full version The Time Value Of Money Essay

The Time Value Of Money

Category: Business

Autor: Antonio 07 January 2010

Words: 17421 | Pages: 70


This chapter develops and applies time value formulas. The focus is on using time value concepts to solve business problems.


A brief explanation of the fact that money promised in the future is worth less than money in hand today.
Four kinds of problems and using time lines.

A. The Future Value of an Amount
The expression relating future and present values of a single amount is developed in terms of the future value factor and the associated table.
Problem solving technique is introduced.
Applications include deferred payment terms as the equivalent of cash discounts and the opportunity cost rate.
Financial calculators are introduced and instruction is provided on their use in solving time value problems.
B. The Expression for The Present Value of an Amount
An alternate formulation emphasizing the present value in terms of the future value. More on technique.

A. Annuities
The concept of an annuity and its present and future values.
B. The Future Value of an Annuity - Developing a Formula
The FVA expression, factor and tables are developed.
C. The Future Value of an Annuity - Solving Problems
Problem solving technique, the sinking fund concept, using a financial calculator for annuity problems.
D. Compound Interest and Non-Annual Compounding
Compound interest concepts and how to handle non-annual compounding in problems. The EAR and APR.
E. The Present Value of an Annuity
The formula, factor, and table are developed.
Computer spreadsheet techniques for solving time value problems are introduced and explained.
F. The Present Value of an Annuity - Solving Problems
Applications include discounting a stream of payments, amortized loans and amortization schedules, and working with mortgages.
G. The Annuity Due
The concept and formula are developed and applied.
H. Perpetuities
The idea of a never-ending stream with a finite present value is developed intuitively.
Applications include the capitalization of earnings and preferred stock.
I. Multi-Part Problems
Dealing with situations in which the solutions to problems become inputs to other problems.
Visualizing and time lining complex problems.
J. Uneven Streams and Imbedded Annuities
Recognizing and dealing with annuities imbedded in uneven payment streams.


1. Why are time value concepts important in ordinary business dealings, especially those involving contracts?

ANSWER: Business contracts and agreements generally specify payments that are due at future times. If such payments are more than a few months into the future, the correct analysis of the value of the agreement depends on a recognition of the time value of money.

2. Why are time value concepts crucial in determining what a bond or a share of stock should be worth?

ANSWER: All securities derive their value solely from the future cash flows that come from owning them. The only way to value a future cash flow today is through the present value concept. Therefore, the value of a security depends entirely on time value ideas.

3. In a retail store a discount is a price reduction. What's a discount in finance? Are the two ideas related?

ANSWER: Discounting in finance means taking the present value of a sum promised in the future. The present value process always results in an figure that's less than the future amount, so in a sense, present valuing reduces the price of the future payment.

4. Calculate the present value of one dollar 30 years in the future at 10% interest. What does the result tell you about very long-term contracts?

ANSWER: PV = FV30[PVF10,30]
= $1 [.0573]
= $0.0573
= A little less than six cents.

Money promised in a very long-term contract isn't worth much today, even if its receipt is certain. Therefore, we should be careful about what we give up today for a commitment in the distant future.

5. Write a brief, verbal description of the logic behind the development of the time value formulas for annuities.

ANSWER: To develop a time value formula for an annuity we take an annuity with a finite number of payments and develop its present or future value by treating each payment as an amount. We then examine the resulting expression looking for a pattern that can be extended to longer streams of payments. Recognizing such a pattern allows us to generalize the expression to an arbitrary number of annuity payments, n.

6. Deferred payment terms are equivalent to a cash discount. Discuss and explain this idea.

ANSWER: Deferred payment means the seller will accept the promise of a future payment instead of the full price today (with no interest charged on the amount deferred). Since the present value of the deferred amount is less than its nominal value, the transaction is actually being conducted at a price (in present value terms) that's less than the stated price of the article. In essence this is a sale at a discount.
Looking at it another way, the buyer could put the deferred amount in the bank until it was due. At that time she could withdraw the amount deposited, pay the bill, and keep the interest that would be approximately the amount of the discount.

