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Asset Pricing Under Complete Certainty A. Introduction and Notation Consider a world consisting of the present and an indefinite number of future time periods in which all future events, including securities prices, are known with certainty at the present time. For simplicity, we will assume that no taxes are levied in this world and we will measure time in years. It is currently date 0. Exactly 1 year from today it will be date 1; exactly 2 years from today it will be date 2; and so on. The price of any security S at any date t will be denoted as pSt. The first letter in the subscript will identify which security the price applies to. The final number in the subscript will denote the date at which the price appears. Consider a security X that will deliver to its owner a stream of dollar payouts over the next N years. To keep things simple, I will restrict consideration initially to the case where N=3 so that security X has a maturity of 3 years. Let x0, x1, x2 denote the dollar payouts to be delivered by this security over the next three years. For simplicity we will assume that any security makes payments only once per year and that payments are always made at the end of the year. Suppose security X currently (at date 0) sells for a price px0. Because both the amounts and dates of its future payouts are known with certainty at date 0 , security X is very much like a real world coupon bond with a maturity of 3 years (albeit the annual coupons might vary in size from year to year). Indeed, all securities in a world of certainty look like bonds. Even though perfect certainty is a far cry from reality, the fact that securities in this world are very much like bonds will enable us to learn a lot from this analysis about how the prices of bonds and other fixed income securities (e.g. mortgages) are determined in the real world. (See Bodie 6th ed. Ch. 13, sections 13.1-13.4 and Ch. 14,, section 14.1 or Bodie 5th ed. Ch. 11, sections 11.1 – 11.4 and Ch. 12, section 12.1). In fact, security X differs from a real world bond in only one respect: all future prices of security X are known with certainty, whereas in the real world one cannot know with certainty the price of a 3-period bond at any future date. px1 is the price at which security X will sell at date1 and px2 is the price at which security X will sell at date 2 . We will assume that the payouts x0, x1, x2 are made at the end of each year just prior to the start of the next year. Thus, each future price is the selling price of the security after the preceding year's payout has been made; i.e. px1 is the price of the security at date 1 after the payout x0 has been distributed. If an investor buys security X for price px1 at date 1, he/she is entitled to receive only the future payment stream x1, x2. (Observe that the security expires with the payment x2 = $500 just prior to date 3, so px3 = $0). For purposes of numerical examples, the following values will be assigned to security X. x0 = $100 x1 = $200 x2 = $500 px0 = $708.56

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The values for px1 and px2 will be deduced later. In addition to security X, our numerical example will introduce Treasury Bills with varying terms to maturity. (A T-Bill is a zero coupon bond that makes a one-time payment of its par value only at maturity –see Bodie 6th ed, pp.27-30 or Bodie 5th ed. Pp28-32). The notation p(m)Tt will be used to denote the price at date t of a T-Bill with m years to maturity. For example p(1)T0 is the date 0 price of a T-Bill that will make a one-time payment of its par value at the end of the first year (just prior to date 1). B. Yields, Interest Rates, Present Discounted Values, and Holding Period Returns Definition 1: The yield to maturity (ytm) is a measure of the average annual compound rate of return to be earned on a bond if it is purchased now and held to maturity. For any security S with a maturity of N years, a current price pS0, and known future payouts s0, s1, s2,………,sN-1 to be made over the next N years, respectively, the yield to maturity is the discount rate rytm that solves

.)1(

.......)1()1()1(

13

22

100 N

ytm

N

ytmytmytmS r

srs

rs

rs

p+

+++

++

++

= −

The definition is valid for all maturities N ≥ 1 and applies to all securities in our world of perfect certainty and also to all bonds in the real world --see Bodie pp. 404-405. In the real world the intended future payments (the values for the si's) are known for any bond but, except for federal government bonds, are subject to default risk. (The federal government has the power to print money and will never default). Because of this risk investors will not pay as much for a bond issued by an entity that might default (e.g. a corporation or a Provincial/local government) as they will for a federal government bond with similar maturity and intended future payments. As a consequence the yield to maturity on a "risky" bond will always exceed that on a comparable federal government bond. The difference between the yield to maturity on a "risky" bond and that on a comparable federal government bond is termed a risk premium. The higher the deemed risk of default, the greater will be the risk premium. ________________________________________________________________________ Using the numerical values given earlier, the yield to maturity for security X satisfies

32 )1(500$

)1(200$

)1(100$56.708$

ytmytmytm rrr ++

++

+= .

