Workshop 7 Answers - Useful for course. Will help in passing the exam. PDF

Preview

Summary

Useful for course. Will help in passing the exam. ...

Description

Finance 261 Workshop 7 Answers 1. Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 6%, and the market’s average return was 14%. Performance is measured using an index model regression on excess returns. Index model regression estimates R-square SD of excess returns

Stock A 1%+1.2(rM-rf)

Stock B 2%+0.8(rM-rf)

0.576 21.6%

0.436 24.9%

Stock A 1% 0.4907

Stock B 2% 0.3373

8.833

10.5

Calculate: a. Jensen’s alpha b. Sharpe ratio c. Treynor index Answer: Alpha=regression intercept Sharpe ratio =(r-rf)/� Treynor measure=(r-rf)/β Stock A: E(r-rf)= 1%+1.2(rM-rf)=1+1.2(14-6)=10.6 Sharpe ratio: E(r-rf)/�=10.6/21.6=0.4907 Treynor measure: = E(r-rf)/β=10.6/1.2=8.833

Stock B: E(r-rf)= 2%+0.8(rM-rf)=2+0.8(14-6)=8.4 Sharpe ratio: E(r-rf)/�=8.4/24.9=0.3373 Treynor measure: = E(r-rf)/β=8.4/0.8=10.5

2. Consider the following data for a particular sample period:

Average return Beta Standard deviation Tracking error (nonsystematic risk, �e)

Portfolio P

Market M

35% 1.20 42% 18%

28% 1 30% 0

a) Calculate the following performance measures for portfolio P and the market: Sharpe ratio, Jensen’s alpha, and Treynor index. The T-bill rate during the period was 6%. Answer: Sharpe:

( ´r −´r f ) σ

SRP = (35-6)/42=0.69 SRM = (28-6)/30=0.733

Alpha:

´r −[ ´r f +β ×( r´ M −´r f ) ]

aP=35-[6+1.2*(28-6)]=2.6 aM=0 (by definition)

Treynor:

( ´r −´r f ) β

TIP = (35-6)/1.2 =24.2 TIM = (28-6)/1 = 22

b) Calculate the M2 measure. Answer: To match 30% standard deviation with P and risk-free, the weight on P is 30/42. Hence, the adjusted portfolio is formed by mixing bills and portfolio P with weights 30/42=0.714 in P and 1-0.714=0.286 in bills. The return on this portfolio is (0.714*35%) + (0.286*6%) =26.7%. This is 1.3% less than the market return. M2 measure = -1.3 Notice that the negative M2 coincides with P’s Sharpe ratio lower than that of M.

c) Calculate the information ratio. Answer: Information ratio = Alpha / Residual standard deviation IRP = 2.6 / 18 = 0.1444 IRM = 0 (by definition)

3. Bonds of X Corporation with a par value of $1000 sell for $960, mature in 5 years, and have a 7% annual coupon rate paid semiannually. Calculate the: a) Current yield Answer: Current yield=coupon/price=$70/$960=0.0729

b) Yield to maturity Answer: On a financial calculator, enter: N=10; PV=-960; FV=1000; PMT=35 → I/YR = 3.993% semiannually. YTM = 7.986% (annualized)

c) Realized compound yield for an investor with a 3-year holding period and a reinvestment rate of 6% over the period. At the end of 3 years the 7% coupon bonds with 2 years remaining will sell to yield 7%. Answer: First find the future value of reinvested coupons and principal. There will be 6 payments of $35 each, reinvested semiannually at 3% per period. On a financial calculator enter: PV=0; PTM=35; n=6; i=3%. Compute: FV=226.39 Three years from now, the bond will be selling at the par value of $1000 because the yield to maturity is forecast to equal the coupon rate. Total proceeds in three years will be: $226.39+$1000=$1226.39 Find the rate that makes the FV of the purchase price equal to =$1226.39 $960*(1+RCY)6=$1226.39 => RCY=4.166% (semiannual) or 8.33% p.a.

4. Consider a 5-year semiannual bond with 10% coupon rate that has a present yield to maturity of 12%.

a) What is the current price? Answer: N=10; I/YR=6; FV=1000; PMT=50 → PV = 926.40 b) Would the price change as time goes by when the yield to maturity stays the same? Answer: The bond is selling at a discount so the price will increase as it gets closer to the maturity. c) Suppose that at the end of Year 1, the yield to maturity is 11%. What is the price of the bond? Answer: N=8; I/YR=5.5; FV=1000; PMT=50 → PV = 968.33 d) Comparing the answers in a) and c), how much of the change was anticipated and unanticipated? Answer: If the YTM stayed the same: N=8; I/YR=6; FV=1000; PMT=50 → PV = 937.90 937.90-926.40=11.50 was anticipated. 968.33 – 937.90 = 30.43 was the unanticipated change in price due to the change in YTM. 5.

Find the Macaulay duration of a 6% coupon bond making annual coupon payments if it has 3 years until maturity and has YTM of 6%. Answer: YTM=6%

(1) Time until payment (years)

Cash flow

1 2 3

$60 $60 $1060

(2)

(3) PV of CF (discount rate=6%) $56.6 $53.4 $890

(4) PV of CF/price of the bond

(5) Column (1)*column (4)

0.0566 0.0534 0.8900

0.0566 0.1068 2.6700

Column sums Duration=2.833 years

$1000

1.000

2.8334

6. A 9-year annual bond has a yield to maturity of 10% and duration of 7.194 years. If the market yield changes by 50 basis points, what is the % change in the bond’s price using the approximation with modified duration? Answer: The approximate percentage change in the bond’s price is: -(Macaulay duration)/(1+y)×(change in y) =-(7.194/1.10)*0.005=-0.0327...