MGCR 341: Finance 1
Summer 2010
Vadim di Pietro
Assignment 1: Solutions
Topic: Time value of money
1) Today is July 1, 2010. You just graduated university. You plan to take a year off to travel and then
start work one year from today. Your first monthly salary of $5,000 will be paid on August 1, 2011.
Assume your monthly salary will increase by 1% each month thereafter, until you retire. Suppose that
you plan to retire on July 1, 2041, right after receiving your last pay check on that same day. For each
pay check, you save a fraction of your salary and the rest is used to pay off your bills. You expect to
live for another 40 years after the day you retire. Your goal is to save enough of each pay check such
that in retirement you can afford to purchase each month the same amount of goods that $1,000 can
buy today. Assume that in retirement, your purchases are made each month, with the first purchase on
August 1, 2041, and the last purchase on July 1, 2081.
The inflation rate is 0.5% per month, and the nominal interest rate is 12% APR, with monthly
compounding.
a)
What is the per-month real interest rate (i.e., the rate at which the purchasing power of
a risk free investment grows per month)? (5 points)
The nominal monthly interest rate is given by APR/12 = 12%/12 = 1%.
If nominal dollars grow by 1% per month, but prices increase by 0.5% per month, then the
purchasing power of a risk free investment increases by the real monthly interest rate of
= +
+ -
rr 1 r1 i 1
= + .
+ .
- = .
%
1 0 011 0 005 1 0 498
b)
What is the PV of the amount of money you need in retirement? Solve this in two
ways: first, by using formulas that involve nominal cash flows and nominal interest
rates, and second, by using formulas that involve real cash flows and real interest
rates. (Hint: one way involves a forward starting growing annuity, and the other way
involves a forward starting constant annuity). (20 points)
Nominal Approach:
The first step is to figure out what the nominal cash flows will have to be in retirement. In order
to buy on Aug 1, 2041 what $1000 can buy today, you need to have
( + )
*
+ =
.
1000 1 i 31 12 1 6426 01
on Aug 1, 2041. Each subsequent month, you will need 0.5% more than the previous month in
order to purchase the same real $1000. Thus, the required cash flows are a forward starting
growing annuity, where the first cash flow C is $6426.01, the growth rate is g = i = 0.005, and