7. What's an opportunity cost interest rate?

ANSWER: An opportunity rate is the rate that would be earned on money given up for some purpose. For example, in the previous question, if the seller finances his business at 12%, his cost of deferring payment is 12%, because he could have paid off some 12% debt if he'd have received the money at the time of the sale.

8. Discuss the idea of a sinking fund. How is it related to time value?

ANSWER: Lenders are often concerned that borrowers won't be able to raise the cash to pay off loan principals even though they're able make interest payments during the lives of loans. A sinking fund is an arrangement in which a borrower is required to put aside money periodically during the life of a loan, so that at maturity funds are available to pay off the principal.
In one such arrangement, funds are deposited at interest to accumulate into the principal amount by the loan's maturity. Time value is involved because it takes a future value of an annuity problem to calculate the periodic payment that will just pay off the loan at maturity.

9. The amount formulas share a closer relationship than the annuity formulas. Explain and interpret this statement.

ANSWER: The two amount formulas are really the same expression written differently for convenience. Both expressions involve the present value and the future value of the same money, and either can be used to solve any amount problem.
The annuity formulas are two distinct, separately derived expressions. They deal with two different things, the present and future value of the annuity in question. Annuity problems must be classified as either present or future value, and the correct expression has to be used to solve either.

10. Describe the underlying meaning of compounding and compounding periods. How does it relate to time value? Include the idea of an effective annual rate (EAR). What is the annual percentage rate (APR)? Is the APR related to the EAR?

ANSWER: Compounding relates to earning interest on previously earned interest. Money initially deposited at interest earns interest on that amount until the end of a compounding period. At that time the interest is "credited" to the account, and future interest is earned on the sum of the original deposit plus the interest earned in the first period. Interest earned in the second period is credited at its end, so interest earned in the third period is based on the original deposit plus the first and second period's interest. And so on. The more frequently (shorter compounding periods) interest is compounded, the more interest is earned. Interest rates are generally stated in annual terms (the nominal rate) and must be adjusted to reflect the compounding periods in use.
Time value assumes compound interest. In problems, compounding periods and rates must be consistently specified.
The Effective Annual Rate (EAR) is the rate of annually compounded interest that is just equivalent to a nominal rate compounded more frequently.
The APR is the nominal rate. The APR and the EAR are not directly related.

11. What information are we likely to be interested in that’s contained in a loan amortization schedule?

ANSWER: A loan amortization schedule details the interest and principal components of every loan payment as well as the beginning and ending loan balance for every payment period.

12. Discuss mortgage loans in terms of the time value of money and loan amortization. What important points should every homeowner know about how mortgages work? (Hint: Think about taxes and getting the mortgage paid off.)

ANSWER: A mortgage is an amortized loan, generally of fairly long term. 30 years is common. Payments are made monthly so there are 360 payments in a 30-year mortgage.
Amortized loan payments are generally constant in amount, but the split between interest and principal within the payments varies during the life of the loan. Early payments contain relatively more interest and later payments relatively more principal repayment.
This phenomenon is extreme in long term loans like mortgages. Early payments are nearly all interest while later payments are nearly all principal repayment. This leads to two important facts for homeowners. First, because interest is tax deductible, early mortgage payments save a lot on taxes while later payments save only a little. Second, a loan is not reduced by much during the early years of its life. That is, half way through the loan’s life, a great deal more than half of the original loan balance is still unpaid.

13. Discuss the idea of capitalizing a stream of earnings in perpetuity. Where is this idea useful? Is there a financial asset that makes use of this idea?

ANSWER: A constant stream of earnings that can be expected to go on forever has a finite present value, which is known as the capitalized value of the stream.
The idea is useful in valuing businesses. Essentially, any firm is worth the capitalized value of its expected future earnings. Where the best estimate of future earnings is simply a continuation of current earnings, capitalizing a stream of that magnitude gives an estimate of value.
Preferred stock is a security that pays a constant dividend indefinitely. Its value is the capitalization of the stream of its dividends.

14. When an annuity begins several time periods into the future, how do we calculate its present value today? Describe the procedure in a few words.

