What is the Pv of a Continuous Stream of Cash Flows
Compounding Intervals and Bond Prices
In calculating the value of the 7 percent Treasury bonds, we made two approximations. First, we assumed that interest payments occurred annually. In practice,
13Early in 2001 the Turkish overnight rate exceeded 20,000 percent.
most U.S. bonds make coupon payments semiannually, so that instead of receiving $70 every year, an investor holding 7 percent bonds would receive $35 every half year. Second, yields on U.S. bonds are usually quoted as semiannually compounded yields. In other words, if the semiannually compounded yield is quoted as 4.8 percent, the yield over six months is 4.8/2 = 2.4 percent.
Now we can recalculate the value of the 7 percent Treasury bonds, recognizing that there are 10 six-month coupon payments of $35 and a final payment of the $1,000 face value:
1035
The difficult thing in any present value exercise is to set up the problem correctly. Once you have done that, you must be able to do the calculations, but they are not difficult. Now that you have worked through this chapter, all you should need is little practice.
The basic present value formula for an asset that pays off in several periods is the following obvious extension of our one-period formula:
You can always work out any present value using this formula, but when the interest rates are the same for each maturity, there may be some shortcuts that can reduce the tedium. We looked at three such cases. The first is an asset that pays C dollars a year in perpetuity. Its present value is simply
The second is an asset whose payments increase at a steady rate g in perpetuity. Its present value is
The third is an annuity that pays C dollars a year for t years. To find its present value we take the difference between the values of two perpetuities:
Our next step was to show that discounting is a process of compound interest. Present value is the amount that we would have to invest now at compound interest r in order to produce the cash flows Cx, C2, etc. When someone offers to lend us a dollar at an annual rate of r, we should always check how frequently the interest is to be compounded. If the compounding interval is annual, we will have to repay (1 + r)t dollars; on the other hand, if the compounding period is continuous, we will have to repay 2.718rt (or, as it is usually expressed, ert) dollars. In capital budgeting we often assume that the cash flows occur at the end of each year, and therefore we discount them at an annually compounded rate of interest.
PART I Value
Sometimes, however, it may be better to assume that they are spread evenly over the year; in this case we must make use of continuous compounding.
It is important to distinguish between nominal cash flows (the actual number of dollars that you will pay or receive) and real cash flows, which are adjusted for inflation. Similarly, an investment may promise a high nominal interest rate, but, if inflation is also high, the real interest rate may be low or even negative.
We concluded the chapter by applying discounted cash flow techniques to value United States government bonds with fixed annual coupons.
We introduced in this chapter two very important ideas which we will come across several times again. The first is that you can add present values: If your formula for the present value of A + B is not the same as your formula for the present value of A plus the present value of B, you have made a mistake. The second is the notion that there is no such thing as a money machine: If you think you have found one, go back and check your calculations.
FURTHER READING
The material in this chapter should cover all you need to know about the mathematics of discounting; but if you wish to dig deeper, there are a number of books on the subject. Try, for example:
R. Cissell, H. Cissell, and D. C. Flaspohler: The Mathematics of Finance, 8th ed., Houghton Mifflin Company, Boston, 1990.
QUIZ
1. At an interest rate of 12 percent, the six-year discount factor is .507. How many dollars is $.507 worth in six years if invested at 12 percent?
2. If the PV of $139 is $125, what is the discount factor?
3. If the eight-year discount factor is .285, what is the PV of $596 received in eight years?
4. If the cost of capital is 9 percent, what is the PV of $374 paid in year 9?
5. A project produces the following cash flows:
| Year | Flow |
| 1 | 432 |
| 2 | 137 |
| 3 | 797 |
If the cost of capital is 15 percent, what is the project's PV?
6. If you invest $100 at an interest rate of 15 percent, how much will you have at the end of eight years?
7. An investment costs $1,548 and pays $138 in perpetuity. If the interest rate is 9 percent, what is the NPV?
8. A common stock will pay a cash dividend of $4 next year. After that, the dividends are expected to increase indefinitely at 4 percent per year. If the discount rate is 14 percent, what is the PV of the stream of dividend payments?
9. You win a lottery with a prize of $1.5 million. Unfortunately the prize is paid in 10 annual installments. The first payment is next year. How much is the prize really worth? The discount rate is 8 percent.
