II.
ON THE ARITHMETIC OF COMPOUND INTEREST: THE TIME VALUE OF
MONEY
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
8
Finance Theory
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 rate
is 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 any
payments. 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 a
given period is equal to the interest rate at which he can borrow in that same
period. 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 particular
individual independent of the amount he borrows or lends. I.e., the choices made
by 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
9
Robert C. Merton
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 by
individuals, "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
rV o ,
the initial amount V o plus the interest earned,
redeposits
V1
dollars
for
the
second
2
V 2 = (1 + r) V 1 = (1 + r)[(1 + r) V o ] = (1 + r ) V o .
or
period
V 1 = Vo + rV o = (1+ r)V o . If he
at
rate
r,
then
Similarly, at the end of period (t - 1), he will
t
have V t -1 and redeposited, he will have V t = (1 + r) V t -1 = (1 + r ) V o at the end of period t.
Therefore, the compound value is given by
(II.1)
n
V n = (1 + r ) V o ,
n
and (1 + r ) 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 n = 2 V o ). Substituting into (II.1), we have that the number of periods required, n , is
given by
10
Finance Theory
*
n = log(2)/ log(1 + r) = .69315/ log(1 + r)
(II.2)
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)
*
n ≈ 72/100r
(" Rule of 72" )
and
(II.4)
*
n ≈ 0.35 + 69/100r
(" Rule of 69" )
Of the two, the Rule of 69 is the more precise although the Rule of 72 has the virtue of requiring
*
only 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, PV n (x),
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 V n = x in
n
n
(II.1), and solving for V o = V n /(1 + r ) = x/(1 + r ) . I.e.,
(II.5)
n
PV n (x) = x/(1 + r ) ,
11
Robert C. Merton
n
and 1/(1 + r ) 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 PV n (x). To see this, suppose that
the cost of the future claim were P > PV n (x). 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
n
n periods, he would have $P(1 + r ) which by hypothesis is larger than PV n (x)(1 + r ) = $x.
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 PV n (x). Suppose that the
price offered for the future claim today were P < PV n (x). If he sells, then he will have $P
today. Suppose that, instead of selling the future claim, he borrows $ PV n (x) today for one
period. At the end of the first period, he will owe PV n (x) plus interest, rPV n (x), for a total of
(1 + r) PV n (x). If he pays off this loan and interest by borrowing $(1 + r) PV n (x) for another
period (i.e., he "refinances" the loan), then at the end of this (the second) period, he will owe
(1 + r) PV n (x) plus interest, r(1 + r) PV n (x) for a total of (1 + r )2 PV n (x). If he continues to
refinance the loans in the same fashion of n periods, then at the end of period n, he will owe
(1 + r )n PV n (x) or $x which he can exactly pay off with the $x payment from the claim he
owns. The net of these transactions is that he will have received $ PV n (x) initially which by
hypothesis is larger than $P. I.e., he would have more money initially by borrowing the money
"against" the future claim rather than by selling the future claim for $P, and therefore he would
be better off not to sell the future claim.
In summary, if the price of the future claim, P, exceeds its present value, PVn(x), then
the individual would prefer to sell the claim rather than hold it (or if he did not own it, he would
not buy it). If the price of the future claim, P, is less than its present value, PV n (x), then the
12
Finance Theory
individual would prefer to hold it rather than sell it (or if he did not own it, he would buy it).
Therefore, at P = PV n (x), the individual would have no preference between buying, holding, or
selling the future claim. Hence, the present value of a future payment is such that the individual
would be indifferent between having that number of dollars today or having a claim on the future
payment.
Present Value of Multiple Future Payments
The present value of a stream of payments with a schedule of $ xt paid at the end of
period t for t = 1,2,..., 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 $ xt could be made at the end of
period t for each period t, t = 1,2,..., N.
We denote this present value by PV( x1 , x 2 ,..., x N ).
To derive the formula for its present value, we proceed as follows: Suppose that we establish
today N separate bank accounts where in "Account #t," we deposit
PV t ( x t )
dollars,
t = 1,2,..., N. If we let the interest payments accumulate in Account #t until the end of period t,
then the amount of money in the account at that time will equal the compound value of
PV t ( x t ).