Solving this equation for the yield to maturity requires that we solve a third-order polynomial equation – not a simple task and one that will yield 3 solution values for rytm. Here two of the three are imaginary numbers (involving the mathematical concept

1−=i ). The one real solution is rytm = 0.05, or a yield to maturity of 5%.

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(Students should be advised that as long as all of the annual payouts are 0 or positive numbers there will always be single real solution value for the yield to maturity of any security. This will always be the case for real world bonds). For reasons that will soon be demonstrated, yield to maturity is not a particularly useful measure for purposes of comparing the performances of alternative bonds. Nonetheless, the measure often appears in bond analysis in the real world, primarily because it is the only return concept that can be computed for coupon bonds. (As we will see, other concepts of rate of return require knowledge of future values for security prices). Look at any table of "bond yields" published daily by the financial press (e.g. The Financial Post) and the figures that appear are yields to maturity of various corporate and government coupon bonds. Or look at the frequently published chart called "the yield curve" (see Bodie, Ch.12, section 12.1) and you will observe a graph showing how the yield to maturity on various government bonds varies with the term to maturity. Suppose that at date 0 there are three Treasury Bills outstanding in our world of certainty, each with the same par value of $1 but with differing dates of maturity and selling prices as described below. T-Bill Type Maturity Date Par Value Price at Date 0 Yield 1-year (m = 1) date 1 $1 p(1)T0 = $0.9615 4.0%

2-year (m = 2) date 2 $1 p(2)T0 = $0.9184 4.35% 3-year (m = 3) date 3 $1 p(3)T0 = $0.8575 5.26% In each case the "yield" shown in the table is the yield to maturity of the T-Bill, computed using the definition given above. The student should apply the ytm definition to verify that the values in the table are correct. Observe each of the three T-Bills has a different yield to maturity and all of the yields differ from the 5% yield to maturity of security X. Because both security X and the 3-year T-Bill have identical maturity dates, one might compare their yields to maturity and conclude that the 3-year T-Bill is the more attractive investment. One is tempted to infer that if an investor invested equal amounts in the 3-year T-Bill and security X at date 0, the T-Bill would yield a greater payoff at date 3. But remember, we are operating in a world of perfect certainty. If the 3-year T-Bill would yield a greater payoff with certainty, then there must be a profitable arbitrage opportunity here: an investor could short sell security X, invest the proceeds in the 3-year T-Bill and earn a certain profit without investing any of his/her wealth. Is that possible here? The answer is no for the following reason. If an investor short sells 1 unit of security X, he/she will receive proceeds of px0 = $708.56 but be obligated to deliver the future payments x0 = $100, x1 = $200, and x2 = $500. The investor can cover the first two of these payments by purchasing (at date 0) 100 of the 1-year T-Bills and 200 of the 2-year T-Bills for a total cost of $279.83. This will leave the investor with $708.56 - $279.83 = $428.73 to invest in 3-year T-Bills. This is exactly sufficient for the investor to purchase 500 of the 3-year T-Bills. Just prior to date 3, the investor will

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receive a total payout of $500 from the 3-year T-Bills but be obligated to pay x2 = $500 to cover the short sale of security X, leaving a net profit of exactly $0. There is no profitable arbitrage opportunity here and an investor should be indifferent between investing in security X or 3-year T-Bills. The illustration underscores the earlier statement that yields to maturity are not particularly useful criteria for comparing different securities. The explanation for this is that the formula used to calculate a yield to maturity implicitly assumes that all payments received prior to maturity can be re-invested at a rate of interest equal to the yield to maturity. This is not generally the case and is certainly not the case in our numerical example. To fully understand why, we need to define what a rate of interest is in this setting. (Students should note that the following definition does not appear in Bodie). Definition 2: The 1-year rate of interest at any date t is the interest rate earned on a 1-year loan that will be repaid as principal plus all accrued interest just prior to date t+1. This is equal to the yield to maturity on a 1-year T-Bill (or zero coupon bond) at date t. The 2-year rate of interest at any date t is the annual interest rate earned on a 2-year loan that will be repaid as principal plus all accrued interest just prior to date t+2. This is equal to the yield to maturity on a 2-year T-Bill (or zero coupon bond) at date t. . . . The N-year rate of interest at any date t is the annual interest rate earned on an N -year loan that will be repaid as principal plus all accrued interest just prior to date t+N. This is equal to the yield to maturity on an N-year T-Bill (or zero coupon bond) at date t. ________________________________________________________________________ We will adopt the following notation: rnt will denote the current n-year rate of interest prevailing at date t. In our numerical example r10 = 0.04, r20 = 0.0435, and r30 = 0.0526; that is, at date 0 the prevailing 1-year rate of interest is 4 %, the prevailing 2-year rate of interest is 4.35 % and the prevailing 3-year rate of interest is 5.26 %. We are now in a position to derive a key result for asset pricing in a world of perfect certainty. Refer back to the arbitrage discussion of the last full paragraph. We argued in effect that owning 1 unit of security X is equivalent to owning a portfolio consisting of 100 one-year T-Bills, 200 two-year T-Bills, and 500 three-year T-Bills, because both security X and that portfolio will deliver the exact same payouts over each of the next three years. Then, for there to be no profitable arbitrage opportunity, the current price of security X must be equal to the date 0 cost of acquiring the portfolio: px0 = 100 × p(1)T0 + 200 × p(2)T0 + 500 × p(3)T0 .

