# Present Value Examples

2. Your life span is two periods. You are endowed with $500 today. You have a production technology which can transform an investment of $I today into $40

√ I next year. Also,

you can borrow at 331 3 % per annum and lend at 25% per annum.

(a) What is the maximum feasible consumption today?

(b) What is the maximum feasible consumption next year?

(c) What is the optimal consumption if U(C0, C1) = min(C0, C1)?

(d) What is the “Fisher Separation Theorem?” Is it valid under the assumption of different borrowing and lending rates?

3. Present Value Examples (a) The present value of 200 paid at the end of n years, plus the present value of 100 paid at the end of 2n years is 200. Determine the annual effective rate of interest (express the answer as a function of n).

(b) Someone offers you a security which pays $n at the end of the nth year until forever (i.e., it pays $1 at the end of the first year, $2 at the end of the second year, and so on). If the annually compounded interest rate is 10% per year, what is the fair price of such security? (Hint: Let

S = 1

1 + r +

2

(1 + r)2 +

3

(1 + r)3 + · · · (1)

Multiply (1) by 1 + r and then subtract one from the other.)

(c) (Rule of 69) People in the banks have a quick way of finding out how long it takes to double your money. The trick is to divide 69 by the continuously compounded interest rate (in percentage). For example, if the continuously compounded interest rate is 10% per year, then you know it takes 69/10 = 6.9 years to double your money. Why 69? Can you figure out a similar rule to find out the number of years it takes to triple your money?

4. You just signed a 30-year lease agreement for a business property. The monthly rent for the first year is $1,000/month, with the first month’s rent due today. Starting from the second year onward, the monthly rent will be increased by 5%/year (i.e., the monthly rent for the second year will be $1,050, the monthly rent for the third year will be $1,000(1.05)2 = $1,102.5, and so on). Assuming the annually compounded interest rate is 12%/year, what is the present value of the 360 rental payments.

5. You obtain a $200,000 mortgage loan from TD bank to buy a house. The mortgage has a 5-year fixed rate of 5%/year (using Canadian mortgage convention), and the amortization period of the mortgage is 20 years.

(a) What is the monthly mortgage payment?

(b) How much do you owe the bank after the 60th monthly payment?

2

(c) For the 24th monthly payment, how much of it is for interest, and how much of it is for principal repayment?

(d) What is the present value of the interest portion of the first 60 payments?

3

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