chpt2 - time value of money

8/19/2019 Chpt2 - Time Value of Money

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II. ON THE ARITHMETIC OF COMPOUND INTEREST: THE TIME VALUE OFMONEY

From our everyday experiences, we all recognize that we would not be indifferent to a

choice between a dollar to be paid to us at some future date (e.g., three years from now) or a

dollar paid to us today. Indeed, all of us would prefer to receive the dollar today. The

assumption implicit in this common-sense choice is that having the use of money for a period of

time, like having the use of an apartment or a car, has value. The earlier receipt of a dollar is

more valuable than a later receipt, and the difference in value between the two is called the time

value of money. This positive time value of money makes the choice among various

intertemporal economic plans dependent not only on the magnitudes of receipts and expenditures

associated with each of the plans but also upon the timing of these inflows and outflows.

Virtually every area in Finance involves the solution of such intertemporal choice problems, and

hence a fundamental understanding of the time value of money is an essential prerequisite to the

study of Finance. It is, therefore, natural to begin with those basic definitions and analytical tools

required to develop this fundamental understanding. The formal analysis, sometimes called the

arithmetic of compound interest, is not difficult, and indeed many of the formulas to be derived

may be quite familiar. However, the assumptions upon which the formulas are based may not be

so familiar. Because these formulas are so fundamental and because their valid application

depends upon the underlying assumptions being satisfied, it is appropriate to derive them in a

careful and axiomatic fashion. Then, armed with these analytical tools, we can proceed in

subsequent sections with the systematic development of finance theory. Although the emphasis

of this section is on developing the formulas, many of the specific problems used to illustrate

their application are of independent substantive importance.

A positive time value of money implies that rents are paid for the use of money. For goods

and services, the most common form of quoting rents is to give a money rental rate which is the

dollar rent per unit time per unit item rented. A typical example would be the rental rate on an

apartment which might be quoted as "$200 per month (per apartment)." However, a rental rate

can be denominated in terms of any commodity or service. For example, the wheat rental rate


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would have the form of so many bushels of wheat rent per unit item rented. So the wheat rental

rate on an apartment might be quoted as "125 bushels of wheat per month (per apartment)."

In the special case when the unit of payment is the same as the item rented, the rental rateis called the own rental rate, and is quoted as a pure percentage per unit time. So, for example, if

the wheat rental rate on wheat were ".01 bushels of wheat per month per bushel of wheat rented,"

then the rental rate would simply be stated as "1 percent per month." In general, the own rental

rate on an item is called that item's interest rate, and therefore, an interest rate always has the

form of a pure percentage per unit time.

Because it is so common to quote rental rates in terms of money, the money rental rate

(being an own rental rate) is called the money interest rate, or simply the interest rate, and the

rents received for the use of money are called interest payments. Moreover, as is well known, to

rent money from an entity is to borrow, and to rent money to an entity is to lend . If one borrows

money, he is a debtor , and if he lends money, he is a creditor .

Throughout this section, we maintain four basic assumptions:

(A.II.1) Certainty: There is no uncertainty about either the magnitude or timing of anypayments. In particular, all financial obligations are paid in the amounts and at

the time promised.

(A.II.2) No Satiation: Individuals always strictly prefer more money to less.

(A.II.3) No Transactions Costs: The interest rate at which an individual can lend in agiven period is equal to the interest rate at which he can borrow in that sameperiod. I.e., the borrowing and lending rates are equal.

(A.II.4) Price-Taker : The interest rate in a given period is the same for a particularindividual independent of the amount he borrows or lends. I.e., the choices madeby the individual do not affect the interest rate paid or charged.

In addition, we will frequently make the further assumption that the rate of interest in each

period is the same, and when such an assumption is made, that common per period rate will be


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denoted by r . Although no specific institutional structure for borrowing or lending is presumed,

the reader may find it helpful to think of the described financial transactions as being between an

individual and a bank. Indeed, for expositional convenience, we will call loans made byindividuals, "deposits."

Compound Interest Formulas

Compound Value

Let V n denote the amount of money an individual would have at the end of n periods if he

initially deposits V o dollars and allows all interest payments earned to be left on deposit (i.e.,

reinvested). V n is called the compound value of V o dollars invested for n periods. Suppose

the interest rate is the same each period. At the end of the first period, the individual would have

the initial amount V o plus the interest earned, ,rV o or 1 o oo= V + = (1+ r) .V rV V If he

redeposits V 1 dollars for the second period at rate r , then

.V )r +(1=]V r)+r)[(1+(1=V r)+(1=V o2

o12 Similarly, at the end of period 1),-(t he will

have V 1-t and redeposited, he will have V )r +(1=V r)+(1=V ot 1-t t at the end of period t .