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Making use of our interest rate definitions, we know that the price of any T-Bill is its future payout discounted by the appropriate interest rate. That is,

p(1)T0 = )1(

1$

10r+; p(2)T0 = 2

20 )1(1$r+

; p(3)T0 = 330 )1(1$r+

.

Substituting these relationships into the previous equation yields

(1) 330

22010

0 )1(500$

)1(200$

)1(100$

rrrpx +

++

++

= .

Note that the numerators on the RHS of Equation (1) correspond to the future payouts of security X. Thus, we can re-write the equation as

(2) 330

22

20

1

10

00 )1()1()1( r

xrx

rx

px ++

++

+= .

What Equation (2) says is that the current price of security X is equal to the present discounted value of its future payouts, where the discount rates are current 1-, 2- and 3-year interest rates. While we derived this result using a specific numerical example, the "No Arbitrage" condition requires that Equation (4) be satisfied for any set of payouts x0, x1, x2 and for any set of interest rates r10 , r20 , r30. Further, the result generalizes. Result 1: In a world of perfect certainty asset S has known future payouts s0, s1, s2,………,sN-1 to be made to be made over the next N years , respectively. For any maturity N ≥ 1, the "No Arbitrage" condition requires that the current price the asset be equal to the following present discounted value of those payouts:

NN

NS r

sr

srs

rs

p)1(

............)1()1()1( 0

13

30

22

20

1

10

00 +

+++

++

++

= − .

________________________________________________________________________ One can apply the above equation to determine prices only for federal government bonds. It does not apply in the real world to "risky" bonds issued by corporations or by municipalities and Provincial governments because the discount rates (the rt0 's) do not incorporate the risk premiums commanded by such bonds. Having established Result 1, we can begin to examine future prices and interest rates. Suppose there is an investor with initial wealth of W0 at date 0 who wishes to invest this in a single security and cash in all proceeds one year later at date 1. Given the four securities introduced in our analysis thus far, the investor has four options.

(1) Purchase 0

0

)1( TpW

of the 1-year T-Bills and collect $1 for each just prior to date 1.

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(2) Purchase 0

0

)2( TpW

of the 2-year T-Bills and re-sell each for price p(1)T1, at date1, at

which point they will have 1 year remaining to maturity and be 1-year T-Bills.

(3) Purchase 0

0

)3( TpW

of the 3-year T-Bills and re-sell each for price p(2)T1 at date1, at

which point they will have 2 years remaining to maturity and be 2-year T-Bills.

(4) Purchase 0

0

xpW

units of security X, collect the payout of x0 dollars for each of these at

the end of the year, then re-sell the securities for a price px1 per unit at date 1. The following table shows the date1 payoffs associated with each of the alternatives. ________________________________________________________________________ Option (1) Option (2) Option (3) Option (4)

Dollar Payoff at date 1 0

0 )1(1$

TpW

0

10 )2(

)1(

T

T

ppW

0

10 )3(

)2(

T

T

ppW )(

0

100

x

x

ppx

W+

________________________________________________________________________ Since the initial investment of W0 is the same for all of the options, all of these payoff must be equal; otherwise there exists a profitable arbitrage opportunity. (To see this,

suppose 0

0 )1(1$

TpW <

0

10 )2(

)1(

T

T

ppW . Then an investor could sell short 1-year T-Bills and

invest the proceeds in 2-year T-Bills. The investor would have invested none of her own wealth but would earn a positive profit with certainty at date 1. This would create a huge excess supply of 1-year T-Bills and a huge excess demand for 2-year T-Bills, causing

p(1)T0 to fall and p(2)T0 to rise until 0)1(

1$

Tp =

0

1

)2()1(

T

T

pp ). Equality of the payoffs

implies

(3) 0)1(

1$

Tp =

0

1

)2()1(

T

T

pp =

0

1

)3()2(

T

T

pp = )(

0

10

x

x

ppx +

.