Do not use the Appendix tables for these questions. The interest rate is 10 percent.
a. What is the PV of an asset that pays $1 a year in perpetuity?
b. The value of an asset that appreciates at 10 percent per annum approximately doubles in seven years. What is the approximate PV of an asset that pays $1 a year in perpetuity beginning in year 8?
c. What is the approximate PV of an asset that pays $1 a year for each of the next seven years?
d. A piece of land produces an income that grows by 5 percent per annum. If the first year's flow is $10,000, what is the value of the land?
Use the Appendix tables at the end of the book for each of the following calculations:
a. The cost of a new automobile is $10,000. If the interest rate is 5 percent, how much would you have to set aside now to provide this sum in five years?
b. You have to pay $12,000 a year in school fees at the end of each of the next six years. If the interest rate is 8 percent, how much do you need to set aside today to cover these bills?
c. You have invested $60,476 at 8 percent. After paying the above school fees, how much would remain at the end of the six years?
You have the opportunity to invest in the Belgravian Republic at 25 percent interest. The inflation rate is 21 percent. What is the real rate of interest? The continuously compounded interest rate is 12 percent.
a. You invest $1,000 at this rate. What is the investment worth after five years?
b. What is the PV of $5 million to be received in eight years?
c. What is the PV of a continuous stream of cash flows, amounting to $2,000 per year, starting immediately and continuing for 15 years?
You are quoted an interest rate of 6 percent on an investment of $10 million. What is the value of your investment after four years if the interest rate is compounded: a. Annually, b. monthly, or c. continuously?
Suppose the interest rate on five-year U.S. government bonds falls to 4.0 percent. Recalculate the value of the 7 percent bond maturing in 2006. (See Section 3.5.)
What is meant by a bond's yield to maturity and how is it calculated?
Use the discount factors shown in Appendix Table 1 at the end of the book to calculate the PV of $100 received in:
a. Year 10 (at a discount rate of 1 percent).
b. Year 10 (at a discount rate of 13 percent).
c. Year 15 (at a discount rate of 25 percent).
d. Each of years 1 through 3 (at a discount rate of 12 percent).
Use the annuity factors shown in Appendix Table 3 to calculate the PV of $100 in each of:
a. Years 1 through 20 (at a discount rate of 23 percent).
b. Years 1 through 5 (at a discount rate of 3 percent).
c. Years 3 through 12 (at a discount rate of 9 percent).
a. If the one-year discount factor is .88, what is the one-year interest rate?
b. If the two-year interest rate is 10.5 percent, what is the two-year discount factor?
c. Given these one- and two-year discount factors, calculate the two-year annuity factor.
d. If the PV of $10 a year for three years is $24.49, what is the three-year annuity factor?
e. From your answers to (c) and (d), calculate the three-year discount factor.
PRACTICE
QUESTIONS
Brealey-Meyers: Principles of Corporate Finance, Seventh Edition
I. Value
3. How to Calculate Present Values
© The McGraw-H Companies, 2003
PART I Value
A factory costs $800,000. You reckon that it will produce an inflow after operating costs of $170,000 a year for 10 years. If the opportunity cost of capital is 14 percent, what is the net present value of the factory? What will the factory be worth at the end of five years?
Harold Filbert is 30 years of age and his salary next year will be $20,000. Harold forecasts that his salary will increase at a steady rate of 5 percent per annum until his retirement at age 60.
a. If the discount rate is 8 percent, what is the PV of these future salary payments?
b. If Harold saves 5 percent of his salary each year and invests these savings at an interest rate of 8 percent, how much will he have saved by age 60?
c. If Harold plans to spend these savings in even amounts over the subsequent 20 years, how much can he spend each year?
A factory costs $400,000. You reckon that it will produce an inflow after operating costs of $100,000 in year 1, $200,000 in year 2, and $300,000 in year 3. The opportunity cost of capital is 12 percent. Draw up a worksheet like that shown in Table 3.1 and use tables to calculate the NPV.
Halcyon Lines is considering the purchase of a new bulk carrier for $8 million. The forecasted revenues are $5 million a year and operating costs are $4 million. A major refit costing $2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at $1.5 million. If the discount rate is 8 percent, what is the ship's NPV?
As winner of a breakfast cereal competition, you can choose one of the following prizes:
b. $180,000 at the end of five years.
c. $11,400 a year forever.
d. $19,000 for each of 10 years.
e. $6,500 next year and increasing thereafter by 5 percent a year forever. If the interest rate is 12 percent, which is the most valuable prize? Refer back to the story of Ms. Kraft in Section 3.1.
a. If the one-year interest rate were 25 percent, how many plays would Ms. Kraft require to become a millionaire? (Hint: You may find it easier to use a calculator and a little trial and error.)
b. What does the story of Ms. Kraft imply about the relationship between the one-year discount factor, DF1, and the two-year discount factor, DF2?