Therefore, the compound value is given by

(II.1) ,V )r +(1=V on

n

and )r +(1n is called the compound value of a dollar invested at rate r for n periods.

Problem II.1. "Doubling Your Money": Given that the interest rate is the same each period, how

many periods will it take before the individual doubles his initial deposit? This is the same as

asking how many periods does it take before the compound value equals twice the initial deposit

(i.e., V 2=V on ). Substituting into (II.1), we have that the number of periods required, n* , is

given by


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(II.2) r)+(1.69315/ =r)+(1(2)/ =n* logloglog

where "log" denotes the natural logarithm (i.e., to the base e). Two "rules of thumb" used to

approximate n* in (II.2) are:

(II.3) )72"of Rule(" 72/100r n* ≈

and

(II.4) )69"of Rule(" 69/100r +0.35n* ≈

Of the two, the Rule of 69 is the more precise although the Rule of 72 has the virtue of requiringonly one number to remember. Both rules provide reasonable approximations to n

*. For

example, if r equals 6 percent per annum, to one decimal place, the Rule of 72 gives n* = 12.0

years while the Rule of 69 and the exact solution gives n* = 11.9 years. Moreover, in this day of

hand calculators, any more accurate estimates should simply be computed using (II.2). For

further discussion of these rules, see Gould and Weil (1974).

Present Value of a Future Payment

The present value of a payment of $ x, n periods from now, (x),PV n

is defined as the smallest number of dollars one would have to deposit today so that with it and

cumulated interest, a payment of $ x could be made at the end of period n. It is therefore, equal

to the number of dollars deposited today such that its compound value at the end of period n is

$ x. If one can earn at the same rate of interest r per period on all funds (including cumulated

interest) for each of the n periods, then the present value can be computed by setting x=V n in

(II.1), and solving for .)r + x/(1=)r + /(1V =V nn

no I.e.,

(II.5) ,)r + x/(1=(x)PV n

n


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and )r +1/(1n is the present value of a dollar to be paid n periods from now.

If one were offered a payment of $ x, n periods from now, what is the most that he would

pay for this claim on a future payment today? The answer is (x).PV n To see this, suppose that

the cost of the future claim were (x).PV >P n Further, suppose that instead of buying the future

claim, he deposited $P today and reinvested all interest payments for n periods. At the end of

n periods, he would have )r +$P(1n which by hypothesis is larger than $x.=)r +(x)(1PV

nn

I.e., he would have more money at the end of n periods by simply depositing the money rather

than by purchasing the future claim for P. Therefore, he would be better off not to purchase the

future claim.

If one owned a future claim on a payment of $ x, n periods from now, what is the least

amount that he would sell this claim for today? Again, the answer is (x).PV n Suppose that the

price offered for the future claim today were (x).PV


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funds (including cumulated interest) for each of the N periods, then from (II.5) and (II.6), we

have that

(II.7) .)r + /(1 x=) x ,..., x , xPV(t

t

N

1=t

N 21 ∑

As this derivation demonstrates, a claim on a stream of future payments is formally

equivalent to a set of claims with one claim for each of the future payments. As was shown, an

individual would be indifferent between having ) x(PV $ t t today or a payment of x$ t at the

end of period t . It, therefore, follows that he would be indifferent between having

) x ,..., x , x$PV( N 21 today or a claim on the stream of future payments with the schedule of x$ t

paid at the end of period t for . N 1,2,...,=t

As may already be apparent, the present value concept is an important tool for the

solution of intertemporal choice problems. For example, suppose that one has a choice between

two claims: the first, call it "claim Y ," provides a stream of payments of y$ t at the end of

period t for N,1,2,...,=t and the second, call it "claim X ," provides a stream of payments of

x$ t at the end of period t for . N 1,2,...,=t Which claim would one choose? We have

already seen that one would be indifferent between having a claim on stream of future payments

or having its present value in dollars today. So one would be indifferent between having claim Y

or ) y ,..., y , y$PV( N 21 today, and similarly, one would be indifferent between having claim X

or ) x ,..., x , x$PV( N 21 today. Hence to make a choice between having ) y ,..., y , y$PV( N 21

today or ) x ,..., x , x$PV( N 21 today is formally equivalent to making a choice between claim Y

or claim X . But, as long as one prefers more to less, the former choice is trivial to make:

Namely, one would always prefer the larger of ) y ,..., y , y$PV( N 21 or ) x ,..., x , x$PV( N 21 today.

