5 MATHEMATICS OF FINANCE Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 5.1 Compound Interest Copyright © Cengage Learning. All rights reserved.

Continuous Compounding of Interest

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 2.71828… (4)

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

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

Example 5 – Solution  1271.22 or $1271.22. cont’d  1271.22 or $1271.22. b. Here, we use Formula (5) with P = 1000, r = 0.08, and t = 3, obtaining A = 1000e(0.08)(3)  1271.25 or $1271.25.

Effective Rate of Interest

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.

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

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.

Effective Rate of Interest

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) – 1 = 0.08

Example 6 – Solution cont’d b. Let r = 0.08 and m = 2. Then Formula (6) yields reff = – 1 = (1.04)2 – 1 = 0.0816 so the effective rate is 8.16% per year.

Example 6 – Solution cont’d c. Let r = 0.08 and m = 4. Then Formula (6) yields reff = – 1 = (1.02)4 – 1  0.08243 so the corresponding effective rate in this case is 8.243% per year.

Example 6 – Solution cont’d d. Let r = 0.08 and m = 12. Then Formula (6) yields reff = – 1  0.08300 so the corresponding effective rate in this case is 8.3% per year.

Example 6 – Solution cont’d e. Let r = 0.08 and m = 365. Then Formula (6) yields reff = – 1  0.08328 so the corresponding effective rate in this case is 8.328% per year.

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

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