ANSWER: The formula for the present value of an annuity gives a value at the beginning of the annuity. If that time is in the future, the "present value" of the annuity has to be brought back in time to the true present as an amount. Hence two consecutive calculations are required. First take the present value of the annuity, then treat that figure as an amount and take its present value.


1. A business can be valued by capitalizing it’s earnings stream (see example 6.15). How might you use the same idea to value securities, especially the stock of large publicly held companies? Is there a way to calculate a value that could be compared to the stock’s market price that would tell an investor whether it’s a good buy? (If the market price is lower than the calculated value, the stock is a bargain.) What financial figures associated with shares of stock might be used in the calculation. Consider the per share figures and ratios discussed in chapter 3 including EPS, dividends, book value per share etc. Does one measure make more sense than the others? What factors would make a stock worth more or less than your calculated value.

Answer: A privately or closely held company is valued by capitalizing a stream of earnings (net income) because all of a firm’s earnings are available to its owners. The analogous figure for the stocks of publicly held companies is dividends, because they represent cash received by stockholders. Although all earnings technically belong to owners, the stockholders of larger companies generally can’t influence how much of those earnings they receive. Hence dividends, which stockholders do receive, are the best measure for mathematically calculating value. (However, practitioners do use EPS regularly for less precise estimates.)
The starting point should be the capitalized value of the current dividend, essentially assuming it will go on forever. The rate used for capitalization should be consistent with the riskiness of the company involved.
This starting estimated value should be factored up or down based on expectations about future dividends. If the stream is expected to grow or shrink, value will be higher or lower respectively. Further, a stable stream should be worth more than one that varies substantially from year to year.
Expectations about future performance usually come from performance in the recent past so a stock with a record of consistent dividend (or EPS) growth should be worth more than one whose dividends have been constant for some time or erratic.


Amount Problems

1. The Lexington Property Development Company has a $10,000 note receivable from a customer due in three years. How much is the note worth today if the interest rate is
a. 9%?
b. 12% compounded monthly?
c. 8% compounded quarterly?
d. 18% compounded monthly?
e. 7% compounded continuously?


a. PV = $10,000 [PVF9,3]
= $10,000 (.7722)
= $7,722

b. PV = $10,000 [PVF1,36]
= $10,000 (.6989)
= $6,989

c. PV = $10,000 [PVF2,12]
= $10,000 (.7885)
= $7,885

d. PV = $10,000 [PVF1.5,36]
= $10,000 (.5851)
= $5,851

e. FV = PV (ekn)
$10,000 = PV [e.07(3)]
$10,000 = PV [1.2337]
PV = $8,105.70

2. What will a deposit of $4,500 left in the bank be worth under the following conditions:
a. Left for nine years at 7% interest?
b. Left for six years at 10% compounded semiannually?
c. Left for five years at 8% compounded quarterly?
d. Left for 10 years at 12% compounded monthly?


a. FV = $4,500 [FVF7,9] = $4,500 (1.8385) = $8,273.25
b. FV = $4,500 [FVF5,12] = $4,500 (1.7959) = $8,081.55
c. FV = $4,500 [FVF2,20] = $4,500 (1.4859) = $6,686.55
d. FV = $4,500 [FVF1,120] = $4,500 (3.3004) = $14,851.80

3. What interest rates are implied by the following lending arrangements?
a. You borrow $500 and repay $555 in one year.
b. You lend $1,850 and are repaid $2,078.66 in two years.
c. You lend $750 and are repaid $1,114.46 in five years with quarterly compounding.
d. You borrow $12,500 and repay $21,364.24 in three years under monthly compounding.
(Note: In c and d, be sure to give your answer as the annual nominal rate.)

a. $555 = $500 [FVFk,1]
FVFk,1 = 1.1100
k = 11%

b. $2,078.66 = $1,850.00 [FVFk,2]
FVFk,2 = 1.1236
k = 6%
c. $1,114.46 = $750.00 [FVFk,20]
FVFk,20 = 1.4859
k = 2%
knom = 8%

d. $21,364.24 = $12,500.00 [FVFk,36]
FVFk,36 = 1.7091
k = 1.5%
knom = 18%

4. How long does it take for the following to happen?
a. $856 grows into $1,122 at 7%.
b. $450 grows into $725.50 at 12% compounded monthly.
c. $5,000 grows into $6724.44 at 10% compounded quarterly.