Thus, one would prefer claim Y to claim X if ), x ,..., x , xPV(>) y ,..., y , yPV( N 21 N 21 and

would prefer claim X to claim Y if .) x ,..., x , xPV(


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In the formal notation, both claim X and claim Y had the same number of payments:

namely N . However, nowhere was it assumed that some of the y x t t or could not be zero.

Thus, the timing of the payments need not be the same. Moreover, nowhere was it assumed that

some of the y x t t or could not be negative. Since the y x t t or represent cash payments to

the owner of the claim (i.e., a receipt) a negative magnitude for these variables is interpreted as a

cash payment from the owner of the claim (i.e., an expenditure). Indeed, it is entirely possible for

the present value of a stream of payments to be negative which simply means one would be

willing to make an expenditure and pay someone to take the claim. Hence, the present value tool

provides a systematic method for comparing claims whose schedules of payments can differ

substantially both with respect to magnitude and timing. While our illustration applied it tochoosing between two claims, it can obviously be extended to the problem of choosing from

among several claims. Its use in this intertemporal choice problem can be formalized as follows:

Present Value Rule:

If one must choose among several claims, then proceed by: first, computing the present

values of all the claims. Second, rank or order all the claims in terms of their present values from

the highest to the lowest. Third, if one must choose only one claim, then take the first claim (i.e.,

the one with the highest present value). More generally, if one must choose k claims out of a

larger group, then take the first k claims in the ordering (i.e., those claims with the k largest

present values in the group). This procedure for choosing among several claims is called the

Present Value Rule.

Note that if the rate of interest in every period were zero, then the present value of a

stream of payments is just equal to the sum of all the payments (i.e.,.

N

t 1 2 N

t=1

PV( , ,..., ) = x x x x

)

∑

In this case, the Present Value Rule would simply say "choose that claim which pays one the

most money in total (without regard to when the payments are received)." However, because of


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the time value of money, the interest rate will not be zero, and no such simple rule will apply.

That one cannot rank or choose between alternative claims without taking into account the

specific interest rate available is demonstrated by the following problem:

Problem II.2. Choosing Between Claims: Suppose that one has a choice between "claim X "

which pays $100 at the end of each year for ten years or "claim Y " which provides for a single

payment of $900 at the end of the third year. Given that the interest rate will be the same each

year for the next ten years, which one should be chosen? The Present Value Rule says "Choose

the one with the larger present value." However, as the following table demonstrates, the claim

chosen depends upon the interest rate.

Interest Rate, r Present Value of Claim X Present Value of Claim Y

0% $1000 $9002% 898 8485% 772 7778% 671 71410% 614 67612% 565 641

While the present values of both claims decline as one moves in the direction of higher interestrates, the rate of decline in the present value of Claim Y is smaller than the rate of decline for

Claim X . Hence, for interest rates below 5 percent, one should choose Claim X and for rates

above 5 percent, one should choose Claim Y .

The result obtained here that one claim is chosen over the other for some interest rates

and the reverse choice is made for other interest rates often occurs in choice problems and is

called the switching phenomenon. It is called this because an individual would "switch" his

choice if he were faced with a sufficiently different interest rate. Hence, without knowing the

interest rate, the choice between two claims will, in general, be ambiguous. So, in general,

unqualified questions like "which claim is better?" will not be well posed without reference to

the specific environment in which the choice must be made. Note, however, that for a specified


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interest rate, the present value of each claim is uniquely determined, and therefore the choice

between them at that interest rate level is always unambiguous.

In Problem II.2, it was stressed that, in general, the solution to the problem of choosingamong alternative claims will depend upon the interest rate at which the individual can borrow or

lend. However, it is equally important to stress that the solution depends only upon that interest

rate. Specifically, given that rate of interest, the solution is not altered by the existence of other

claims that an individual owns (i.e., his endowment ). Moreover, the solution does not depend

upon whether he plans to use the payments received for current consumption or to save them for

consumption in the future. That is, the solution does not depend upon the individual's

preferences or tastes for future consumption. While this demonstrated independence of the

solution to either the individual's tastes or endowments has far-ranging implications for the

theory of Finance, further discussion is postponed to Section III where the general intertemporal

choice problem for the individual is systematically examined.

Continuous Compounding

It is not uncommon to see an interest rate quoted as " R% per year, compounded n times

a year." For example, a bank might quote its rate on deposits as "7% per year, compoundedquarterly (i.e., every three months or four times a year)" or "7% per year, compounded monthly

(i.e., every month or twelve times a year)." Provided that funds are left on deposit until the end

of a compounding date, such quotations can be interpreted to mean that n times a year, the

account is credited with cumulated interest earned at the rate, ( R/n), per period of (1/n) years.