In our numerical example values have been given for all of the date 0 prices appearing in the denominators of Equality (3). We can use those to compute the date 1 values of the prices appearing in the numerators of the last 3 terms. The student shou1d verify that these values are p(1)T1 = $0.9551, p(2)T1 = $0.8918, and px1 = $636.90. We can extend our findings to future date 2 by repeating the arguments that led to Equality (3). If at date 1 an investor wishes to invest her entire wealth in a single asset and cash in all proceeds a year later, the "No Arbitrage" condition requires that all assets yield identical payoffs at date 2. This implies the following:

7

(4) 1)1(

1$

Tp=

1

2

)2()1(

T

T

pp = )(

1

21

x

x

ppx +

.

Only three securities appear in Equation (4) because the 1-year T-Bill of date 0 reaches maturity at date 1 and cease to exist thereafter. The student can use the previously calculated values for the denominator prices to verify that p(1)T2 = $0.9337 and px2 = $466.84.. By similar reasoning, a given amount of wealth invested at date 2 in either the remaining T-Bill or security X must yield identical payouts at date 3, which implies

(5) 2

2

2)1(1$

xT px

p= .

Using the values of the date 2 prices from above, the student should verify that Equation (5) is satisfied in our numerical example. Equalities (3) and (4) and Equation (5) constitute a second main result of our enquiry into asset pricing in a world of perfect certainty, but at this point the result applies only to the four securities identified in our analysis. In order to generalize it we must introduce a new rate of return concept called the "one-year holding period return". Definition 3: If an investor buys a security at some date t, holds it for 1 year, then liquidates all proceeds at date t+1, the investor is said to have a one-year holding period spanning the interval between dates t and t+1. The rate of return earned by the investor over this interval is the security's 1-year holding period return between dates t and t+1. (See Bodie p. 410). For any security S that is purchased for price pSt at date t, delivers a cash payout of st at the end of the year and is then sold at date t+1 for price pSt+1, the 1-year holding period return is the value for rS,t-t+1 that solves the following equation:

St

SttttS p

psr 1

1, )1( ++−

+=+ .

It follows (from Definition 2) that if security S is a T-Bill (or zero coupon bond) that will mature at date t+1, its holding period return over the interval t to t+1 will be equal to r1t, the 1-year rate of interest prevailing at date t. ________________________________________________________________________ The notation introduced in the Definition 3 is a little awkward. In rS,t-t+1 the first letter in the subscript identifies the security S. Following the comma, the interval t -t+1 in the subscript identifies the time interval for which the holding period return applies. Thus, rx,1-2 would denote the 1-year holding period return earned from purchasing security X at date 1 and re-selling the security at date 2. For a T-Bill, we will use the notation r(m)T,t-t+1 to identify the 1-year holding period return on an m-year T-Bill that is purchased at date t and re-sold at date t+1. Thus, r(2)T0-1 denotes the holding

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period return earned by purchasing a 2-year T-Bill at date 0 and re-selling it at date 1 (at which point it will have a remaining maturity of 1 year and be a 1-year T-Bill). While the notation is awkward, the concept of a 1-year holding period return is pretty straight forward. It is simply the rate of return earned on a security that is purchased at some date and sold 1 year later. If the security reaches maturity 1 year after its purchase, then its 1-year holding period rate of return is the same as its yield to maturity, which is, in turn, equal to the 1-year rate of interest. ------------------------------------------------------------------------------------------------------------ Generalization: While we are dealing here only with 1-year holding periods, it should be obvious that we could consider holding periods of any duration. For example, an individual with two years to go until retirement might be interested in the performances of various securities over the next 2 years. We can illustrate how to compute a 2-year holding period return between dates 0 and 2 by using security X. This return measures the average annual rate of return earned by the security between dates 0 and 2 under the assumption that all payouts made by the security between these dates are re-invested at prevailing market interest rates. Security X may be purchased at date 0 for price px0. At the end of the first year the security will deliver the payout x1 which can be re-invested for one year at the prevailing 1-year rate of interest r11. At the end of the second year the security will deliver the payout x2 and immediately thereafter it becomes date 2 and the security may be re-sold for price px2. Therefore, the total liquidation value of the security, including all cumulated earnings from payouts, will be ( x1 (1+r11) + x2 + px2). The 2-year holding period return on the security is the value of rx 0-2 that solves the following

220

221110 )1(

)1(

−++++

=x

xx r

pxrxp .

