1. Mathematics of Finance
It’s all about the $$$ in Sec. 3.6a

2. Interest Compounded Annually
Suppose a principal of P dollars is invested in an account bearing
an interest rate r expressed in decimal form and calculated at the
end of each year. The value of the investment then follows the
growth pattern shown below:
Time in years
Amount in the account 1 2 n

3. Interest Compounded Annually
Interest computed in this way is called compound interest,
because interest is eventually earned on the interest itself !!!
If a principal P is invested at a fixed annual interest rate r,
calculated at the end of each year, then the value of the
investment after n years is
where r is expressed as a decimal.

4. Interest Compounded k Times per Year
What happens when the interest rate r is compounded multiple
times per year??? (say, “k” times per year…)
Then r/k is the interest rate per compounding period, and
kt is the number of compounding periods.
The amount A in the account after t years is
Now, what happens when k gets really, really, really, really,
really, really, really, enormously, gigantically, really big???

5. Interest Compounded Continuously
Recall that
When k approaches infinity, we say that the interest is being
compounded continuously. The amount A after t years in
such a situation is

6. Guided Practice
Suppose you invest $500 at 7% interest compounded annually.
Find the value of your investment 10 years later.
with P = 500, r = 0.07, and n = 10

7. Guided Practice
Suppose you now invest $500 at 9% annual interest which is
compounded monthly (12 times a year). What is the value of
your investment 5 years later?
with P = 500, r = 0.09,
k = 12, and t = 5

8. Guided Practice
Now you’re investing $100 at 8% annual interest compounded
continuously. Find the value of your investment at the end of
each of the years 1, 2,…, 7.
with P = 100, r = 0.08,
and t = 1,2,…,7
After 1 year:
To find the other values, let’s use a table!!!
Let

9. It will take 20 years, 8 months
Guided Practice
Determine how much time is required for an investment to
quadruple in value if interest is earned at the rate of 6.75%,
compounded monthly.
It will take 20 years, 8 months

10. Annual Percentage Yield
With so many methods for compounding interest, how do
we compare different investment plans?
For example, would you prefer an investment earning 8.75%
annual interest compounded quarterly or one earning 8.7%
compounded monthly?
We use…
Annual Percentage Yield (APY) – the percentage rate that,
compounded annually, would yield the same return as the
given interest rate with the given compounding period.

11. Computing APY Uma invests $2000 with Crab Key Bank at 5.15% annual
interest compounded quarterly. What is the equivalent APY?
Let x = the equivalent APY
Then the value of the investment at the end of 1 year using
this rate is
Thus, equating the two investment values:

12. Computing APY Uma invests $2000 with Crab Key Bank at 5.15% annual
interest compounded quarterly. What is the equivalent APY?

13. Computing APY Uma invests $2000 with Crab Key Bank at 5.15% annual
interest compounded quarterly. What is the equivalent APY?
In other words, Uma’s $2000 invested at 5.15%
compounded quarterly for 1 year earns the same
interest and yields the same value as $2000 invested
elsewhere paying 5.250% interest once at the end
of the year.

14. Comparing APYs
Which investment is more attractive, one that pays 8.75%
compounded quarterly or another that pays 8.7% compounded
monthly?
= the APY for the 8.75% rate
= the APY for the 8.7% rate
The 8.7% rate compounded monthly is more attractive b/c
its APY is higher than that for the 8.75% rate compounded
quarterly…

15. Whiteboard Practice
Judy has $500 to invest at 9% annual interest rate compounded
monthly. How long will it take for her investment to grow to
$3000?
with P = 500, r = 0.09,
k = 12, and A = 3000

16. Whiteboard Practice Now, confirm our answer graphically… years
Judy has $500 to invest at 9% annual interest rate compounded
monthly. How long will it take for her investment to grow to
$3000?
with P = 500, r = 0.09,
k = 12, and A = 3000
Now, confirm our
answer graphically…
years

17. Whiteboard Practice
Stephen has $500 to invest. What annual interest rate,
compounded quarterly (4 times per year) is required to double
his money in 10 years?
with P = 500, k = 4,
t = 10, and A = 1000

18. Whiteboard Practice Now, confirm our answer graphically…
Stephen has $500 to invest. What annual interest rate,
compounded quarterly (4 times per year) is required to double
his money in 10 years?
Now, confirm our
answer graphically…

19. Whiteboard Practice Plan A
Joey has to choose between three investment options. Plan A
has a 5% APR compounded monthly, Plan B gives a 4.7% APR
compounded continuously, and Plan C provides a 5.1% APR
compounded annually. How long does it take for each of the
investment options to double Joey’s money?
Plan A

20. Whiteboard Practice Plan B
Joey has to choose between three investment options. Plan A
has a 5% APR compounded monthly, Plan B gives a 4.7% APR
compounded continuously, and Plan C provides a 5.1% APR
compounded annually. How long does it take for each of the
investment options to double Joey’s money?
Plan B

21. Whiteboard Practice Plan C
Joey has to choose between three investment options. Plan A
has a 5% APR compounded monthly, Plan B gives a 4.7% APR
compounded continuously, and Plan C provides a 5.1% APR
compounded annually. How long does it take for each of the
investment options to double Joey’s money?
Plan C

22. Returning to the “Joey”
Which of Joey’s three plans offers the better APY? Does your
answer agree with the results from our initial calculations for
doubling times?
Plan A
Plan B
Doubling Time = yr.
APY = 5.116%
Doubling Time = yr.
APY = 4.812%
Plan C
Clearly, Plan A is
the best value!!!
Doubling Time = yr.
APY = 5.1%