The "true" annual rate of interest, call it in, when there are n such compoundings per year can

be derived using the compound value formula (II.1). From that formula, one dollar will grow to

) R/n+$(1n in one year, and therefore,

(II.8) .) R/n+(1=i+1n

n


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By inspection of (II.8), for a given value of R, more frequent compoundings (i.e., larger n)

result in a larger "true" annual interest rate, .in The limiting case of n → ∞ is called

continuous compounding, and the limit of (II.8) is

(II.9) e=i+1 R

∞

where "e" is a constant equal to 2.7183..., and e R

is called the exponential factor . The

difference between the true or effective annual rate i∞ and the stated rate R will be larger, the

larger is R although for typical interest rates, this difference will not be large. For example, at a

stated rate of R = 5%, i∞ = 5.13%. However, the cumulative difference in compound value for

higher interest rates and over several years can be significant as is illustrated in the following

table:

Compound Value of $100 at the End of N Years

At 10% At 10% per Year, N per Year Compounded Continuously

1 $ 110.00 $ 110.52

2 121.00 122.145 161.05 164.87

10 259.37 271.8315 417.72 448.1720 672.75 738.9130 1,744.93 2,008.55

One can, of course, invert the original question and ask "What continuously-compounded

rate, ,r c will produce a "true" annual interest rate, r ?" From (II.9), we have that

(II.10) ,r +1er ≡c

or by taking (natural) logarithms of both sides of (II.10), we can rewrite (II.10) as


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(II.11) .r)+(1r c log≡

In the analysis of interest rate problems, it is frequently more convenient to work with thecontinuously-compounded rate, ,r c rather than the actual rate, r . For example, in Problem II.1,

we derived a formula for the number of periods required to double our money, n*. Substituting

from (II.11) into (II.2), we have that

II.12) .r .69315/ =r (2)/ =n cc* log

If, in addition, one approximates the stream of payments from a claim, , } x{ t by a

continuous stream of payments, {x(t)}, then the discrete-time formula for the present value of a

stream of payments, (II.7), can be approximated by the integral formula,

(II.13) c1 2

N - t r

n

0

PV(x , x ,..., x ) x(t) dt,e≈ ∫

and in some cases, the integral expression in (II.13) provides an easier way to compute formula

for the present value than its discrete-time counterpart in (II.7).

Annuity Formulas

A claim which provides for a stream of payments of equal fixed amounts at the end of

each period for a specified number of periods is called an annuity. Suppose that one owned an

annuity claim which pays $ y at the end of each year for N years. How much money would one

have at the end of year N if payments are immediately deposited in an account which earns r %


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per year (on both cumulated interest and the initial deposit) in each year? Using the compound

value formula, (II.1), we have that:

year 1's payment will grow to )r + y(11- N



. . . . . .. . . . . .. . . . . .year ( N-1)'s payment will grow to r)+ y(1

year N 's payment will grow to y .

Hence, the total amount accumulated, N ,S will be the sum of all N terms. I.e., S N =

.)r +(1 y=)r + y(1t

1- N

0=t

t - N N

1=t

∑∑ To further simplify the formula, we make a brief digression to

develop a mathematical formula. The sum of a geometric progression,

, x= x+...+ x+ x+1 t 1- N

=0t

1- N 2 ∑ is given by the formula

(II.14) . N -1

t

t=0

N x = ( - 1)/(x - 1) x∑

From (II.14), we also have that

(II.14a) 1).-1)/(x- x x(= x N t

N

1=t

∑

Applying (II.14) with r +1= x to the expression for ,S N we can rewrite it as

(II.15) 1]/r.-)r + y[(1=S N

N


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S N is called the compound value of an annuity, and 1]/r -)r +[(1 N

is called the annuity

compound value factor .

Maintaining the assumption that the interest rate is the same each year, what is the present

value of an annuity (denoted by A N )? From (II.7), we have that

.)r +1/(1 y=)r + y/(1= At

N

1=t

t N

1=t

N ∑∑ From (II.14a), we can rewrite the expression for the

present value as

(II.16) ]/r )r +1/(1- y[1= A N

N

and N

[1 - 1/(1+ r) /r ] is called the annuity present value factor .

Formula (II.16) could have been derived by a different (but equivalent) method. From

(II.15), we know that a N -year annuity paying $ y per year is equivalent to a claim which

provides a single payment of S $ N paid at the end of year N . From (II.5), we have that

.)r + /(1S =)S (PV N

N N N But, the present values of two equivalent streams are the same, and

therefore N

N N = /(1 r .)S A + The reader may verify that this is the case by inspection of (II.16).

