling for ROE), a higher PE implies a lower required return.

 

20. For an out-of-the-money put option, as the volatility of the stock increases (all else equal), the absolute value of the hedge ratio/delta

a. Increases

b. Decreases

c. Stays the same

d. Not enough information to tell

If the put is out-of-the-money, increasing volatility increases the probability of finishing in the money, pushing the hedge ratio towards -1 (increasing in absolute value).

 


Numerical problems

 

21. A stock had earnings over the last year of $2/share (E0 = 2). The earnings retention (plowback) rate was 40%, and it is expected to continue at this rate forever. Due to the financial crisis, ROE (return on equity) over the next year is expected to be 10%. Subsequently (from time 1 onwards), ROE is expected to return to its normal level of 15%. If the appropriate discount rate (required return) is 10%, what is the current value of the stock?

 

This is a 2-stage growth model. Growth rates in the first year and thereafter

g=b(ROE) g1=0.4(10%)=4% g=0.4(15%)=6%

Dividend at time 1

D1=E1(1-b)=E0(1+g1)(1-b)=2(1+4%)(1-0.4)=1.248

Price at time 1 (at which time the Gordon growth formula will hold)

P1=D2/(k-g)= D1(1+g)/(k-g)=1.248(1+6%)/(10%-6%)=33.072
Current value (price)
P0=( P1+ D1)/(1+k)=(1.248+33.072)/(1+10%)=31.20

 

22. The current 1-year, risk-free yield is 2% (y1 =2%). The 1-year forward rate is 3.0% (f1 = 3.0%). What is the price today of a 2-year, annual pay, 4% coupon bond with face amount $1,000? Assume annual compounding.

(Note: This question uses a different notation than we used in class this year. Instead of f1, it should be f2 (in our notation).)
The current 2-year yield
y2=((1+y1)(1+f1))0.5-1=2.499%
Therefore, the price of the bond, which pays 40 at time 1 and 1040 at time 2, is
P=40/(1+y1)+1040/(1+y2)2=40/(1+2%)+1040/(1+2.499%)2=1,029.13

 

23. Assume the risk-free rate is 4% (rf = 4%) and the market risk premium is 4% (E[rM]- rf  = 4%). A stock with a beta of 1.5 and an expected perpetual growth rate of 5% has a current dividend of $1 (D0 = 1). If the CAPM holds and the stock is fairly priced, what is the expected stock price in 1 year (P1)?

The required return on the stock is
k= rf +β(E[rM]- rf)=4%+1.5(4%)=10%
There are now 3 equivalent approaches.


Approach 1:
Current price
P0=D1/(k-g)=(1+g)D0/(k-g)=(1+5%)1/(10%-5%)=21
Expected price (realizing that the price grows at the perpetual growth rate)
P1=(1+g)P0=(1+5%)21=22.05

Approach 2:
Dividend next year
D1= D0(1+g)=1(1.05)=1.05
Price next year
P1=D2/(k-g)=(1+g)D1/(k-g)= (1+5%)1.05/(10%-5%)=22.05


Approach 3:
Current price
P0=D1/(k-g)=(1+g)D0/(k-g)=(1+5%)1/(10%-5%)=21
Dividend next year
D1= D0(1+g)=1(1.05)=1.05
The price next year must generate an expected HPR equal to the required return of 10%
P1= P0(1+k)- D1=21(1+10%)-1.05=22.05

 

24. A 1-year, risk-free, zero-coupon bond with a face amount of $1,000 sells for $970.87. A 2-year, annual pay, 4% coupon, risk-free bond sells at par. Under the expectations hypothesis, what is the expected 1-year rate at time 1?

There are 2 equivalent approaches.

Approach 1:
1-year yield
y1=(1000/970.87)-1=3%
Value of first coupon payment
Vc=40/(1+ y1)= 40/1.03=38.835 OR Vc=40(1000/970.87)=38.835
Therefore, the value of payment of 1040 at maturity (given the bond price of 1000) must be
V=P- Vc =1000-38.835=961.165
The implied 2-year, zero-coupon yield is
y2=(1040/961.165)0.5-1=4.02%
Forward rate
f1=(1+ y2)2/(1+ y1)-1=(1+ 4.02%)2/(1+ 3%)-1=5.05%
Under the expectations hypothesis the expected 1-year rate is the forward rate: 5.05%

Approach 2:
Value of first coupon payment
Vc=40(1000/970.87)=38.835
Value of payment of 1040 at maturity (given bond price of 1000)
V=P- Vc =1000-38.835=961.165
Therefore, the forward rate, based on prices of 1- and 2-year zero-coupon bonds, is
f1=P1/P2-1=0.97087/(961.165/1040)-1=5.05%
Under the expectations hypothesis the expected 1-year rate is the forward rate: 5.05%

25. 
Consider a stock with a current price of $25 that will be worth either $20 or $30 1 year from now. Assume rf = 0% (annual compounding). You have invented a new derivative security called a “square”, whose payoff in 1 year is the square of the stock price, i.e., payoff = S12. What is the current value of this security?

(Note: See Problem Set #6.)
Stock tree


30

25



20

Square tree


900

SQ



400

Hedge ratio
H=(900-400)/(30-20)=50

Borrowing
B=30H-900=30(50)-900=600
Value

SQ=HS-B=50(25)-600=650


 

 1.C 2.b 3.a 4.b 5.d 6.c 7.b 8.c 9.c............................................................

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