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# Calculating Present and Future Value of Annuities

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5-1 How long will it take \$ 200 to double if it earns the following rates? Compounding occurs once a year.

5-2 Find the present values of these ordinary annuities. Discounting occurs once a year.
a. \$ 400 per year for 10 years at 10%
b. \$ 200 per year for 5 years at 5%
c. \$ 400 per year for 5 years at 0%
d. Rework Parts a, b, and c assuming they are annuities due.

5-3 PRESENT VALUE OF A PERPETUITY
What is the present value of a \$ 100 perpetuity if the interest rate is 7%? If interest rates doubled to 14%, what would its present value be?

5-4 EFFECTIVE INTEREST RATE
You borrow \$ 85,000; the annual loan payments are \$ 8,273.59 for 30 years. What interest rate are you being charged?

5-5 Your client is 40 years old; and she wants to begin saving for retirement, with the first payment to come one year from now. She can save \$ 5,000 per year; and you advise her to invest it in the stock market, which you expect to provide an average return of 9% in the future.
a. If she follows your advice, how much money will she have at 65?
b. How much will she have at 70?
c. She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70. If her investments continue to earn the same rate, how much will she be able to withdraw at the end of each year after retirement at each retirement age?

#### Solution Preview

Solutions

5-1 How long will it take \$ 200 to double if it earns the following rates? Compounding occurs once a year.

No interest rate is given, let us assume interest rates as 5%, 10%

Case 1, i=5%
Future Value =FV=\$400
Present Value=PV=\$200
We know that
FV=PV*(1+i)^n
400=200*(1+5%)^n
400/200=1.05^n
2=1.05^n
Taking ln (natural log) both sides
ln(2)=n*ln(1.05)
n=ln(2)/ln(1.05)=14.21 years

Case 2, i=10%
Future Value =FV=\$400
Present Value=PV=\$200
We know that
FV=PV*(1+i)^n
400=200*(1+10%)^n
400/200=1.10^n
2=1.10^n
Taking ln (natural log) both sides
ln(2)=n*ln(1.10)
n=ln(2)/ln(1.10)=7.27 years
Similarly we can find period for any rate of interest.

5-2 Find the present values of these ordinary annuities. Discounting occurs once a year.

a. \$ 400 per year for 10 years at 10%
Annual payments=R=\$400
Interest rate=i=10%
Number of payments=n=10
PV of ordinary annuity=R*((1-1/(1+i)^n)/i)
=400*((1-1/(1+10%)^10)/10%)
=\$2457.83

b. \$ 200 per year for 5 years at 5%
Annual payments=R=\$200
Interest rate=i=5%
Number of payments=n=5
PV of ordinary annuity=R*((1-1/(1+i)^n)/i)
=200*((1-1/(1+5%)^5)/5%)
...

#### Solution Summary

There are 5 problems. Solutions to these problems explain the methodology to calculate PV & FV of annuities and rate of interest.

\$2.19

## Need Help with formulas on problems

These are just exercises. I need help with figuring out the formulas on all 5 questions.

Thanks!

(Complete problem also found in attachment)

1. Annuity Values.
a. What is the present value of a 3-year annuity of \$100 if the discount rate is 6 percent?

b. What is the present value of the annuity in (a) if you have to wait 2 years instead of 1 year for the payment stream to start?

2. Annuity Due. Recall that an annuity due is like an ordinary annuity except that the first payment is made immediately instead of at the end of the first period.

a. Why is the present value of an annuity due equal to (1 + r) times the present value of an ordinary annuity?

b. Why is the future value of an annuity due equal to (1 + r) times the future value of an ordinary annuity?

3. Annuity Due Value. Reconsider the previous problem. What if the lease payments are an annuity due, so that the first payment comes immediately? Is it cheaper to buy or lease?

4. Bond Yields. An AT&T bond has 10 years until maturity, a coupon rate of 8 percent, and sells for \$1,100.
a. What is the current yield on the bond?
b. What is the yield to maturity?

5. Bond Pricing. A General Motors bond carries a coupon rate of 8 percent, has 9 years until maturity, and sells at a yield to maturity of 7 percent.
a. What interest payments do bondholders receive each year?
b. At what price does the bond sell? (Assume annual interest payments.)
c. What will happen to the bond price if the yield to maturity falls to 6 percent?

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