Question 1: Compare the interest rate risk of a noncallable 10-year "treasury coupon bearing bond with a mortgage-backed pass-through security with prepayments related to the level of Interest rates -lower market interest rates raise the rate of prepayments.

Discuss how the changes in cash flows, arising from prepayments, on a mortgage-backed security affect the duration of such securities.

HINT: consider the coupon effect on duration.

Macaulay Duration Measure:

D_{m = ∂P/∂y (1 + y)/P}  For a coupon - bearing bond :

A more complete approximation to the proportional change in price of a bond with respect to a change in yield to maturity takes into account the convexity of the price-yield relationship for the bond:

dP/P = ∂P/∂y 1/P dy + 1/2 ∂²P/∂y² 1/P dy²

where P = Price, C = coupon, F = Face value, y = Yield to maturity, M = maturity (years), t = time (year), dP is the total change in price, and ∂P/∂y is the partial change in price with respect to a change in yiehito maturity.

The second term, excluding the dy², is the convexity effect.

Quesiton 2: Consider the following bank balance sheet (fixed rates and pure discount securities unless indicated otherwise). Interest rates on liabilities (yuare 4 percent and on assets (yA) are 6 percent.

a. What is the duration of assets and the duration of liabilities?

b. The bank will (benefit)/(be hurt) if all interest rates rise. Define your terms and state clearly your assumptions.

ΔE = -Δy [D_A(1+y_A) - L/A D_L/(1 + Y_L) ]A

Setting ΔE equal zero implies:
D_A/1 + Y_A  = L/A D_L/(1 + y_L)

ΔE = change in the market value of equity.

D_A = duration of assets

D_L = duration of liabilities.

L = market value of liabilities.

A = market value of assets, and

Δy = change in interest rates.

c. What is the duration of assets that would be necessary to immunize the market value of equity from interest rate changes for this bank's portfolio holding the D₁ constant?

Question 3: Data on weekly stock prices for Microsoft Corporation, Exxon Mobil Corporation and the S&P 500 Index were used to compute the following historical volatility and expected return. Using these results, answer the following questions:

The historical returns volatility (standard deviation of returns) at an annualized rate for each stock are: MSFT= 0.182, XOM= 0.120 and S&P500 = 0.108. The expected return for each stock and the index at an annual rate for MSFT, XOM and S&P500 is respectively:
      0.136        0.308     0.105
a. Using the computed β (beta) of 0.997 for MSFT and 0.508 for XOM and expected return for each stock and the S&P 500 index over the past year, draw the Securities Market Line (SML) using a risk-free rate of 5.37 percent based on the 3-month Libor rate at the time.

Do each of the stocks fall on the SML? Analyze their relative position and which stock is the best buy and why.

HINT: Use the average return of the S&P 500 stock index provided above as the expected return for the market.

b. Using the data in above and in a. and assuming the average return on the S&P 500 index is a representation of market expected return and risk, compute the slope of the Capital Market Line (CML) when the risk-free rate is approximated by the Libor rate given in a. From your results, what is the market price of risk?