finance theory by r merton

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

Finance Theory

Robert C Merton

Table of Contents

I Introduction 1

II On the Arithmetic of Compound Interest: The Time Value of Money 8

III On the Theory of Accumulation and Intertemporal Consumption Choice by Households in an Environment of Certainty 34

IV On the Role of Business Firms, Financial Instruments and Markets in an Environment of Certainty 57

V The "Default-Free" Bond Market and Financial Intermediation in Borrowing and Lending 76

VI The Value of the Firm Under Certainty 115

VII The Firm's Investment Decision Under Certainty: Capital Budgeting and Ranking of New Investment Projects 134

VIII Forward Contracts, Futures Contracts and Options 151

IX The Financing Decision by Firms: Impact of Capital Structure Choice on Value 165

X The Investor's Decision Under Uncertainty: Portfolio Selection 185

XI Implications of Portfolio Theory for the Operation of the Capital Markets: The Capital Asset Pricing Model 225

XII Risk-Spreading via Financial Intermediation: Life Insurance 241

XIII Optimal Use of Security Analysis and Investment Management 249

XIV Theory of Value and Capital Budgeting Under Uncertainty 270

XV Introduction to Mergers and Acquisitions: Firm Diversification 287

XVI The Financing Decision by Firms: Impact of Dividend Policy on Value 296

XVII Security Pricing and Security Analysis in an Efficient Market 312

Copyright © 1982 by Robert C Merton These Notes are not to be reproduced without the author’s

written permission All rights reserved

Financial Intermediaries

Labor Markets

Households Consumption

Financial Intermediaries

Labor Markets

Households Consumption

Domain of Finance

This course is an introduction to the theory of optimal financial management of households, business firms, and financial intermediaries For the term "optimal" to have meaning, a criterion for measuring performance must be established For households, it is assumed that each consumer has

a criterion or "utility" function representing his preferences among alternatives, and this set of preferences is taken as "given" (i.e., as exogenous to the theory) This traditional approach to

households and their tastes does not extend to economic organizations and institutions That is,

they are regarded as existing primarily because of the functions they serve instead of functioning primarily because they exist Economic organizations and institutions, unlike households and their

tastes, are endogenous to the theory Hence, in the theory of the firm, it is not a fruitful approach to

treat the firm as an "individual" with exogenous preferences Rather, it is assumed that firms are created as means to the ends of consumer-investor welfare, and therefore, the criterion function for judging optimal management of the firm will be endogenous

In a modern large-scale economy, it is neither practical nor necessary for management to

"poll" the owners of the firm to make decisions Instead certain data gathered from the capital markets can be used as "indirect" signals for the determination of the optimal investment and financing decisions What the labor and product markets are to the marketing, production and

product-pricing managers, the capital markets are to the financial manager Hence, a good financial manager must understand how capital markets work

Since the capital markets are central, it is quite natural to begin the study of Finance with the theory of capital markets To derive the functions of financial markets and institutions, we investigate the behavior of individual households Using portfolio selection theory, the households' demand functions for assets and financial securities are derived to develop the demand side of capital markets Taking as given the supply of available assets (i.e., the investment and financing decisions of business firms), the demands of households are aggregated and equated to aggregate supplies to determine the equilibrium structure of returns of assets traded in the capital market Inspection of the structure of these demand functions leads in a natural way to an introductory theory for the existence and optimal management of financial intermediaries

In the second part of the course, the supply side of the capital markets is developed by studying the optimal management of business firms (given the demand functions of households)

The two elements which make Finance a nontrivial subject are time and uncertainty

Capital investments often require substantial commitments of resources to earn uncertain cash flows which may not be generated before some distant future date It is the financial manager's responsibility to determine under what conditions such investments should be taken and to ensure that sufficient funds will be available to take the investments Because future flows and rates of return are not known with certainty, to make good decisions, the financial manager must have a thorough understanding of the tradeoff between risk and return

While the basic mode of approach has universal application, it should be understood that the assumed environment is the (reasonably) large corporation in a large-scale economy with well-developed capital markets and institutions similar to those in the United States Although the emphasis is on the private sector, most of the analysis can be applied directly to public sector financing and investment decisions However, certain assumptions made in developing the theory (which are quite reasonable in the assumed environment) will require modification before being applied to small businesses with limited access to the capital markets or to foreign countries with significantly different institutional and social structures

