# Lecture on Annuity Calculations

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### Review of Compound Interest

Yn = Y0(1+r)n

Remember to express r as a decimal: 10 percent = 0.10

Suppose that interest is compounded periodically, at t times per year.

• quarterly: t = 4
• monthly: t= 12
1. Divide r by t
2. Multiply the number of years, n, by t

Example 1: Initial balance of \$1000, interest rate of 12 percent, compounded monthly. How much will you have after three years?

1. interest rate as a decimal = .12
2. t = 12
3. r/t = .12/12 = .01
4. number of years = n = 3; number of periods = nt = 3(12) = 36
5. Ending balance = \$1000(1+.01)36 = \$1430.77

Continuous compounding. We use e, the number of fingers on an alien's hand (it's between 2 and 3).

Yn = Y0enr

Example 2: \$1000 initial balance, continuous compounding, interest rate of 12 percent, for three years. So r = .12

Yn = \$1000e(.12)(3)

In Microsoft Excel, you would write +1000*EXP(.12*3)

### Solving for n

In Example 1, how long will it take you to have \$3000?

Have to take logs: log(Yn/(Y0) = nt(log(1+r))

log(3000/1000) = log(3) = 12n(log(1.01))

.4771 = 12n(.004321)

n = 9.2 years

In example 2, how long will it take to have \$2500?

ln(2500/1000) = nr = .12n

.916 = .12n

n = 7.6 years

### Annual Bank Deposits

Instead of one initial deposit, you deposit \$C every year. How much will you end up with after n years?

B = C[(1+r)n-1]/r

Example: deposit \$1000 every year for 3 years at an interest rate of 4 percent.

B = \$1000[(1+.04)3-1)]/.04 = \$3121.60

### Solving for C

Use algebra to get

C = rB/[(1+r)n-1]

If the goal is to have \$50,000 for retirement, you have 3 years until you retire, and the interest rate is 4 percent, then you need to save

C = (.04)(\$50,000)/[(1.04)3-1] = \$16017.43

### Dissaving: taking an annuity

If you start with \$50,000 and the interest rate is 4 percent, how much can you take out each year for three years to just exhaust your money?

The formula changes, because we are earning interest while we take money out. The formula is:

C = rB/[(1-(1+r)-n]

C = .04(\$50,000)/[1 -(1.04)-3] = \$18017.43

### Inflation

If the interest rate is 4 percent, and the inflation rate is 1 percent, then the real interest rate is 3 percent.

With a real annuity, the first year payment would be calculated using the first year real interest rate. The second year payment goes up at the rate of inflation.

In our example, the real annuity payment would be

C = .03(\$50,000)/[1-(1.03)-3] = \$17676.52

Next year, C goes up by one percent: \$17676.52(1.01) = \$17853.28

### Monthly Installment Loans

With a monthly installment loan, such as a mortgage or an auto loan, the annuity formula determines the monthly payment. The decimal interest rate gets divided by 12 to make it a monthly rate. And the number of years is multiplied by 12.

Example: \$50,000 loan, ten years, interest rate of 6 percent. Monthly interest rate = .06/12. Number of months = 120.

C = (.005)(\$50,000)/[1-(1.005)-120] = \$555.10