Note that if one has a N -period annuity at time (t=) zero, then this same claim will

become a ( N -1) period annuity at time 1,=t and at time t , it will be an ( N–t ) period annuity.

Hence, the change in the present value of an N -period annuity over one period is equal to

, A- A N 1- N and from (II.16), can be written as

(II.17) .)r + y/(1-= A- A N

N 1- N

Inspection of (II.17) shows that the present value of an annuity declines each period until at time

t = N (called its expiration date), its present value is zero. Note further that the rate of decline is

larger the closer the annuity is to its expiration date. However, in the special limiting case of a


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perpetual annuity or perpetuity where N = ∞, the present value remains unchanged through

time, and is given by

(II.18) y/r.= A∞

Problem II.3. Mortgage Payment Calculations: Probably the annuity claim with which

households are most familiar is the mortgage which is a specific form of loan used to finance the

purchase of a house. The terms of a standard or conventional mortgage call for the borrower to

repay the loan with interest by making a series of periodic payments of equal size for a specified

length of time. In effect, the house buyer "issues" to the lender (usually a bank) an annuity claim

in exchange for cash today. Typically, the length of time, the periodicity of the payments, and

the interest rate are quoted by the bank. Given this information, one can then determine the size

of the periodic payments as a function of the amount of money to be borrowed. Suppose the

bank quotes its mortgage terms as follows: the length of the mortgage's life or term is 25 years;

the periodicity of the payments is once a year; and the interest rate charged is 8 percent per year.

If the amount of money to be borrowed is $30,000, then what will be the annual payments

required? To solve this problem, we use formula (II.16). The amount of money received in

return for the annuity, $30,000, equals the present value of the annuity, . A N The number of

payments, N , equals 25, and the annual interest rate, r , equals .08. Thus, the required annual

payments, y, are given by the formula

(II.19) ].)r +1/(1- /[1rA= y N

N

The annuity present value factor for r = .08 and N = 25 equals 10.675. Therefore, y =

$30,000/10.675 or approximately $2810 per year.

Although the size of the payments remains the same over the life of the mortgage, the

amount of money actually borrowed (called the principal of the loan) does not. In addition to


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covering interest payments, a portion of each year's payment is used to reduce the principal. In

the example above, during the first year of the mortgage, the amount of money borrowed is

$30,000, and therefore, the interest part of the payment is .08 × $30,000 or $2,400. However,because the total payment made is $2,810, the balance after interest, $410, is used to reduce the

principal. Hence, for the second year in the life of the mortgage, the amount actually borrowed is

not $30,000, but $29,590. The following table illustrates how the level of payments are

distributed between interest payments and principal reduction over the life of the mortgage.

25-Year 8% Mortgage: Distribution of Payments

Interest Payments Principal Reduction Amount of LoanYear Total Payment Amount % of Total Amount % of Total Outstanding

1 $2,810 $2,400 85.4% $ 410 14.6% $29,5902 2,810 2,367 84.2 443 15.8 29,1475 2,810 2,252 80.1 558 19.9 27,58910 2,810 1,990 70.8 820 29.2 24,05215 2,810 1,605 57.1 1,205 42.9 18,85520 2,810 1,039 37.0 1,771 63.0 11,22025 2,810 208 7.4 2,602 92.6 0

Note that early in the life of the mortgage, almost all of the total payment goes for interest

payments. However, by the seventeenth year, the distribution of the payment is approximately

half interest payment and half principal reduction, and as the mortgage approaches its expiration

date, virtually all the payment goes for the reduction of principal.

The general case for the distribution of the payments between interest and principal

reduction can be solved by using formulas (II.16) and (II.17). Because the amount of the

mortgage outstanding always equals its present value, the principal at time t , , A t - N is given by

]/r.)r +1/(1- y[1= At - N

t - N We can rewrite this expression in terms of the initial size of the

mortgage, , A N as


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(II.20) 1].-)r +]/[(1)r +(1-)r +[(1 A= A N t N

N t - N

Moreover, the change in principal between t and 1+t is equal to A- A t - N 1-t - N which from

(II.17) can be written as

(II.21) ,)r + y/(1-= A- At - N

t - N 1-t - N

and the percentage of the total payment used to reduce principal between t and 1+t can be

written as

(II.22) .)r +1/(1=]/y A- A[t - N

1-t - N t - N

Problem II.4. Saving for Retirement : A bank recently advertised that if one would deposit $100

a month for twelve years, then at that time, the bank would pay the depositor $100 a month

forever. This is an example of a regular saving plan designed to produce a perpetual stream of

income later, and frequently arises in analyses of retirement plans. For example, how many years

in advance of retirement should one begin to save $ X a year so that at retirement, one would

receive $C a year forever?

