Suggestive Solutions

Suggestive Solutions

Second Midterm Examination

Fall 2000

BA 180 and BA 280

Question 2

(a)  The maximum you should pay for mortgage A or mortgage B would occur when the

NPVA =o and NPVB = o.

Find CAs.t.NPVA = o= r = 7%

By inspection, $70 per year, with a return of $1,000 at the end of the tenth year implies a 7% yield on a $1,000 purchased investment.

Hence, CA = $1,000

Similarly, Find CB s.t.

r = 7%

Using the provided information, CB = $1,421.41

Note: 7% is (by definition) the IRR for each mortgage investment at the respective purchases prices of CA = $1,000 and CB = $1,421.41

(b) 

dA = 7.52

Note: The P.V. of the income is $1,000, using IRR = 7% (i.e., CA = $1,000).

Similarly, dB =

Note: 7% is the IRR for each mortgage investment (from part (a), and the analysis assumes the opportunity cost of money term structure is flat.

Question 2 (continued)

Since dA > dB “Project” B’s cash flow “pay-off” on average sooner than “Project “ A’s; that is, the average time until mortgage A returns capital to the investor is larger than mortgage B.

(c)  The simplest way to evaluate the risk and returns of the two mortgages is to conceptualize by using the FMRR framework.

Where is the reinvestment rate for cash throw-off from the respective mortgages. The reinvestment rate affects the overall return for mortgage B more than mortgage A, and so forth. Hence, the risk of each mortgage investment is affected differently by reinvestment risk versus “internal” mortgage performance return risk, etc. In particular, mortgage B is subject to more reinvestment risk than mortgage A because dA >dB, etc. You, also, might discuss how the risks for each mortgage might be affected by the inter-correlation of the mortgage payments with reinvestment rates.

(d)  Recall that duration measures the interest rate sensitivity of value:

If one were to rearrange terms, then

Question 2 (continued)

For mortgage A:

Question 2 (continued)

For mortgage B:

Note that mortgage A’s value changes by a larger percentage of the original value vis-à-vis mortgage B because dA > dB, etc.

(e)  Prepayments tend to increase (i.e., the mortgage principal is paid back faster) as interest rates decrease. As market interest rates decline relative to the mortgage coupon rate, at some point, prepayments will start to accelerate, as shown in diagram below:

The precise nature of the prepayment speed on mortgages depends upon the coupon rate versus current mortgage market rates, prior pre-payments (“seasoned” or not), socio-economics of household/borrower, and/or the property market, etc. Hence, one might expect that a portfolio of mortgages like mortgage A and mortgage B to prepay at different “speeds” because they appear to have different coupon rates. Also, the effective durations of each mortgage type (A and B) would be expected to decline as interest rates decline, etc. because prepayments cause the structure of all cash flows to occur sooner, etc.