# Annuities, present values, future values

4.3 Calculate the following values, assuming a discount rate of 8%:

a. present value of a perpetuity (also called a perpetual annuity) of $50 received each year at the end of each year

b. present value of an annuity of $50 received at the end of each year for 5 years

c. present value of an annuity of $50 received at the end of each year for 10 years, with the first payment to be received at the end of the 6th year

d. present value of a perpetuity of $50, with the first payment received at the end of the 16th year.

4.4 a. Show (with a time line, for example) that the perpetuity in 4.3a. is exactly the same as the sum of the annuities and perpetuities in 4.3b. to 4.3d.

b. Show that their present values add up to the same amount.

4.5 a. Jane is 20 years old today. Jane is going to put $1,000 into her savings account on her 21st birthday and again on every birthday for 20 payments (i.e., till her 40th birthday). She will earn 5%, paid annually. How much money will be in the account after she collects her interest and makes her 20th payment?

b. Calculate how much money she could take out each year for the 20 years from her 41st birthday till her 60th birthday, assuming she still earns 5% and takes out the same amount each year, leaving exactly $0 in the account after removing her 20th payment.

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#### Solution Preview

4.3

a. 50/0.08= 625

b. (50/1.08)+(50/1.08^2)+(50/1.08^3)+(50/1.08^4)+(50/1.08^5)= 199.64

c. using financial ...

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

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(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?