Deck 13: Annuities: Special Situations

Question 1

A perpetuity and an annuity both have the same values for PMT and i. Which has the larger present value? Give a brief explanation.

Answer 1

The perpetuity has the larger present value. The great the number of payments in an annuity, the larger the annuity's present value. Hence PV(perpetuity) > PV(annuity). Alternatively, the perpetuity may be viewed as a combination of an annuity identical to the given annuity and a deferred perpetuity. Then PV of given perpetuity = (PV of given annuity) + (PV of deferred perpetuity). Therefore, PV of given perpetuity > PV of given annuity.

Question 2

If market interest rates rise, will it require a larger endowment to sustain a perpetuity with a particular payment size? Give a brief explanation.

Answer 2

No. Each payment is the interest earned during the preceding payment interval. At a higher market increase rate, a smaller endowment can generate the same payment size.

Question 3

Will the market value of a perpetual preferred share (paying a fixed periodic dividend) rise or fall if the rate of return (dividend yield) required by investors declines? Give a brief explanation.

Answer 3

Yes, the market value will rise. The rate of return (dividend yield) is the (fixed) annual dividend calculated as a percentage of the market value. If investors will accept a lower rate of return they will pay a higher price for the shares.

Question 4

A perpetuity is to pay $10,000 at the end of every six months. How much less money is required to fund the perpetuity if the money can be invested to earn 5% compounded semiannually instead of 4% compounded semiannually?

Question 5

In 1752, the British government converted all of its outstanding bonds to perpetual bonds that paid a fixed interest rate. These bonds paid only the interest every three months-the principal amount of the debt would never be repaid. The perpetual bonds have come to be known as "Consols," and they trade in the British financial markets. The owner of a ≤1000 face-value (or denomination) Consol receives ≤6.25 every three months.
a) What is the fixed interest rate (on the face value) paid by the Consols?
b) If the prevailing long-term interest rate in the British financial markets is 4.5% compounded quarterly,
what is the fair market value of a ≤1000 face-value Consol? (Assume that you will receive the first interest payment in three months.)

Question 6

An old agreement requires a town to pay $500 per year in perpetuity to the owner of a parcel of land for a water well dug on the property in the 1920s. The well is no longer used, and the town wants to buy out the contract, which has become an administrative nuisance. What amount (including the regular scheduled payment) should the landowner be willing to accept on the date of the next scheduled payment if long-term low-risk investments now earn 5.8% compounded annually?

Question 7

A company's perpetual preferred shares pay a semiannual dividend of $3.00. The next dividend will be paid tomorrow.
a) At what price would the shares provide an investor with a 4.5% semiannually compounded rate of return? The investor will receive tomorrow's dividend.
b) If the shares are trading at $70, what nominal rate of return will they provide to a purchaser?

Question 8

In this section, does constant growth mean that each successive payment increases by the same dollar amount? If not, what does it mean?

Question 9

Randall wants to accumulate $750,000 in his RRSP by the end of his 30-year working career. What should be his initial year-end contribution if he intends to increase the contribution by 3% every year and the RRSP earns 10% compounded annually?

Question 10

How much will it cost to purchase a 20-year indexed annuity in which the end-of-quarter payments start at $5000 and grow by 0.5% every quarter? Assume that the money used to purchase the annuity earns 6% compounded quarterly.

Question 11

Mrs. Sippy (age 65) is about to begin receiving a CPP retirement pension of $11,000per year. This pension is indexed to the Consumer Price Index (CPI). Assume that the annual pension will be paid in a single year-end payment, the CPI will rise 3% per year, and money is worth 6% compounded annually. What is the current economic value of:
a) 20 years of pension benefits?
b) 25 years of pension benefits?

Question 12

Dean has already implemented the first stage of his financial plan. Over a 30-year period, he will continue to increase his annual year-end RRSP contributions by 3% per year. His initial contribution was $2000. At the end of the 30 years, he will transfer the funds to an RRIF and begin end-of-month withdrawals that will increase at the rate of 1.8% compounded monthly for 25 years. Assume that his RRSP will earn 9% compounded annually and his RRIF will earn 6% compounded monthly. What will be the size of his initial RRIF withdrawal?

Question 13

Maritime Bank recently announced that its next semiannual dividend (to be paid six months from now) will be $1.00 per share. A stock analyst's best estimate for the growth in future dividends is 5% compounded semiannually.
a) If you require a rate of return of 10% compounded semiannually on the stock, what maximum price should you be willing to pay per share? Ignore the present value of dividends beyond a 50-year time horizon.
b) What price do you obtain if you do not ignore dividends beyond 50 years? (Hint: Use a large value, say 99,999, for n in the present value calculation.)

Question 14

Using the Constant-Growth Annuity Chart The NEAssets feature at the beginning of this section describes how to access the Constant-Growth Annuity Chart available in this student textbook's Online Learning Centre (OLC). Use this chart to answer the following questions. Assume in every case that you make annual end-of-year contributions to an RRSP for 25 years. The RRSP earns 7.5% compounded annually. Compared to a "base case" of constant contributions of $3000, how much larger (in percentage terms) will the value of your RRSP be after 25 years if you increase the payments by:
a) 1% per year?
b) 2% per year?
c) 4% per year?

Question 15

The common shares of Bancorp Ltd. are forecast to pay annual dividends of $3 at the end of each of the next five years, followed by dividends of $2 per year in perpetuity. What is the fair market value of the shares if the market requires a 10% annually compounded rate of return on shares having a similar degree of risk?

Question 16

What amount can be paid at the end of every month in perpetuity from an endowment of $350,000 which is earning 5.4% compounded monthly?

A) $1,575
B) $1,890
C) $2,100
D) $15,750
E) $18,900

Question 17

What amount can be paid at the end of every six months, in perpetuity, from an endowment of $475,000 earning 7% compounded annually?

A) $16,625
B) $16,344
C) $15,682
D) $14,971
E) $11,579

Question 18

What is the present value of a trust fund that earns 8.8% compounded annually and pays out $3,500 every three months. The next payment is due to be made today.

A) $164,249
B) $745,820
C) $167,749
D) $529,722
E) $526,222

Question 19

If an endowment fund of $2,327,000 is to be used to pay out grants of $175,000 at the end of every year in perpetuity what effective rate of interest must the funds earn?

A) 11.61%
B) 10.06%
C) 8.93%
D) 7.52%
E) 5.44%

Question 20

How much money would accumulate over 35 years if the investor makes an investment of $3,500 at the end of the first year and always increases the size of the annual investment by 5% over the previous year. The investments are expected to earn 11% compounded annually.

A) $1,195,563
B) $1,928,432
C) $1,010,619
D) $1,624,915
E) $2,311,587