 # 1. Consider the single-factor APT. Assume that X, Y and Z are well-...

## Question

1. Consider the single-factor APT. Assume that X, Y and Z are well-diversified portfolios and the risk-free rate is 8%.

Portfolio, Expected Return, (%), Beta
X 16 1.00
Y 12 0.5
Z 12 0.25

a. In this situation you would conclude that T-bills and portfolios X and Y offer an arbitrage opportunity? Why? If yes, then find an arbitrage strategy

No. Check the ratio of the risk premium to beta for X and Y

b. Now consider T-bills and portfolios X and Z. Do they offer an arbitrage opportunity? Why? If yes, then find an arbitrage strategy
Yes. Check the ratio of the risk premium to beta for X and Z.
Example of arbitrage strategy: Buy portfolio C with weights wz=4, wT-bill=-3 and sell short X

c. Now Consider T-bills and portfolios Y and Z where portfolio Z has portfolio specific risk. Do they offer an arbitrage opportunity? Why? If yes, then find an arbitrage strategy

2. Assume that expectation hypothesis holds and use the following to answer questions a- d:

Year 1-Year Forward Rate
1 5.8%
2 6.4%
3 7.1%
4 7.3%
5 7.4%

a. What should the purchase price of a 2-year zero coupon bond be if it is purchased at the beginning of year 2 and has face value of \$1,000?
\$877.54

b. What would the yield to maturity be on a four-year zero coupon bond purchased today?
6.648%

c. Calculate the price at the beginning of year 1 (today) of a 10% annual coupon bond with face value \$1,000 and 5 years to maturity.
\$1135.32

d. What should be the holding period return of a 9% annual coupon bond with face value \$1000 and five years to maturity if it is purchased at the beginning of year 1 (today) and sold at the beginning of year 2 (in one year), assuming that rates do not change.
5.8%

3. A stock price is currently trading at \$50. Paul Tripp, CFA wants to value a half-a-year option using a one period binomial model. The stock will either increase in value by 20% or fall in value by 20%. The stock does not pay dividends. The annual risk-free effective interest rate is 3%.

a. Calculate the value of a half-a-year European call option on the index with an exercise price of \$35.
\$15.51

b. Calculate the value of a half-a-year European put option on the index with an exercise price of \$35
\$0

c. Confirm that your solutions for the values of the call and the put satisfy put-call parity
Confirmed

4. Consider investor John who can lend at risk-free rate of 2% and borrow at risk-free rate of 4%. He can also trade portfolio P that has the expected return of 12% and the standard deviation of return of 10%. For what range of the coefficient or risk aversion John will neither borrow nor lend?
8≤A≤10

5. The spot price of oil is \$100 per barrel and the cost of storing a barrel of oil for one year is \$5, payable at the end of the year. The risk-free interest rate is 2% per annum. What is an upper bound for the one-year futures price of oil? What is your best strategy in the market if the observed futures price is \$112?
\$107

Today: Short futures, borrow \$100 and buy 1 barrel of oil. In one year: Sell oil for \$112, pay \$102 for the loan and \$2 for storage. The realized arbitrage profit is \$5.

## Solution Preview

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1.

(a) There is an arbitrage opportunity if the risk premium (excess return over the risk-free rate) over the risk (beta) is greater than the risk risk-free rate. This ratio for X is (16%-8%)/1=8%; which means that any additional 1 unit of risk taken incurs a return of 8%, just equal to the risk-free rate and hence there is no opportunity for arbitrage involving X.

On the other hand, for Y the ratio is (12%-8%)/0.5=8% and we have the same case as above. Any portfolio consisting of X and Y will also have this ratio and thus no arbitrage opportunity exists involving these two assets.

(b) For Z, the ratio is given by (12%-8%)/0.25=16%. This means that any additional risk taken gives back a return greater than the risk-free rate of 8%. Thus one such arbitrage opportunity is sell short 3 units of the t-bil and a unit of X and use the proceeds to buy Z....

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