Siegfried Basset is 65 years of age and has a life expectancy of 12 more years. He wishes to invest $20,000 in an annuity that will make a level payment at the end of each year until his death. If the interest rate is 8 percent, what income can Mr. Basset expect to receive each year?
James and Helen Turnip are saving to buy a boat at the end of five years. If the boat costs $20,000 and they can earn 10 percent a year on their savings, how much do they need to put aside at the end of years 1 through 5?
Kangaroo Autos is offering free credit on a new $10,000 car. You pay $1,000 down and then $300 a month for the next 30 months. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price. If the rate of interest is 10 percent a year, which company is offering the better deal?
Recalculate the NPV of the office building venture in Section 3.1 at interest rates of 5, 10, and 15 percent. Plot the points on a graph with NPV on the vertical axis and the discount rates on the horizontal axis. At what discount rate (approximately) would the project have zero NPV? Check your answer.
14. a. How much will an investment of $100 be worth at the end of 10 years if invested at
15 percent a year simple interest?
b. How much will it be worth if invested at 15 percent a year compound interest?
c. How long will it take your investment to double its value at 15 percent compound interest?
15. You own an oil pipeline which will generate a $2 million cash return over the coming year. The pipeline's operating costs are negligible, and it is expected to last for a very long time. Unfortunately, the volume of oil shipped is declining, and cash flows are expected to decline by 4 percent per year. The discount rate is 10 percent.
a. What is the PV of the pipeline's cash flows if its cash flows are assumed to last forever?
b. What is the PV of the cash flows if the pipeline is scrapped after 20 years?
[Hint for part ((b): Start with your answer to part (a), then subtract the present value of a declining perpetuity starting in year 21. Note that the forecasted cash flow for year 21 will be much less than the cash flow for year 1.]
16. If the interest rate is 7 percent, what is the value of the following three investments?
a. An investment that offers you $100 a year in perpetuity with the payment at the end of each year.
b. A similar investment with the payment at the beginning of each year.
c. A similar investment with the payment spread evenly over each year.
17. Refer back to Section 3.2. If the rate of interest is 8 percent rather than 10 percent, how much would our benefactor need to set aside to provide each of the following?
a. $100,000 at the end of each year in perpetuity.
b. A perpetuity that pays $100,000 at the end of the first year and that grows at 4 percent a year.
c. $100,000 at the end of each year for 20 years.
d. $100,000 a year spread evenly over 20 years.
18. For an investment of $1,000 today, the Tiburon Finance Company is offering to pay you $1,600 at the end of 8 years. What is the annually compounded rate of interest? What is the continuously compounded rate of interest?
19. How much will you have at the end of 20 years if you invest $100 today at 15 percent annually compounded? How much will you have if you invest at 15 percent continuously compounded?
20. You have just read an advertisement stating, "Pay us $100 a year for 10 years and we will pay you $100 a year thereafter in perpetuity." If this is a fair deal, what is the rate of interest?
21. Which would you prefer?
a. An investment paying interest of 12 percent compounded annually.
b. An investment paying interest of 11.7 percent compounded semiannually.
c. An investment paying 11.5 percent compounded continuously. Work out the value of each of these investments after 1, 5, and 20 years.
22. Fill in the blanks in the following table:
| Nominal Interest | Inflation | Real Interest | ||||||||||
| Rate(%) | Rate (%) | Rate (%) | ||||||||||
| 6 | 1 | — | ||||||||||
| — | 10 | 12 | ||||||||||
| 9 | — | Sometimes real rates of return are calculated by subtracting the rate of inflation from the nominal rate. This rule of thumb is a good approximation if the inflation rate is low. How big is the error from using this rule of thumb to calculate real rates of return in the following cases?