Summary of Different Parts of Finance

Households (Personal Finance)

programs

2 Optimal allocation of savings (portfolio selection)

Manufacturing or Business Firms (Corporate Finance)

financial intermediaries]

2 Proper management is to operate the firm in the best interests of the

owners or shareholders

3 The technology or "blueprints" of available projects (including cost

and revenue forecasts) are known either as point values (certainty) or

as probability distributions

To be Determined: 1 An operation criterion for measuring good management

2 Investment decision in physical assets (capital budgeting)

a Which assets to invest in

b How much to invest in total

3 The long-term financing decision

b Capital structure decisions and the cost of capital

4 The short-term financing decision

a Management of working capital and cash

5 Mergers and Acquisitions: Firm diversification

6 Taxation and its impact on 2-5 (above)

Financial Intermediaries (Financial Institutions)

through other financial intermediaries]

2 Proper management is to operate the intermediary in the best

interests of the owners or shareholders

2 How the management of financial intermediaries differs from the

management of business firms

3 Efficient management and measurement of performance

4 The role of market makers

Capital Markets and Financial Instruments (Capital Market Finance)

2 The characteristics of an "efficient" capital market

3 How an efficient capital market permits decentralization of decision

making

4 The role of capital markets as a source of information (or "signals")

for efficient decision making by households and managers of business firms and financial intermediaries

5 The empirical testing of finance theories using capital market

data

Basic Methodology and Approach of the Course

1 How should the system work?

2 Does it work that way?

3 If not, is there an opportunity for improvement (and hence, a profit opportunity)?

4 If you and the market "disagree," then who is right?

Frequently-Used Concepts

Equilibrium: To understand each element of the system, one must frequently analyze the whole

system To do so, we look at the aggregated resultant of the actions of each unit If each unit is choosing the "best" plan possible and the aggregation of the actions implied by these plans are such that the market clears (i.e., supply equals demand for every item), then these "best" plans can be

realized, and the market is said to be in equilibrium In general, it will be assumed that the markets

are in or tending toward equilibrium

Competition: The basic paradigm adopted is that markets operate such that the very best at their

"job" will earn a "fair" return and those that are not will earn a less-than-fair return This is in contrast to the view that anyone can earn a "fair" return and the "smart" people will earn a "super" return In certain situations, it will be assumed that the capital markets satisfy the technical

conditions of pure competition

"Perfect" or "Frictionless" Markets: At times, we will use the abstract concept of a perfect market

That is, there are no transactions costs or other frictions; that there are no institutional restrictions against market transactions of any sort; there are no divisibility problems with respect to the scale of transactions; that equal information is available to all market participants In some cases, actual markets will be sufficiently "close" to this abstraction to use the resulting analysis directly In other cases, it provides a "benchmark" for the study of imperfections

Summary 53-Year Return Experience: Stocks and Bonds (1926–1978)

Source: “Stocks, Bonds, Bills, and Inflation: Historical Returns (1926–1978),”

R.G Ibbotson and R.A Sinquefield, Financial Analysts Foundation (1979)

Type

Average Annual Return

Standard Deviation

Growth of $1000 (Average Compound Return)

Standard Deviation

Growth of $1000 (Average Compound Return)

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

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

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

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

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

V ) r + (1

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

= V

1

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

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

Therefore, the compound value is given by

n n

and (1 + r ) n 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

(II.2) n * =log(2)/log(1 + r) = .69315/log(1 + r)

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 ("Ruleof 72")

and

(II.4) n *≈0.35 + 69/100r ("Ruleof69")

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 (II.1), and solving for V o = V n /(1 + r ) n = x/(1 + r ) n . I.e.,

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

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

(x).

PV

r)

+

(1 n 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

(x)

PV

r)

+

(1 n 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

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, PV n (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

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 $ x t paid at the end of

period t for = 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 $ x t could be made at the end of

period t for each period t, = 1,2, , N. We denote this present value by PV( x 1 , 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,

N.

