Suppose that the Government passes a new tax on the investment proceeds from futures contracts. The tax will equal 40% of the profit or loss earned on the contract. The tax is fully symmetric, meaning that if the investor suffers losses on the futures contract, those losses reduce the investor’s taxes by 40% of the losses.

a. Derive the spot-futures parity condition in the presence of the new tax. Lay out the risk-free arbitrage (i.e. the cash and carry) strategy that would take advantage of any departure of the futures price from the condition you’ve derived.

b. What is the market value of the government’s tax claim on futures profits? Explain briefly. [Hint: absolutely no math is required to answer this section.]

You run a regression of the monthly returns of a stock portfolio, P, on the returns of the S&P 500 and obtain the following results.

rP = .01 + .8 rS&P + e

Standard error of monthly residual = .06 (i.e. 6%)
Standard deviation of S&P 500 monthly return = .05 (i.e. 5%)

a. What is the variance and standard deviation of the monthly return on the portfolio?

b. What would be the standard deviation and variance of a position in the portfolio if you fully hedged out market risk?

c. Using your answers to parts (a) and (b), what must be the R-square of the regression? [Hint: what fraction of risk has been eliminated?]

d. Suppose the S&P futures contract has a maturity of 3 months, and you wish to hedge for only one month. If the risk-free rate is 1% per month, and the S&P index currently is at 980.30, how many contracts would you need to sell to hedge a $10 million position in the portfolio? The contract multiplier is $250.

Suppose that you observe the following quotes in the FRA market. Each contract is based on LIBOR, with payment once a year.

Period FRA Notional Principal
0 - 1 6% 100
1 - 2 7% 200
2 - 3 8% 300

Find the swap fixed rate (SFR) on a swap with notional principal that varies in each year according to the schedule given in the last column.

A bond investor currently can buy 10-year maturity fixed-rate bonds at a yield to maturity of 7.2 %. She then reads that Endrun Corp. has issued 10-year superfloaters, and considers using these bonds together with some financial engineering to create a synthetic fixed-rate bond. The superfloaters pay annual coupons, with a coupon rate of 2L - 8 % (where L is the LIBOR rate) as long as LIBOR is above 4 % (the floor rate). Otherwise, coupons are suspended for that year. The superfloater sells for 97 % of the par value.

Swap and floor prices are as follows:

10-year swaps, annual pay : Bid rate = 7 %, Ask rate = 7.1 %
10-year floors, annual pay (floor rate = 4 %) :
{Bid price = 3 % of notional principal; Ask price = 3.2 % of notional principal}

Note that a Floor is analogous to a put option on LIBOR. That is, the Floor buyer receives max(floor rate – LIBOR, 0). Since it is an option-like security, there is an initial premium to pay when an investors buys it and the bid and ask prices are listed as above.

a. Devise a portfolio to create a synthetic fixed-rate bond using the superfloater, swap, and floor securities.

b. What is the effective yield to maturity on that synthetic bond? Is the synthetic bond a better buy than the fixed-rate bond?

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