a. $856 = $1,122 [PVF7,n]
PVF7,n = .7629
n = 4 years

b. $450.00 = $725.50 [PVF1,n]
PVF1,n = .6203
n = 48 months = 4 years

c. $5,000 = $6,724.44 [PVF2.5,n]
PVF2.5,n = 0.7436
n = 12 quarters = 3 years

5. Sally Guthrie is looking for an investment vehicle that will double her money in five years.
a. What interest rate, to the nearest whole percentage, does she have to receive?
b. At that rate, how long will it take the money to triple?
c. If she can't find anything that pays more than 11%, approximately how long will it take to double her investment?
d. What kind of financial instruments do you think Sally is looking at? Are they risky? What could happen to Sally's investment?


a. 2 = 1 [FVFk,5]
FVFk,5 = 2
k= 15%

b. FVF15,n = 3
n = 7.9 years (approximate with 8 years)

c. FVF11,n = 2
n = 6.6 years (approximate with 7 years)

d. Investments with anticipated returns like these are probably growth-oriented stocks with considerable risk. She could lose money.

6. Branson Inc. has sold product to the Brandywine Company, a major customer, for $20,000. As a courtesy to Brandywine, Branson has agreed to take a note due in two years for half of the amount due, and half in cash.
a. What is the effective price of the transaction to Branson if the interest rate is: (1) 6%, (2) 8%, (3) 10%, (4) 12%?
b. Under what conditions might the effective price be even less as viewed by Brandywine?

a. 1) PV = FV [PVF6,2] = $10,000 (.8900) = $8,900
$8,900 + $10,000 = $18,900
Effective Discount = 5.5%
2) PV = FV [PVF8,2] = $10,000 (.8573) = $8,573
$8,573 + $10,000 = $18,573
Effective Discount = 7.1%
3) PV = FV [PVF10,2] = $10,000 (.8264) = $8,264
$8,264 + $10,000 = $18,264
Effective Discount = 8.7%
4) PV = FV [PVF12,2] = $10,000 (.7972) = $7,972
$7,972 + $10,000 = $17,972
Effective Discount = 10.1%

b. The discount from Brandywine's perspective is calculated as in part a), but using the interest rate at which that firm borrows. If Brandywine's rate is higher than Branson's, it will perceive a greater discount.

7. John Cleaver's grandfather died recently and left him a trunk that had been locked in his attic for years. At the bottom of the trunk John found a packet of 50 World War I "liberty bonds" that had never been cashed in. The bonds were purchased for $11.50 each in 1918, and pay 3% interest as long as they're held. (Government savings bonds like these accumulate and compound their interest unlike corporate bonds, which regularly pay out interest to bondholders.)
a. How much are the bonds worth in 2007?
b. How much would they have been worth if they paid interest at a rate more like that paid during the 1970s and 80s, say 7%?
c. Comment on the difference between the answers to parts a and b.

First notice that
[FVFk,a+b] = [FVFk,a] [FVFk,b]
(1+k)a+b = (1+k)a (1+k)b,
FVFk,n = (1+k)n
[FVFk,89] = [FVFk,50] [FVFk,30] [FVFk,9]
[FVF3,89] = [FVF3,50] [FVF3,30] [FVF3,9]
= (4.3839)(2.4273)(1.3048)
= 13.8844

[FVF7,89] = [FVF7,50] [FVF7,30] [FVF7,9]
= (29.4570)(7.6123)(1.8385)
= 412.26

a. FV = PV [FVF3,89]
= $11.50 (13.8844)
= $159.67 per bond

b. FV = PV [FVF7,89]
= $11.50 (412.26)
= $4,740.99 per bond

c. Over a long period the interest rate makes an enormous difference in investment results!