By the definition of the present value of a single future payment, we will have just
enough money to make a payment of $ xt at the end of period t by liquidating Account #t. If
we follow this procedure for each of the N separate accounts, then we would be able to make
exactly the schedule of payments required. Hence, the present value of the stream of payments
with this schedule is equal to the total amount of deposits required for these N accounts. I.e.,
PV( x1 , x 2 ,..., x N ) = PV 1( x1 ) + PV 2 ( x 2 ) + ... + PV N ( x N )
(II.6)
N
= ∑ PV t ( x t ) .
t =1
So, the present value of a stream of payments is just equal to the sum of the present values of
each of the payments. Hence, if one can earn at the same rate of interest r per period on all
13
Robert C. Merton
funds (including cumulated interest) for each of the N periods, then from (II.5) and (II.6), we
have that
N
PV( x1 , x 2 ,..., x N ) = ∑ xt /(1 + r ) .
t
(II.7)
t =1
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 $ PV t ( x t ) today or a payment of $ xt at the
end of period
t.
It, therefore, follows that he would be indifferent between having
$PV( x1 , x 2 ,..., x N ) today or a claim on the stream of future payments with the schedule of $ xt
paid at the end of period t for t = 1,2,..., N .
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 t = 1,2,..., N, and the second, call it "claim X," provides a stream of payments of
$ xt
at the end of period t for t = 1,2,..., N . 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 $PV( y1 , y 2 ,..., y N ) today, and similarly, one would be indifferent between having claim X
or
$PV( x1 , x 2 ,..., x N ) today. Hence to make a choice between having $PV( y 1 , y 2 ,..., y N )
today or $PV( x1 , x 2 ,..., x N ) 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 $PV( y 1 , y 2 ,..., y N ) or $PV( x1 , x 2 ,..., x N ) today.
Thus, one would prefer claim Y to claim X if PV( y 1 , y 2 ,..., y N ) > PV( x1 , x 2 ,..., x N ), and
would prefer claim X to claim Y if PV( y 1 , y 2 ,..., y N ) < PV( x1 , x 2 ,..., x N ) .
Moreover, if the
two present values are equal, then one would be indifferent between the two claims.
14
Finance Theory
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 xt or y t could not be zero.
Thus, the timing of the payments need not be the same. Moreover, nowhere was it assumed that
some of the xt or y t could not be negative. Since the xt or y t 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 to
choosing 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
N
stream of payments is just equal to the sum of all the payments (i.e., PV( x 1 , x 2 ,..., x N ) = ∑ x t ).
t=1
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
15
Robert C. Merton
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
0%
2%
5%
8%
10%
12%
Present Value of Claim X
$1000
898
772
671
614
565
Present Value of Claim Y
$900
848
777
714
676
641
While the present values of both claims decline as one moves in the direction of higher interest
rates, 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
16
Finance Theory
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 choosing
among 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, compounded
quarterly (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
$(1 + R/n )n in one year, and therefore,
(II.8)
1 + i n = (1 + R/n )n .
17
Robert C. Merton
By inspection of (II.8), for a given value of R, more frequent compoundings (i.e., larger n)
The limiting case of n → ∞ is called
in .
result in a larger "true" annual interest rate,
continuous compounding, and the limit of (II.8) is
1 + i∞ = eR
(II.9)
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
N
At 10%
per Year
1
2
5
10
15
20
30
$ 110.00
121.00
161.05
259.37
417.72
672.75
1,744.93
At 10% per Year,
Compounded Continuously
$ 110.52
122.14
164.87
271.83
448.17
738.91
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)
er c ≡ 1 + r ,
or by taking (natural) logarithms of both sides of (II.10), we can rewrite (II.10) as
18
Finance Theory
r c ≡ log(1 + r) .
(II.11)
In the analysis of interest rate problems, it is frequently more convenient to work with the
continuously-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
*
n = log(2)/ r c = .69315/ r c .
II.12)
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,
N
PV(x1 , x2 ,...,xn ) ≈ ∫ x(t)e - r ctdt,
(II.13)
0
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%
19
Robert C. Merton
per year (on both cumulated interest and the initial deposit) in each year? Using the compound
value formula, (II.1), we have that:
N -1
year 1's payment will grow to y(1 + r )
N -2
year 2's payment will grow to y(1 + r )
N -3
year 3's payment will grow to y(1 + r )
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. .
.
year (N-1)'s payment will grow to y(1 + r)
year N's payment will grow to y .