If it is assumed that the annual rate of interest is the same in each year and if one starts

saving T years prior to retirement, then from formula (II.15), a total of 1]/r -)r +$X[(1T

will

have been accumulated by the retirement date. From formula (II.18), it will take $C/r at that

time to purchase a perpetual annuity of $C per year. Hence, the required number of years of

saving is derived by equating the accumulated sum to the cost of the annuity. By taking the

logarithms of both sides and rearranging terms, we have that

(II.23) r],+[1C/X]/ +[1=T loglog

or alternatively, using (II.11), we can rewrite (II.23) in terms of the equivalent continuously-

compounded interest rate as


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(II.24) .r C/X]/ +[1=T clog

Note that for a fixed ratio of C/X , the length of time required is inversely proportional to the

(continuously-compounded) interest rate. So, if that rate is doubled, then the required saving

period is halved. In the special case where C = X , (II.24) reduces to

(II.25) r 0.69315/ =T c

where 0.69315 ≈ log(2). Comparing (II.25) with (II.2), the number of years of required saving is

exactly equal to the number of years it takes to "double your money," and therefore a "quick"solution for T can be obtained by using either the Rule of 72 or the Rule of 69. Applying (II.25)

to the bank advertisement, we can derive the monthly interest rate implied by the bank to be 0.48

percent per month or 5.93 percent per year.

Problem II.5. The Choice Between a Lump-Sum Payment or an Annuity at Retirement : Having

participated in a pension plan, it is not uncommon for the individual to be offered the choice at

retirement between a single, lump-sum payment or a lifetime annuity. Suppose one is offered a

choice between a single payment of $ x or an annuity of $ y per year for the rest of his life.

Given that the interest rate at which he can invest for the rest of his life is r , which should he

choose? Provided that y > rx, the proper choice depends upon the number of years that the

individual will live. Clearly, if he expects to live long enough, then he should choose the

annuity. Otherwise, he should take the lump-sum payment. We can determine the "switch point"

in terms of life expectancy by solving for the number of years, N * , such that the present value

of the annuity is just equal to the lump-sum payment x. Substituting x for A N in (II.16) and

rearranging terms, we have that

(II.25) r].+[1rx)]/ -[y/(y= N * loglog


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Robert C. Merton

26

Hence, if he expects to live longer than N * years, then he should choose the annuity.

Problem II.6. Tax-Deferred Saving for Retirement : Under certain provisions of the tax code,

individuals are permitted to establish tax-deferred savings plans for retirement (e.g., Individual

Retirement Accounts or Keogh Plans). Contributions to these plans are deductible from current

income for tax purposes and interest on these contributions is not taxed when earned. These

plans are called "tax-deferred" rather that "tax-free" because any amounts withdrawn from the

plan are taxed at that time. Suppose that an individual faces a proportional tax rate of τ which is

the same each period and that the interest rate r is the same each period. Further suppose that

he contributes $ y each year to the plan until he retires N years from now at which time he

begins a withdrawal program on an annuity basis for n years. Assuming that his first

contribution to the plan takes place one year from now, what is the economic benefit of the tax-

deferred saving plan over an ordinary saving plan?

Using formula (II.15), his total before-tax amount accumulated at retirement,

is N N , $y[(1+r -1]/r.)S From formula (II.16), he can generate a withdrawal plan of

n N $q = /[1-1/(1+r ])rS per year for n years from this accumulated sum. However, he must

pay taxes of $τq each year on the withdrawals. Hence, the tax-deferred plan will produce an

after-tax stream of payments for n years beginning at retirement of

(II.26) ].)r +1/(1-1]/[1-)r +)y[(1-(1=q$n N

1 τ

If, instead, he had chosen an ordinary saving plan, he would have had to pay $τ y additional

taxes each year during the accumulation period because contributions to an ordinary saving plan

are not deductible. So, without changing his expenditures on other items during the

accumulation period, he could only contribute )y-$(1 τ each year. Moreover, the interest

earned in an ordinary saving plan is taxable at the time it is earned. Therefore, instead of earning