8. Paladin Enterprises manufactures printing presses for small-town newspapers that are often short of cash. To accommodate these customers, Paladin offers the following payment terms:
1/3 on delivery
1/3 after six months
1/3 after 18 months
The Littleton Sentinel is a typically cash-poor newspaper considering one of Paladin's presses
a. What discount is implied by the terms from Paladin's point of view if it can invest excess funds at 8% compounded quarterly?
b. The Sentinel can borrow limited amounts of money at 12% compounded monthly. What discount do the payment terms imply to the Sentinel?
c. Reconcile these different views of the same thing in terms of opportunity cost.

SOLUTION: Assume a price of $300.
a. PV = $100 + $100 [PVF2,2] + $100 [PVF2,6]
= $100 + $100 (.9612)+ $100 (.8880)
= $284.92
Discount = $15.08/$300 = 5%

b. PV = $100 + $100 [PVF1,6] + $100 [PVF1,18]
= $100 + $100 (.9420)+ $100 (.8360)
= $277.80
Discount = $22.20/$300 = 7.4%

c. The buyer is avoiding more financing cost than the seller is giving up because funds are available to both of them at different opportunity rates.

9. Charlie owes Joe $8,000 on a note which is due in five years with accumulated interest at 6%. Joe has an investment opportunity now that he thinks will earn 18%. There’s a chance, however, that it will earn as little as 4%. A bank has offered to discount the note at 14% and give Joe cash that he can invest today.
a. How much ahead will Joe be if he takes the bank’s offer and the investment does turn out to yield 18%?
b. How much behind will he be if the investment turns out to yield only 4%?

a. First calculate the amount Charlie’s note will pay in five years.
FV = PV[FVF6,5] = $8,000(1.3382) = $10,705.60
If the bank discounts that at 14%, Joe will receive
PV = FV[PVF14,5] = $10,705.60(.5194) = $5,560.49
This amount invested at 18% for five years will grow into
FV = PV[FVF18,5] = $5,560.49(2.2878) = $12,721.29.
And Joe will be ahead by the difference
$12,721.29 - $10,705.60 = $2,015.69

b. If the investment yields only 4%, the last time value calculation will be
FV = PV[FVF4,5] = $5,560.49(1.2167) = $6,765.45,
And Joe will be behind by
$10,705.60 - $6,765.45 = $3,940.15

10. Ralph Renner just borrowed $30,000 to pay for a new sports car. He took out a 60 month loan and his car payments are $761.80 per month. What is the effective annual interest rate (EAR) on Ralph’s loan?

First, calculate the periodic interest rate
n = 60; PV = 30,000; PMT = (761.80); FV = 0 CPT I/Y = 1.5%
EAR = (1 + .015)12 – 1 = .1956 or 19.56%

Annuity Problems

11. How much will $650 per year be worth in eight years at interest rates of
a. 12%
b. 8%
c. 6%

a. FVA = $650 [FVFA12,8] = $650 (12.2997) = $7,994.81
b. FVA = $650 [FVFA8,8] = $650 (10.6366) = $6,913.79
c. FVA = $650 [FVFA6,8] = $650 (9.8975) = $6,433.38

12. The Wintergreens are planning ahead for their son's education. He's eight now and will start college in 10 years. How much will they have to set aside each year to have $65,000 when he starts if the interest rate is 7%?

$65,000 = PMT [FVFA7,10] = PMT (13.8164)
PMT = $4,704.55

13. What interest rate would you need to get to have an annuity of $7,500 per year accumulate to $279,600 in 15 years?

$279,600 = $7,500 [FVFAk,15]
FVFAk,15 = 37.28
k = 12%

14. How many years will it take for $850 per year to amount to $20,000 if the interest rate is 8%? Interpolate and give an answer to the nearest month.

$20,000 = $850 [FVFA8,n]
FVFA8,n = 23.529
n = 13.75 yrs = 13 yrs 9 mos

15. What would you pay for an annuity of $2,000 paid every six months for 12 years if you could invest your money elsewhere at 10% compounded semiannually?