Hence, the total amount accumulated, S N , will be the sum of all N terms. I.e., SN =
N
∑ y(1 + r )
N -1
N -t
t =1
develop
a
= y ∑ (1 + r ) . To further simplify the formula, we make a brief digression to
t
t =0
mathematical
formula.
The
sum
of
a
geometric
N -1
1 + x + x2 + ... + x N -1 = ∑ xt , is given by the formula
t=0
N -1
(II.14)
∑ x = (x
t
N
- 1)/(x - 1).
t=0
From (II.14), we also have that
N
(II.14a)
∑ x = x( x
t
N
- 1)/(x - 1).
t =1
Applying (II.14) with x = 1 + r to the expression for S N , we can rewrite it as
(II.15)
N
S N = y[(1 + r ) - 1]/r.
20
progression,
Finance Theory
SN
N
is called the compound value of an annuity, and [(1 + r ) - 1]/r
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
N
annuity
(denoted
by
AN )?
From
(II.7),
we
have that
N
A N = ∑ y/(1 + r ) = y ∑ 1/(1 + r ) .
t
t =1
t
From (II.14a), we can rewrite the expression for the
t =1
present value as
N
A N = y[1 - 1/(1 + r ) ]/r
(II.16)
N
and [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
N
PV N ( S N ) = S N /(1 + r ) . But, the present values of two equivalent streams are the same, and
N
therefore A N = S N /(1 + r ) . 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 t = 1, 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
AN -1 - AN , and from (II.16), can be written as
(II.17)
N
AN -1 - A N = - y/(1 + r ) .
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
21
Robert C. Merton
perpetual annuity or perpetuity where N = ∞, the present value remains unchanged through
time, and is given by
A∞ = y/r.
(II.18)
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, AN .
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)
y = rA N /[1 - 1/(1 + r )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
22
Finance Theory
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
Year
1
2
5
10
15
20
25
Interest Payments
Amount % of Total
Principal Reduction
Amount % of Total
$2,400
2,367
2,252
1,990
1,605
1,039
208
$ 410
443
558
820
1,205
1,771
2,602
Total Payment
$2,810
2,810
2,810
2,810
2,810
2,810
2,810
85.4%
84.2
80.1
70.8
57.1
37.0
7.4
14.6%
15.8
19.9
29.2
42.9
63.0
92.6
Amount of Loan
Outstanding
$29,590
29,147
27,589
24,052
18,855
11,220
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, AN -t , is given by
A N - t = y[1 - 1/(1 + r )
N -t
]/r.
We can rewrite this expression in terms of the initial size of the
mortgage, AN , as
23
Robert C. Merton
N
t
N
A N - t = A N [(1 + r ) - (1 + r ) ]/[(1+ r ) - 1].
(II.20)
Moreover, the change in principal between t and t + 1 is equal to AN -t -1 - AN -t which from
(II.17) can be written as
N -t
AN -t -1 - AN -t = - y/(1 + r ) ,
(II.21)
and the percentage of the total payment used to reduce principal between t and t + 1
can be
written as
[ AN -t - AN -t -1 ]/y = 1/(1 + r )N -t .
(II.22)
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
T
saving T years prior to retirement, then from formula (II.15), a total of $X[(1 + r ) - 1]/r 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)
T = log [1 + C/X]/ log [1 + r],
or alternatively, using (II.11), we can rewrite (II.23) in terms of the equivalent continuouslycompounded interest rate as
24
Finance Theory
(II.24)
T = log [1 + C/X]/ r c .
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)
T = 0.69315/ r 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 AN in (II.16) and
rearranging terms, we have that
(II.25)
*
N = log[y/(y - rx)]/ log[1 + r].
25
Robert C. Merton
*
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 taxdeferred saving plan over an ordinary saving plan?
Using formula (II.15), his total before-tax amount accumulated at retirement,
N
S N , is $y[(1+ r ) - 1]/r.
From formula (II.16), he can generate a withdrawal plan of
$q = rS N /[1- 1/(1+ r )n ] 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)
$ q1 = (1 - τ )y[(1 + r )N - 1]/[1 - 1/(1 + r )n ].