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27

at the rate r each year on invested money, he only receives rate )r -(1 τ after tax. Again using

formula (II.15), his total amount accumulated at retirement from the ordinary saving plan, ,S 2

is )r.-1]/(1-))r -(1+)y[(1-$(1 N τ τ τ Because he has paid the taxes on contributions and

interest along the way, the S $ 2 accumulated is not subject to further tax. However, any interest

earned on invested money during the subsequent withdrawal period is taxed at rate τ. Thus,

from formula (II.16), he can generate an after-tax withdrawal plan of

]))r -(1+1/(1- /[1rS )-(1=q$n

22 τ τ per year for n years which can be rewritten as

(II.27) ].))r -(1+1/(1-1]/[1-))r -(1+)y[(1-(1=q$n N

2 τ τ τ

Clearly, the tax-deferred plan provides a positive benefit because q1 > q2. Inspection of

(II.26) and (II.27) shows that this differential can be expressed in terms of a higher effective

interest rate on accumulations in the tax-deferred plan. Specifically, the tax-deferred plan is

formally equivalent to having an ordinary saving plan where the interest earned is not taxed.

Problem II.7. The Choice Between Buying or Renting a Consumer Durable: For most large

consumer durables (e.g., a house or car), the individual can either choose to buy the good or rent

it. Suppose an individual faces the decision of whether to buy a house for $ I or rent it where the

annual rental charge is $ X per year. If he buys the house, then he must spend $ M for

maintenance and $PT for property taxes each year. These are both included in the rent.

Suppose that the individual faces a proportional tax rate of τ which is the same each period and

that the interest rate r is the same each period. His problem is to choose the method of

obtaining housing services with the lowest (present value of) cost.

The present value of cost equals the discounted value of the after-tax outflows discounted

at the after-tax rate of interest, )r.-(1 τ Because property taxes can be deducted from income


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Robert C. Merton

28

for federal income tax purposes, the after-tax outflow for property taxes each year is )PT.-(1 τ

Hence, the cost of owning the house, PCO, can be written as

(II.28)

)r - M/(1+PT/r + I =

))r -(1+)PT]/(1-(1+[M + I =PCOt

1=t

τ

τ τ ∑∞

where we have assumed that the (properly-maintained) house continues in perpetuity and applied

the annuity formula. Similarly, the cost of renting the house, PCR, can be written as

(II.29)

)r.- X/(1=

))r -(1+ X/(1=PCRt

1=t

τ

τ ∑∞

Hence, if PCR > PCO, then it is better to own rather than rent. Of course, the relationship

between PCR and PCO depends upon the rent charged. In a competitive market, the rent

charged should be such that the landlord earns a return competitive with alternative investments.

Hence, X should be such that the present value of the after-tax cash flows to the landlord equals

the cost of his investment I . The pretax net cash flow to the landlord each year is ( X-M-PT ). In

computing his tax liability, the landlord can deduct depreciation, D, a non-cash item. Hence,

his taxes are (X-M-PT-D) whereτ τ is his proportional tax rate. Therefore, his after

tax cash flow is (X-M-PT)(1 - ) + Dτ τ . Discounting these after-tax cash flows at his after-

tax interest rate, (1 - )r, τ we have that X must satisfy

I = [(X-M-PT)(I - )+ D]/(I - )r τ τ τ or

(II.30) X = rI + M + PT - D/(1 - ).τ τ

From (II.28), (II.29), and (II.30), we have that the cost saving of owning over renting can be

written as


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Finance Theory

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(II.31) PCR - PCO = [I + PT/r]/(1- ) - D/[(1- )(1- )r].τ τ τ τ τ

The advantage to ownership is that one is not taxed on the rent paid to oneself. The disadvantage

is that one cannot take a tax deduction for the (non-cash) depreciation item. So if the

depreciation rate on the property is high or the individual is in a low tax bracket, then renting is

less costly. On the other hand, if property taxes are high and the individual is in a high tax

bracket, then owning is probably less costly.

"Pure" Discount Loan

A pure discount loan calls for the borrower to repay the loan with interest by making a

single lump-sum payment to the lender at a specified future date called the maturity or expiration

date. Hence, unlike an annuity-type loan, there are no interim payments made to the lender. This

form of loan is most common for short maturity loans, and the best known examples are U.S.

Treasury Bills and corporate commercial paper. If it is assumed that the interest rate is the same

each period, then the present value of a discount loan (denoted by D N ) which has a promised

payment of $ M to be paid N periods from now can be written as

(II.32) .)r + M/(1= D N

N

If one has a N -period discount loan at time (t=) zero, then this same loan will become a ( N – 1)

period discount loan at time t = 1, and at time t , it will be a ( N - t ) period discount loan.

Hence, the change in the present value of a N -period discount loan over one period is equal to

, D- D N 1- N and from (II.32), can be written as

(II.33)

N

N -1 N

N

- = rM/(1+ r ) D D= rD .


