1. 5 MATHEMATICS OF FINANCE
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5.1
Compound Interest
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3. Continuous Compounding of Interest

4. Continuous Compounding of Interest
What happens to the accumulated amount over a fixed period of time if the interest is computed more and more frequently?
Let us consider the compound interest formula:
A = P
As m gets larger and larger, A approaches P(e)rt = Pert where e is an irrational number approximately equal to …
(4)

5. Continuous Compounding of Interest
In this situation, we say that interest is compounded continuously. Let’s summarize this important result.

6. Example 5
Find the accumulated amount after 3 years if $1000 is invested at 8% per year compounded (a) daily (assume a 365-day year) and (b) continuously.
Solution:
a. Use Formula (3) with P = 1000, r = 0.08, m = 365, and t = 3.
Thus, i = and n = (365)(3) = 1095, so
A = 1000

7. Example 5 – Solution  1271.22 or $1271.22.
cont’d 
or $
b. Here, we use Formula (5) with P = 1000, r = 0.08, and t = 3, obtaining
A = 1000e(0.08)(3) 
or $

8. Effective Rate of Interest

9. Effective Rate of Interest
The effective rate is the simple interest rate that would produce the same accumulated amount in 1 year as the nominal rate compounded m times a year.
The effective rate is also called the annual percentage yield.
To derive a relationship between the nominal interest rate, r per year compounded m times, and its corresponding effective rate, R per year, let’s assume an initial investment
of P dollars.

10. Effective Rate of Interest
Then the accumulated amount after 1 year at a simple interest rate of R per year is
A = P(1 + R)
Also, the accumulated amount after 1 year at an interest rate of r per year compounded m times a year is
A = P(1 + 𝑟 𝑚 )mt
= P

11. Effective Rate of Interest
Equating the two expressions gives
P(1 + R) = P
1 + R =
If we solve the preceding equation for R, we obtain the following formula for computing the effective rate of interest.
Divide both sides by P.

12. Effective Rate of Interest

13. Example 6
Find the effective rate of interest corresponding to a nominal rate of 8% per year compounded (a) annually, (b) semiannually, (c) quarterly, (d) monthly, and (e) daily.
Solution:
a. The effective rate of interest corresponding to a nominal rate of 8% per year compounded annually is, of course, given by 8% per year.
This result is also confirmed by using Formula (6) with r = 0.08 and m = 1. Thus,
reff = ( ) – 1 = 0.08

14. Example 6 – Solution
cont’d
b. Let r = 0.08 and m = 2. Then Formula (6) yields
reff = – 1
= (1.04)2 – 1 =
so the effective rate is 8.16% per year.

15. Example 6 – Solution
cont’d
c. Let r = 0.08 and m = 4. Then Formula (6) yields
reff = – 1
= (1.02)4 – 1 
so the corresponding effective rate in this case is 8.243% per year.

16. Example 6 – Solution
cont’d
d. Let r = 0.08 and m = 12. Then Formula (6) yields
reff = – 1 
so the corresponding effective rate in this case is 8.3% per year.

17. Example 6 – Solution
cont’d
e. Let r = 0.08 and m = 365. Then Formula (6) yields
reff = – 1 
so the corresponding effective rate in this case is 8.328% per year.

18. Effective Rate of Interest
If the effective rate of interest reff is known, then the accumulated amount after t years on an investment of P dollars may be more readily computed by using the formula
A = P(1 + reff)t

19. Practice
p. 278 #22, 24, 30, 51