Lecture 25 with solutions

# Lecture 25 with solutions - Lecture 25 Outline Prepayment...

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Lecture 25 Outline Prepayment valuation example Lender’s Loan Decisions

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Example 1: Mortgage Preliminaries One year risk free rate, r 0,1 = 5% Next year, the one year risk free, r 1,1 , rate will equal 4% or 6%, with equal probability Assume investors are risk neutral What is the annual payment and interest rate on a 2- yr, fully amortizing, constant payment, fixed rate \$1M mortgage with no default risk, and no prepayment option, with annual payments at t = 1 and t = 2? What is the mortgage interest rate? Assume annual rate with annual compounding
Example 1: Solution Step 1: express the value of the mortgage at t = 1 (after first payment is made) if interest rates go down to 4% as a function of C: à C/1.04 Step 2: express the value of the mortgage at t = 1 (after first payment is made) if interest rates go up to 6% as a function of C: à C/1.06 Step 3: value of mortgage at t = 0 is 1M and is also equal to: C/1.05 + E[value of mortgage at t = 1]/1.05 Note: to discount (expected) cash flows from t = 1 to t = 0, you use r 0,1 1M = C/1.05 + [0.5(C/1.04) + 0.5(C/1.06)]/1.05 à C = 537,781 Alternative approach: find r 0,2 using (local) expectations hypothesis, then use 1M = C/1.05 + C/(1+r 0,2 ) 2 Mortgage interest rate, i, satisfies 1M = C/(1+i) + C(1+i) 2 à i = 5% (approximately)

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Example 2: More Preliminaries For the mortgage considered in Example 1 a) How much of the t = 1 payment is considered interest? b) How much of the t = 1 payment is considered principal? c) What is the loan balance at t = 1 after the t = 1 payment is made?
Example 2: Solutions a) How much of the t = 1 payment is considered interest? (1M)(5%) =50,000 b) How much of the t = 1 payment is considered principal? 537,781 – 50,000 = 487,781 c) What is the loan balance at t = 1 after the t = 1 payment is made? 1M – 487,781 = 512,219 (note: 512,219(1.05) = 537,781 = C) (you may get slightly different results due to rounding) Note the loan balance at any year t is also equal to the remaining promised cash flows discounted back to time t using the mortgage interest rate.

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Example 3: Prepayment Consider the same mortgage (1M present value), but
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