If, instead, he had chosen an ordinary saving plan, he would have had to pay $τy 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 $(1 - τ )y
each year. Moreover, the interest
earned in an ordinary saving plan is taxable at the time it is earned. Therefore, instead of earning
26
Finance Theory
at the rate r each year on invested money, he only receives rate (1 - τ )r after tax. Again using
formula (II.15), his total amount accumulated at retirement from the ordinary saving plan, S 2 ,
N
is $(1 - τ )y[(1 + (1 - τ )r ) - 1]/(1 - τ )r.
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
$ q 2 = (1 - τ ) rS 2 /[1 - 1/(1 + (1 - τ )r )n ] per year for n years which can be rewritten as
(II.27)
$ q 2 = (1 - τ )y[(1 + (1 - τ )r )N - 1]/[1 - 1/(1 + (1 - τ )r )n ].
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, (1 - τ )r. Because property taxes can be deducted from income
27
Robert C. Merton
for federal income tax purposes, the after-tax outflow for property taxes each year is (1 - τ )PT.
Hence, the cost of owning the house, PCO, can be written as
∞
PCO = I + ∑ [M + (1 - τ )PT]/(1 + (1 - τ )r )
(II.28)
t
t =1
= I + PT/r + M/(1 - τ )r
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
∞
PCR = ∑ X/(1 + (1 - τ )r )t
(II.29)
t=1
= X/(1 - τ )r.
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)τ where τ
is his proportional tax rate. Therefore, his after
tax cash flow is (X-M-PT)(1 -τ ) + τ D . Discounting these after-tax cash flows at his aftertax
interest
rate,
(1 - τ )r, we
have
that
X
must
satisfy
I = [(X-M-PT)(I - τ )+τ D]/(I - τ )r or
(II.30)
X = rI + M + PT - τ D/(1 - τ ).
From (II.28), (II.29), and (II.30), we have that the cost saving of owning over renting can be
written as
28
Finance Theory
(II.31)
PCR - PCO = τ [I + PT/r]/(1 - τ ) - τ D/[(1 - τ )(1 - τ )r].
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)
N
D N = M/(1 + r ) .
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 N -1 - D N , and from (II.32), can be written as
(II.33)
D N -1 - D N = rM/(1+ r )
= rD N .
29
N
Robert C. Merton
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 and
equal 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 a
lump-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
N
(II.34)
t
N
I N = ∑ C/(1 + r ) + M/(1 + r )
t=1
= C[1 - 1/(1 + r )N ]/r + M/(1 + r )N .
30
Finance Theory
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
IN=M
(II.35)
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 at
which 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
Vo
dollars
invested
for
n
periods,
then
V 1 = (1 + r 1 ) V o ; V 2 = (1 + r 2 ) V 1 = (1 + r 2 )(1 + r 1 ) V o ; and
V t = (1 + r t )V t -1 = (1 + r t )(1 + r t -1 )(1 + r t - 2 )...(1 + r 1 )V o . Hence, the analogous formula to (II.1)
for the compound value is
(II.36)
⎡ n
⎤
=
1+
r
(
)
V n ⎢Π
t ⎥V o
⎣ t=1
⎦
where "Π" is a shorthand notation for the "product of." I.e.,
n
∏ (1+ rt ) ≡ ( 1+ r1 )( 1+ r2 ) ... ( 1+ rn−1 )(1+ rn ) .
For notational simplicity, we define
t=1
the number R n as that rate such that compounding at that (equal) rate each period for n periods
31
Robert C. Merton
will give the same compound value as compounding at the actual (and different) one-period
rates. That is,
n
(1+ R n ) ≡ Π (1+ r t ),
n
(II.37)
t=1
and therefore, 1 + R n is the geometric average of the {1 + r t }, t = 1,2,..., n.
Hence, we can
rewrite (II.36) as
n
V n = (1 + R n ) V o .
(II.38)
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
n
PV n (x) = x/(1 + R n ) ,
(II.39)
and the present value of a stream of payments with a schedule of $ xt paid at the end of period
t, t = 1,2,...,N, can be written as
N
PV( x1 , x 2 ,..., x N ) = ∑ PV t ( xt )
(II.40)
t =1
N
= ∑ x /(1+R
t
t
t
).
t =1
Using the formalism of R n , 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 " .
entire path of interest rates from time t = 1 to time
The former depends upon the
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
32
Finance Theory
(II.41)
>
R n = R n -1
<
if and only if
Hence, R n = r n if and only if R n = R n-1 .
>
r n = R n -1 .
<
Moreover, r n > R n-1 does not imply that r n > r n -1 .
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.
33