The developer is risk averse with the degree of risk

Question

Question 1 - Collateralized Mortgage Obligations

Consider the following Collateralized Mortgage Obligations (CMOs) with three classes of securities, Tranche A, Tranche B and Tranche Z.  

Assets 10-year, 11% fixed rate Mortgages = $75,000,000 Overcollateralization = $3,000,000
Tranche	Stated maturity	Coupon Rate	Amount Issued
A	4 years	9%	$27,000,000
B	5 years	10%	$15,000,000
C	10 years	11%	$30,000,000
All cash flows are annual

It is expected that the mortgage pool will experience constant prepayment rate of either 0% or 10% per year depending on the mortgage interest rate level.

a. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow after the Issue of the securities increases to 15%.

b. Determine the market value of Tranche A, B, and Z securities if the mortgage interest rate tomorrow after the issue of the securities decreases to 6%.

Question 2: Pre-Sale Market Analysis

Consider the supply decision of MFIN Limited, a residential real estate developer facing future price uncertainty. Let q be the number of residential units to be built. The variable cost C(q) is equal to c/2 * q². The fixed cost of construction is equal to B.

The developer can pre-sell some units (h) in the presale market at price p_f today and sell the rest of the supply quantity (q - h) at the spot market price at the end of the construction period. The presale revenue generated will be held as cash only with zero rate of return.

The future spot price is uncertain and is represented by p~ follows:

P~ =  P + ε

where p is the expected (mean) future spot price and ε is the forecast error which has mean equal to zero and variance equal to al. That is,

E(p~) = p and Var(p~) = σ².

The developer is risk averse with the degree of risk aversion given by λ_s.

(a) Determine the optimal supply quantity q* and the optimal pre-sale quantity h*.

Now consider the demand side of the market with a representative short-term investor facing the same market conditions as MFIN Limited. The investor, like the MFIN Limited, is risk averse with degree of risk aversion given by λ_b. Similarly, the investor determines the optimal pre-sale demand quantity (q_b) by maximizing the certainty-equivalent profit with respect to the expected future spot price.

(b) Determine the optimal demand quantity (q_b*).

(c) What is the market-clearing pre-sale price today?