FM213 MT2019 Problem Set Solutions
Class 2
4. Annuity with growth:
a. The present value of Mike Polanski’s future salary payments can be written as an annuity
with growth. Employing the fomula that we have on the slides, the PV of this annuity with
growth is;
=
−1−1+
1+ = 40,000
0.08−0.051−1.05
1.08=$760,662.53
b. The present value of all future savings are equal to PV(salary) x 0.05 = $38,033.13. Since
these savings are invested at an interest rate of 8%, at age 60, Mike will have accumulated:
Future value = $38,033.13 x (1.08)30 = $382,714.30
c. During Mike’s retirement, from age 60 to age 80, Mike plans to consume the same amount
every year. Thus, Mike will use the $382,714.30 that he has accumulated to create a 20-year
annuity cashflow stream. The question is asking you to find the annual cashflow in that
annuity. We have;
=
1− 1
1+
So;
382,714.30=
0.081− 1
1.08 ⇒ = $38.980.29
5. This is an annuity of 30x12 monthly payments whose PV equals $500k. Thus the annuity
formula is appropriate.
a. Stated annual rate of 6% = 0.5% per month. Using the same technique as in part (c) of
question 4 we can find the monthly annuity cashflow that delivers a present value for the
annuity stream of $500k
!"#$%=500,000×0.005
(1−1.005())=$2997.75
b. At the end of the first year of payments, you owe the mortgage company an annuity stream
with 348 monthly payments and a monthly cashflow of $2997.75. The present value of this
stream is the value of your outstanding debt to the mortgage company. It is;
= 2997.75
0.005 1− 1
1.005+,=$493,859.94
c. The total annual payment you made in the first year of the mortgage was 2997.75 × 12 =
$35,973. The loan balance has declined by 500,000 - 493,860 = $6,140. Therefore the
remainder of the payments you have made are interest payments;
Interest = 35,973 – 6,140 = $29,833
