Math 300
13.1 – The Time Value of Money
Name _________________________________________
Interest: The amount paid to borrow money.
Principal: The amount borrowed (or invested)
Interest rate: The percentage of principal that is paid for borrowing the principal
Simple Interest Formula
If principal P is borrowed at an interest rate of r for t years, then the simple interest I is given by
I = Prt
1. Find the simple interest paid to borrow $5000 for 9 months at 4%.
Future Value: The future value of a loan is the total amount repaid (including interest).
Future Value Formula
If principal P is borrowed at simple interest for t years at an annual interest rate of r, then the future
value of the loan is
A = Principal + Interest = P + Prt = P(1 + rt)
2. Suppose you take out a simple interest loan of $500 to purchase a laptop. If the interest rate is 6%,
and you have to repay it in 8 months, find the future value of the loan.
3. Suppose you are granted an 8-month deferral on the $500 to purchase your laptop. What principal
would you need to invest now at 3% simple interest in order to have $500 at the end of 8 months to
pay for your laptop? (In other words, what is the present value of the deferred $500 payment?)
Compound Interest: Interest paid on principal plus interest.
4. Suppose you borrow $10,000 at 5% annual interest, and will need to pay the loan at the end of 5
years. Compare simple and compound interest.
At the end of 5 years, the ending balance (future value) with simple interest is _________________
Use the table below to help you find the ending balance using annual compounding interest.
Year
Beginning Balance
1
2
3
4
5
$10,000
Interest Earned
I = Prt
Ending Balance
Future Value Formula for Compound Interest
If P dollars are invested at annual interest rate r, compounded n times per year, and the money is left on
deposit for t years, the future value is given by
A  P 1  nr 
5.
nt
Suppose the interest in the previous problem was compounded monthly. Find the ending balance.
6. Suppose a couple wants to save for their grandchild’s college expenses. They estimate that it will
cost $100,000 for the child to attend college when they turn 18 in 10 years. What lump sum,
deposited today at 6% compounded quarterly, will produce the amount needed?
Effective Annual Yield: The percentage that would be paid at the end of one year if the interest were
simple.
7.
What is the effective annual yield of an investment paying 4% annual interest, compounded
quarterly?
Effective Annual Yield Formula
A nominal interest rate of r, compounded n times per year, is equivalent to the following effective
annual yield.
n
 r
Y  1    1
 n
Future Value Formula for Continuous Compounding
If an principal P earns continuously compounded interest at annual rate r for a period of t years, then
the future value of the investment is
A  Pert
Inflation: The periodic increase in the cost of living.
8. Suppose you currently make $50,000 per year. Approximately
what will your salary need to be 20 years from now in order to
maintain the same standard of living if inflation were to continue
at the 2012 rate of 2.1%?
9.
The average cost of tuition and fee at U.S. institutions of higher
learning increased from $1626 for the 1982-83 academic year to
$10,683 for the 2012-13 academic year (Source: National Center
for Education Statistics, U.S. Dept. of Education). Compare these
increases to average inflation over the same period.