I. Compound Interest A. The Formula

• P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years.

• Think of P as the present value, and F as the future value of the deposit.

tn

n

rPF

1

Notes: Compound Interest and Annual Yield

Number of Times Compounded

Period Interest

Credited

Times

Credited

per year

Annual year 1

Semiannual 6 months 2

Quarterly quarter 4

Monthly month 12

Ex 1: Suppose you invest $5000 in an IRA account that compounds quarterly at 5.5%. How much money will be in an

account after 1 year?

14

14

055.015000

F

tn

n

rPF

1

72.5280$

After 10 years?

104

104

055.015000

F

tn

n

rPF

1

85.8633$

Decrease in value: When something is decreasing in value

(such as a new car) we can use the compound interest

formula. However, we will need to use subtraction instead of

addition.

Ex 2: Your parents bought a car two years ago for $32,000.

They are going to give it to you when you graduate high

school next year. If the value of the car decreases 15%

each year, how much will it be worth by the time they give it

to you?

)3(1

1

15.132000

F = $19652

If you need to solve for time, use the log or ln function. Use either

one, just be consistent.

Ex 3: Lets say we want to know how long it will take $32,000 to grow to $50,000 invested in an account that has 5.2% annual

interest compounded quarterly.

tn

n

rPF

1

t

4

4

052.13200050000

t

4013.15625.1

t

4013.1ln5625.1ln

013.1ln))(4(5625.1ln t

013.1ln))(4(5625.1ln t

)(

013.1ln4

5625.1lnt

)(64.8 t

B. Continuous Compounding

rtPeA

• P = principal amount invested

•r = the interest rate

• t = the number of years interest is being compounded

• A = the compound amount, the balance after t years

Ex 1: Ten thousand dollars is invested at 6.5% interest compounded continuously. When

will the investment be worth $41,787?

Since the interest rate is 6.5%, r = 0.065. Since ten

thousand dollars is being invested, P = 10,000. And

since the investment is to grow to become $41,787,

A = 41,787. We will make the appropriate

substitutions and then solve for t.

rtPeA

te 065.0000,10787,41

Therefore, the $10,000 investment will grow to $41,787, via 6.5% interest compounded

continuously, in 22 years.

te 065.01787.4 Divide by 10,000.

t065.01787.4ln Rewrite the equation in logarithmic form.

t22 Divide by 0.065 and solve for t.

II. Yield

• One may compare investments with different interest rates and different frequencies of compounding by looking at the values of P dollars at the end of one year, and then computing the annual rates that would produce these amounts without compounding.

• Such a rate is called the effective annual yield, annual percentage yield, or simply yield. This is not to be confused with annual percentage rate (which we will discuss later).

A. Calculating Effective Annual Yield

y = (1 + r/n)n – 1

Ex 1: Find the annual yield for an investment that has an annual interest rate of 8.4% compounded monthly.

• ANSWER: y = (1 + .084/12)12 – 1

• y = (1.007)12 – 1 = 0.087310661 = 8.73%

• The yield will usually be greater than the interest rate.

• Note the interest rate is sometimes called the nominal interest rate.

B. Effective Annual Yield for Continuous Compounding

• The effective annual yield, y, for compounding continuously at an annual interest rate of r is:

P

PePy

r

1 rey

OR

Example: • In our previous example with compound interest, when we

compounded quarterly, after one year we had:

4(1)

1

0.0555000 1 $5280.72

4F

• To find the effective annual yield, y, notice that we gained $280.72 on interest after a year compounded quarterly. That interest represents a gain of 5.61% on $5000:

0561.05000

72.280y

Beware: APY vs APRAPR is the annual rate of interest without taking into account the compounding of

interest within that year. Alternatively, APY does take into account the effects of intra-

year compounding. This seemingly subtle difference can have important implications for

investors and borrowers. Here is a look at the formulas for each method:

For example, a credit card company might charge 1% interest each month;

therefore, the APR would equal 12% (1% x 12 months = 12%). This differs from

APY, which takes into account compound interest. The APY for a 1% rate of interest

compounded monthly would be 12.68% [(1 + 0.01)^12 – 1= 12.68%] a year. If you

only carry a balance on your credit card for one month's period you will be charged

the equivalent yearly rate of 12%. However, if you carry that balance for the year,

your effective interest rate becomes 12.68% as a result of compounding each

month. -Investopedia.com