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Robert C. Merton

30

Inspection of (II.33) shows that unlike an annuity, the present value of a discount loan increases

each period until at t = N , its present value is M . Hence, the amount of money actually

borrowed increases over the life of the loan. The rate of increase each period is the same andequal to the interest rate r .

"Interest-Only" Loans

Another common form for a loan is an "interest-only" loan which calls for the borrower to

make a series of periodic payments equal in amount to the interest payments for a specified

length of time and, in addition, at the end of that length of time, to make a single payment equal

to the initial amount borrowed (i.e., the principal). The periodic payments are called coupon

payments, and the single, lump-sum (or "balloon") payment at the end is called the return of

principal or simply the principal payment . This form of loan is most common for long maturity

loans, and the best known examples are U.S. Treasury Notes and corporate bonds.

The structure of "interest-only" loans is a mixture of the annuity and pure discount forms

of loans. With the exception of the principal payment, the payment patterns are like those of an

annuity because the size of the coupon payments are all the same. Like a discount loan, there is alump-sum payment at the maturity date. However, unlike both the annuity and discount loans,

the amount of the loan outstanding or the principal remains the same throughout the term of the

loan. If it is assumed that the interest rate is the same each period, then the present value of an

interest-only loan (denoted by I N ) which has a coupon payment of $C per period and a

balloon payment of $ M can be written as

(II.34).)r + M/(1+]/r )r +1/(1-C[1=

)r + M/(1+)r +C/(1= I

N N

N t N

1=t

N ∑


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If the initial amount borrowed is $ M and the coupon is set equal to the interest on the amount

borrowed (i.e., C = rM ), then substituting into (II.34), we have that

(II.35) M = I N

independent of N . Hence, the present value of the loan remains the same over the life of the

loan.

Compound and Present Values When the Interest Rate Changes Over Time

To this point, all the formulas were derived using the assumption that the interest rate atwhich the individual can borrow or lend is the same in each period. We now consider the general

case where the interest can vary, and we denote by r t the one-period rate of interest which will

obtain for the period beginning at time (t – 1) and ending at time t . If, as before, V n denotes

the compound value of V o dollars invested for n periods, then

;V )r +)(1r +(1=V )r +(1=V ;V )r +(1=V o12122o11 and

.V )r +)...(1r +)(1r +)(1r +(1=V )r +(1=V o12-t 1-t t 1-t t t Hence, the analogous formula to (II.1)

for the compound value is

(II.36) ( )n

n o

t=1

t 1+ r V V =⎡ ⎤⎢ ⎥⎣ ⎦Π

where "Π" is a shorthand notation for the "product of." I.e.,

( ) ( )( ) ( )( )1 2 1n

t=1

t n n1+ r 1+ r 1+ r ... 1+ r 1+ r .−≡∏ For notational simplicity, we define

the number Rn as that rate such that compounding at that (equal) rate each period for n periods


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Robert C. Merton

32

will give the same compound value as compounding at the actual (and different) one-period

rates. That is,

(II.37)

nn

n t

t=1

(1+ (1+ ),) R r ≡Π

and therefore, R+1 n is the geometric average of the n.1,2,...,=t },r +{1 t Hence, we can

rewrite (II.36) as

(II.38) .V ) R+(1=V on

nn

From (II.38) and the definition of present value, the present value of a payment of $ x, n

periods from now, can be written as

(II.39) ,) R+ x/(1=(x)PV n

nn

and the present value of a stream of payments with a schedule of x$ t paid at the end of period

t , t = 1,2,..., N , can be written as

(II.40)

.) R x

xPV =) x ,..., x , xPV(

t t t

N

1=t

t t

N

1=t

N 21

+ /(1=

)(

∑

∑

Using the formalism of , Rn the compound and present value formulas when interest rates vary

look essentially the same as in the constant interest rate case. However, care should be exercised

to ensure that one does not confuse the " R" n with the ." r " n The former depends upon the

entire path of interest rates from time t = 1 to time t = n while the latter is simply the one-

period rate that obtains between t = n – 1 and t = n. For example, from (II.37), we have that


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Finance Theory

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(II.41) . R

<

=

>

r if onlyand if R

<

=

>

R 1-nn1-nn

Hence, r = R nn if and only if . R= R 1-nn Moreover, R>r 1-nn does not imply that .r >r 1-nn

Further discussion of the relationship between the } R{ t and }r { t is postponed until Section V

where they will be placed in substantive context.

This completes the formal preparation on the time value of money, and, as promised, we

now turn to the systematic development of finance theory.

chpt2 - time value of money

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