1,2, ,

= 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

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.,

(II.6)

) x ( PV

=

) x ( PV +

+ ) x ( PV + ) x ( PV

= ) x , , x , x PV(

t t N

1

= t

N N 2

2 1

1 N

2 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

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

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 $ x t at the

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

) x , ,

x

,

x

$PV( 1 2 N today or a claim on the stream of future payments with the schedule of $ x t

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

$ t at the end of period t for = 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( y 1 , y 2 , , y N ) today, and similarly, one would be indifferent between having claim X

or $PV( x 1 , x 2 , , x N ) today Hence to make a choice between having $PV( y 1 , y 2 , , y N )

today or $PV( x 1 , 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( x 1 , x 2 , , x N ) today

Thus, one would prefer claim Y to claim X if PV( y 1 , y 2 , , y N ) > PV( x 1 , x 2 , , x N ), and

would prefer claim X to claim Y if PV( y 1 , y 2 , , y N ) < PV( x 1 , x 2 , , x N ) . Moreover, if the

two present values are equal, then one would be indifferent between the two claims

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 x t 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 x t or y t could not be negative Since the x t 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

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

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

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

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

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

∞

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

(II.11) r c≡log(1 + r)

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

1 2

N

- t r n

0 PV(x , x , , x ) x(t)≈∫ e dt,

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%

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 y(1 + r ) N - 1

year 2's payment will grow to y(1 + r ) N - 2

year 3's payment will grow to y(1 + r ) N - 3

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 =

) r + (1 y

= )

r

+

1 - N

0

= t

=0 t

1 - N

2 ∑ is given by the formula

N -1 t

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

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

S N is called the compound value of an annuity, and [(1 + r ) N - 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 annuity (denoted by AN)? From (II.7), we have that

) r + 1/(1

y

= ) r +

and [1 - 1/(1 + r) N ] /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 + /(1 S

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, 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

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

perpetual annuity or perpetuity where N = ∞, the present value remains unchanged through time, and is given by

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) 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

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

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 N - t , is given by

]/r.

) r + 1/(1

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 $X[(1 + r ) T - 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 compounded interest rate as

continuously-(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

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 * = [y/(y - rx)]/ [1 + r].

loglog

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 deferred saving plan over an ordinary saving plan?

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

N , $y[(1+ r ) - 1]/r.

S From formula (II.16), he can generate a withdrawal plan of

n N

$q = rS /[1- 1/(1+ r ) ] 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) $ q 1 = (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 each year Moreover, the interest )y

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

at the rate r each year on invested money, he only receives rate (1 -τ after tax Again using )rformula (II.15), his total amount accumulated at retirement from the ordinary saving plan, S 2 ,

is $(1 -τ)y[(1 + (1 -τ)r ) N - 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

] ) )r - (1 + 1/(1 - /[1 rS

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 Because property taxes can be deducted from income )r.

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

(II.28)

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

=

) )r - (1 + )PT]/(1 -

(1 + [M + I

=

1

= t

τ

ττ

∑∞

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

=

1

= t

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) where τ τ is his proportional tax rate Therefore, his after tax cash flow is (X-M-PT)(1 - ) + D τ 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

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

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

(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

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

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 V o dollars invested for n periods, then

; V ) r + )(1 r + (1

= V ) r + (1

= V

; V

+ )(1 r + )(1 r + (1

= V

V t t t - 1 t t - 1 t - 2 1 o Hence, the analogous formula to (II.1)

for the compound value is

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

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

(II.37)

n n

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

x PV

= ) x , , x , x PV(

t t t

N

1

= t

t t N

1

= t N 2 1

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

This completes the formal preparation on the time value of money, and, as promised, we now turn to the systematic development of finance theory

CONSUMPTION CHOICE BY HOUSEHOLDS IN AN ENVIRONMENT OF

CERTAINTY

Begin the study of Finance with the analysis of an economy where all future outcomes are known with certainty, but households receive income (their endowments) and consume at different points in time In particular, it is shown how the consumption-saving decision is made and why the introduction of a capital market and financial securities can improve consumer welfare

As was discussed in the Introduction, the major decisions of the financial manager are to choose which (physical) investments to make and to choose the appropriate means for financing them It is assumed that the "correct" policies chosen will be those that maximize some criterion function (or performance index) specified by the firm We prepare for the study of corporate finance by deducing here and in Section IV a rational criterion function for the firm and the management rules which optimize this criterion function in the simplified world of perfect markets and certainty Despite the simplicity of the model relative to the "real" world, the results derived from this model form a basis for the rationalization of the more complex decision rules developed later Hence, while the manifest functions of the analysis are to show how intertemporal allocations are made and to show what role capital markets play in these allocations, an important latent function of the analysis is to provide a foundation for corporate financial theory

