Question

[Questions 20-21] An investor owns a stock. Daily change in stock price, ∆S, has the standard deviation of 13. To hedge risks of the stock price, the investor considers cross-hedging using one of the following futures contracts. The following table shows each futures contract’s standard deviation σF of futures price change, ∆F, and the correlation coefficient ρ between ∆S and ∆F. Futures contract σF ρ A 22 0.7 B 20 0.9 C 15 0.6 D 10 0.8 If the investor shorts h units of futures, the change in the portfolio value is ∆S −h∆F. 20. Which futures contract results in the smallest variance, Var (∆S − h∆F)? (Assume that for each futures, we use respective minimum-variance hedge ratio). (a) contract A (b) contract B (c) contract C (d) contract D 21. What is the minimum variance if we use the futures contract found in question 20? (a) 5.67 (b) 32.11 (c) 60.84 (d) 86.19

[Questions 22-23] The current term-structure of spot rates is as follows (with continuous compounding): Maturity (years) Zero-rate(%) 1 3.0 2 4.5 3 5.5 22. What is the implied forward rate r0(2, 3)? (a) 6.00% (b) 6.75% (c) 7.50% (d) 7.53% 23. A bank offers a special bond A through which investors can borrow (lend) $100 in year 2 and repay (receive) $100×e 0.07 in year 3. Is there an arbitrage? If so, what is the arbitrage’s net cash flow in year 0? (Consider an arbitrage strategy where we use one unit of bond A and the net cash flows are zero from years 1 through 3) (a) 0.392 (b) 0.456 (c) 0.499 (d) 0.538

Answer 1

SORRY ONLY ONE ALLOWED

Minimum variance=standard deviation of spot^2+h^2*standard deviation of futures^2-2*h*standard deviation of spot*standard deviation of futures*correlation between spot and futures

h=correlation*standard deviation of spot/standard deviation of futures

Minimum variance=standard deviation of spot^2-(correlation*standard deviation of spot)^2=standard deviation of spot^2*(1-correlation^2)

A=13^2*(1-0.7^2)=86.1900

B=13^2*(1-0.9^2)=32.1100

C=13^2*(1-0.6^2)=108.1600

D=13^2*(1-0.8^2)=60.8400

1.
Contract B

2.
32.11