In 1880 five aboriginal trackers were each promised the equivalent of 100 Australian dollars for helping to capture the notorious outlaw Ned Kelley. In 1993 the granddaughters of two of the trackers claimed that this reward had not been paid. The prime minister of Victoria stated that, if this was true, the government would be happy to pay the $100. However, the granddaughters also claimed that they were entitled to compound interest. How much was each entitled to if the interest rate was 5 percent? What if it was 10 percent? A leasing contract calls for an immediate payment of $100,000 and nine subsequent $100,000 semiannual payments at six-month intervals. What is the PV of these payments if the annual discount rate is 8 percent? A famous quarterback just signed a $15 million contract providing $3 million a year for five years. A less famous receiver signed a $14 million five-year contract providing $4 million now and $2 million a year for five years. Who is better paid? The interest rate is 10 percent. In August 1994 The Wall Street Journal reported that the winner of the Massachusetts State Lottery prize had the misfortune to be both bankrupt and in prison for fraud. The prize was $9,420,713, to be paid in 19 equal annual installments. (There were 20 installments, but the winner had already received the first payment.) The bankruptcy court judge ruled that the prize should be sold off to the highest bidder and the proceeds used to pay off the creditors. a. If the interest rate was 8 percent, how much would you have been prepared to bid for the prize? b. Enhance Reinsurance Company was reported to have offered $4.2 million. Use Appendix Table 3 to find (approximately) the return that the company was looking for. You estimate that by the time you retire in 35 years, you will have accumulated savings of $2 million. If the interest rate is 8 percent and you live 15 years after retirement, what annual level of expenditure will those savings support? Unfortunately, inflation will eat into the value of your retirement income. Assume a 4 percent inflation rate and work out a spending program for your retirement that will allow you to maintain a level real expenditure during retirement. You are considering the purchase of an apartment complex that will generate a net cash flow of $400,000 per year. You normally demand a 10 percent rate of return on such investments. Future cash flows are expected to grow with inflation at 4 percent per year. How much would you be willing to pay for the complex if it: a. Will produce cash flows forever? b. Will have to be torn down in 20 years? Assume that the site will be worth $5 million at that time net of demolition costs. (The $5 million includes 20 years' inflation.) Now calculate the real discount rate corresponding to the 10 percent nominal rate. Redo the calculations for parts (a) and (b) using real cash flows. (Your answers should not change.) Vernal Pool, a self-employed herpetologist, wants to put aside a fixed fraction of her annual income as savings for retirement. Ms. Pool is now 40 years old and makes $40,000 a year. She expects her income to increase by 2 percentage points over inflation (e.g., 4 percent inflation means a 6 percent increase in income). She wants to accumulate $500,000 in real terms to retire at age 70. What fraction of her income does she need to set aside? Assume her retirement funds are conservatively invested at an expected real rate of return of 5 percent a year. Ignore taxes. At the end of June 2001, the yield to maturity on U.S. government bonds maturing in 2006 was about 4.8 percent. Value a bond with a 6 percent coupon maturing in June 2006. The bond's face value is $10,000. Assume annual coupon payments and annual compounding. How does your answer change with semiannual coupons and a semiannual discount rate of 2.4 percent? Refer again to Practice Question 31. How would the bond's value change if interest rates fell to 3.5 percent per year? A two-year bond pays a coupon rate of 10 percent and a face value of $1,000. (In other words, the bond pays interest of $100 per year, and its principal of $1,000 is paid off in year 2.) If the bond sells for $960, what is its approximate yield to maturity? Hint: This requires some trial-and-error calculations. Here are two useful rules of thumb. The "Rule of 72" says that with discrete compounding the time it takes for an investment to double in value is roughly 72/interest rate (in percent). The "Rule of 69" says that with continuous compounding the time that it takes to double is exactly 69.3/interest rate (in percent). a. If the annually compounded interest rate is 12 percent, use the Rule of 72 to calculate roughly how long it takes before your money doubles. Now work it out exactly. b. Can you prove the Rule of 69? Use a spreadsheet program to construct your own set of annuity tables. An oil well now produces 100,000 barrels per year. The well will produce for 18 years more, but production will decline by 4 percent per year. Oil prices, however, will increase by 2 percent per year. The discount rate is 8 percent. What is the PV of the well's production if today's price is $14 per barrel? Derive the formula for a growing (or declining) annuity. Calculate the real cash flows on the 7 percent U.S. Treasury bond (see Section 3.5) assuming annual interest payments and an inflation rate of 2 percent. Now show that by discounting these real cash flows at the real interest rate you get the same PV that you get when you discount the nominal cash flows at the nominal interest rate. Use a spreadsheet program to construct a set of bond tables that shows the present value of a bond given the coupon rate, maturity, and yield to maturity. Assume that coupon payments are semiannual and yields are compounded semiannually. CHALLENGE QUESTION/ |
Continue reading here: The Jones Family Incorporated
Was this article helpful?
Source: https://www.ajjacobson.us/corporate-finance-2/compounding-intervals-and-bond-prices.html
0 Response to "What is the Pv of a Continuous Stream of Cash Flows"
Post a Comment