We begin the analysis by solving the two-period problem and then extend it in a natural fashion to the general case of many periods

Consumer Behavior: The Two-Period Case

The four assumptions of Section II (A.II.1) - (A.II.4), are maintained throughout the analysis It is further assumed that each consumer has a well-behaved utility function expressing his preferences between current consumption, C0 , and next period's consumption, C1

Because the emphasis is on the intertemporal allocation of consumption, it is assumed that there

is a single consumption good in each period The consumer's utility function is denoted by

U[C0,C1] Because both period's consumptions are considered goods (in contrast to "bads"), it is

assumed that U1[C0,C1] ≡ ∂U[C0,C1]/ ∂C0 > 0 and U2[C0,C1] ≡ ∂U[C0,C1]/∂C1 > 0 By assuming the strict inequality, we rule out the possibility of satiation I.e., consumers will always

strictly prefer more to less of either C0 or C1 We also assume sufficient regularity and

concavity of U to ensure existence of unique interior maximums

consumption, (C0,C1), such that the consumer is indifferent among these alternative combinations i.e., they are curves of equal utility or iso-utility curves Formally, it is the

functional relationship between C0 and C1 such that U[C0,C1] = , U where U is a constant

Figure 1 illustrates the general shape of the indifference curves, and as they are drawn,

1

U

= U

Case 1 The Simplest Capital Market: Pure Exchange

For this case, we assume that there are no means of physical production I.e., there is no way of using the current period's goods to produce additional goods next period However, suppose there does exist a market for trading current period's goods in return for a claim on

goods next period So, an individual can go to the market and exchange current period goods for

"pieces of paper" which, in turn, can be exchanged next period for goods Alternatively, he can

receive current period goods by issuing "pieces of paper" which he must redeem for goods next period In effect, in the former case, he is lending and in the latter, he is borrowing

If, by convention, the price per unit of current period goods is set equal to one (i.e., a unit

of current period goods is numeraire), then the (current) price per unit of next period goods, P,

is the rate of exchange for claims on next period goods in terms of current period goods So, P

units of current period goods can buy a claim on one unit of next period goods In an intertemporal context, this price is also written as P ≡ 1/(1+ r) where r is the rate of interest

Hence, one unit of current goods can be exchanged for (1 + r) units of goods delivered next

period

Figure III.1 Indifference Curves

number of units of current goods he owns and y1 is the number of units of goods that he will receive next period The consumer's current wealth, W0, is equal to the value of his endowment i.e., W0 = y0 + Py1 The consumer's feasible consumption set is the set of all combinations

(C0,C1) which he can afford to buy Thus, if (C0,C1) are in the consumer's feasible consumption set, then the cost of that consumption program, C0 + PC1, can be no larger than his wealth W0 Moreover, as long as a consumer prefers more consumption to less, he would never choose a program which costs less than his wealth Hence, if it is assumed that the consumer will choose the most preferred feasible consumption program, then he will act so as to maximize U[C0,C1] subject to his budget constraint that W0 = C0 + PC1

Substituting for C0 in U from the budget constraint, we can write the consumer choice

* 1

* 1 0 1 1

where (C0*,C1*) is the optimal consumption program Noting that C0* = W0 - PC1*, we can

rewrite (III.3) as

(III.4) ( dC 1 / dC 0 ) U = U * = - 1/P = - (1 + r) ,

where U*≡ U [C0*,C1*] is the maximum feasible value of utility Hence, the optimum occurs atthe point where an indifference curve is tangent to the budget constraint as shown in Figure 2 Note that in arriving at the optimality condition (III.3), we have used assumption (A.II.4) that the consumer acts as a pure competitor or price-taker So, in solving for his most preferred consumption program, the consumer treats the price (or interest rate) as a given number which does not change in response to the different consumption choices that he might make

Figure III.2

In the absence of an exchange market and without physical storage of goods through time, the optimal consumption program for the consumer will simply be to consume current income I.e., Co = yo and C1 = y1. Hence, if the solution to (III.3) yields C *0 ≠ y0 (and therefore, C *1 ≠ y ),1 then the consumer will be better off as a result of the creation of an exchange market Moreover, he can be no worse off because he always has the option not to use the market and choose C0 = y0 and C1 = y1 which is called the autarky point

Even if physical storage of goods is feasible, then in the absence of an exchange market, the feasible consumption choices are constrained to have C0 ≤ y0 That is, physical storage