PVA = $2,000 [PVFA5,24] = $2,000 (13.7986)
PVA = $27,597.20

16. Construct an amortization schedule for a four-year, $10,000 loan at 6% interest compounded annually.

$10,000 = PMT [PVFA6,4]
PMT = $2,885.92

Year Beg Bal PMT INT Prin Red End Bal

1 $10,000.00 $2,885.92 $600.00 $2,285.92 $7,714.08
2 7,714.08 2,885.92 462.84 2,423.08 5,291.00
3 5,291.00 2,885.92 317.46 2,568.46 2,722.54
4 2,722.54 2,885.92 163.35 2,722.54 0.00

17. A $10,000 car loan has payments of $361.52 per month for three years. What is the interest rate? Assume monthly compounding and give the answer in terms of an annual rate.

$10,000 = $361.52 [PVFAk,36]
PVFAk,36 = 27.661
k = 1.5%
knom = 18%

18. Joe Ferro's uncle is going to give him $250 a month for the next two years starting today. If Joe banks every payment in an account paying 6% compounded monthly, how much will he have at the end of three years?

= $250 [FVFA.5,24](1.005)
= $250 (25.4320) (1.005) = $6,389.79
which stays in the bank for another year:
FV = $6,389.79 [FVF.5,12]
= $6,389.79 (1.0617) = $6,784.04

19. How long will it take a payment of $500 per quarter to amortize a loan of $8,000 at 16% compounded quarterly? Approximate your answer in terms of years and months. How much less time will it take if loan payments are made at the beginning of each month rather than at the end?

$8,000 = $500 [PVFA4,n]
PVFA4,n = 16
n = 26 quarters = 6.5 years = 6 years 6 months

PVAd = PMT [PVFAk,n](1+k)
$8,000 = $500 [PVFA4,n](1.04)
PVFA4,n = 15.3846
n = 24 1/3 quarters = 6 years 1 month
So it will take about 5 months less time.

20. Ryan and Laurie Middleton just purchased their first home with a traditional (monthly compounding and payments) 6% 30-year mortgage loan of $178,000.
a. How much is their monthly payment?
b. How much interest will they pay the first month?
c. If they make all their payments on time over the 30-year period, how much interest will they have paid?
d. If Ryan and Laurie decide to move after 7 years what will the balance of their loan be at that time?
e. If they finance their home over 15 rather than 30 years at the same interest rate, how much less interest will they pay over the life of the loan?

SOLUTION (using a financial calculator):
a. PV = $178,000; n = 360, I/Y = .5; CPT PMT = $1,067.20
b. Interest in the first payment is the opening loan balance times the monthly rate
$178,000 x .005 = $890
c. Total interest is total payments minus the amount borrowed
Total payments are $1067.20 x 360 = $384,192
And total interest is $384,192 - $178,000 = $206,192
d. The loan balance at any time is the PV of the remaining payments.
After 7 years, the Middletons will have made 7 x 12 = 84 monthly payments which leaves 360-84 = 276 remaining.
n = 276, I/Y = .5, PMT = $1,067.20; CPT PV = $159,558.11
e. The payment on a 15 year loan is
n = 180, I/Y = .5, PV = $178,000; CPT PMT = $1,502.07
Total interest is total payments minus the amount borrowed
Total payments are $1,502.07 x 180 = $270,372.60.
And total interest is $270,372.60 - $178,000 = $92,372.60
The difference is $113,819.40

21. What are the monthly mortgage payments on a 30-year loan for $150,000 at 12%? Construct an amortization table for the first six months of the loan.

$150,000 = PMT [PVFA1,360] = PMT(97.2183)
PMT = $1,542.92

Month Beg Bal PMT INT Prin Red End Bal
1 $150,000.00 $1,542.92 $1,500.00 $42.92 $149,957.08
2 149,957.08 1,542.92 1,499.57 43.35 149,913.73
3 149,913.73 1,542.92 1,499.13 43.79 149,869.94
4 149,869.94 1,542.92 1,498.70 44.22 149,825.72
5 149,825.72 1,542.92 1,498.26 44.66 149,781.06
6 149,781.06 1,542.92 1,497.81 45.11 149,735.95

22. Construct an amortization schedule for the last six months of the loan in Problem 21. (Hint: What is the unpaid balance at the end of 29