We were given the values px0 = $708.56, x1 = $100, and x2 = $200, and we later deduced that the value of price at date 2 is px2 = $466.83. In what follows on p. 9 we will also deduce that the value of the 1-year interest rate that will prevail at date 1 is r11 = 0.047. The student should use these figures to verify that the value of the 2-year holding period return is rx 0-2 = 0.0435 (or, 4.35 %). The student should also attempt to derive an expression for the 3-year holding period return for security X between dates 0 and 3. ------------------------------------------------------------------------------------------------------------ Keeping Definition 3 in mind, the student should be able to recognize that each of the individual terms appearing in Equalities (3) and (4), and in Equation (5) is (1 + a holding

period return). For example, the second term in Equality (4) is 1

2

)2()1(

T

T

pp , which 1 + the

1-year holding period return earned on a 2-year T-Bill that is purchased at date 1 and re-sold at date 2 (at which point it has 1 year remaining to maturity). Consequently, we can

replace 1

2

)2()1(

T

T

pp by (1 + r(2)T,1-2). Similarly, we can replace each of the terms in

Equalities (3) and (4) and in Equation (5) by (1+ the appropriate holding period return).

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Then, after canceling the superfluous 1's, we can re-write Equalities (3) and (4) and Equation (5) as follows. (3)' r10 = r(2)T,0-1 = r(3)T,0-1 = rx,0-1 . (4)' r11 = r(2)T,1-2 = rx,1-2 . (5)' r12 = rx,2-3 . Observe that for the first term in each of the above we have used that part of the Definition 3 that states that the 1-year holding period return for any security with exactly 1-year to maturity will be equal to the 1-year interest rate prevailing at the start of the holding period. Equality (3)' says that between dates 0 and 1 the 1-year holding period returns on all existing assets must equal the 1-year rate of interest prevailing at date 0. Equality (4)' says that between dates 1 and 2 the 1-year holding period returns on all existing assets must equal the 1-year rate of interest that will prevail at date 1. And Equation (5)' says that between dates 2 and 3 the 1-year holding period returns on all existing assets must equal the 1-year rate of interest that will prevail at date 2. The student should verify from our numerical example that r11 = 0.047 and r12 = 0.071. We are now in a position to state the general result. Result 2 In a world of perfect certainty, for any 1-year holding period the "No Arbitrage" condition requires that all assets have the same holding period return and this must be equal to the value of the 1-year interest rate that prevails at the start of the holding period. ________________________________________________________________________ (Note that Result 2 does not restrict the 1-year rate of interest to be constant over time. It can vary from one date to the next, as indeed it does in our numerical example). Does Result 2 apply to default risk-free Government of Canada bonds in the real world? No, because for any bond with a maturity longer than 1-year, the one-year holding period return depends on the price at which the security will sell a year into the future. Future prices are known in a world of certainty but they can't be predicted with perfect accuracy in the risky real world. ------------------------------------------------------------------------------------------------------------ Generalization: Result 2 generalizes to any length holding period. Over any specified holding period of length T, all securities must have the same holding period return and this must be equal to the value of the T-year interest rate prevailing at the start of the holding period. The result applies for all T > 0.

10

For a numerical demonstration, recall that in the previous Generalization the 2-year holding period return rx 0-2 was found to be 4.35%. This is equal (as it should be) to the value of the 2-year interest rate r20 given earlier in the table describing T-Bills. ------------------------------------------------------------------------------------------------------------ C. Real versus Nominal Rates of Return Thus far all of the securities we have considered have prices and payouts expressed in dollars. When we compute a security's yield to maturity or its holding period return using its dollar prices and payouts, the result is a nominal return. Nominal returns measure the rates of exchange between dollar amounts at various points in time. If a security has a nominal, 1-year holding period return of r between dates 0 and 1, that tells us that for every $1 we invest in the security at date 0 we will receive $1 ×(1+r) at date 1. In considering buying/selling securities, economic agents are concerned, not with dollar amounts, but with the quantities of goods and services involved in such transactions. When an economic agent buys a security today and redeems it tomorrow, the agent gives up consumption today in exchange for higher consumption tomorrow. Crucial to the agent's decision of whether to undertake such an inter-temporal realignment of consumption is the rate of exchange between current versus future consumption that is offered by a security – what Economists call a security's real rate of return.. (See Bodie, Ch. 5, Section 5.1). We will use the symbol capital R to denote a real rate of return. There is a real return counterpart to each of the nominal return concepts defined earlier in this document. If a security has a real, 1-year holding period return of R, that tells us that for every 1 unit of consumption we give up to invest in the security at date 0 we will receive 1× (1+R) units of consumption at date 1. To formalize the concept of a real rate of return and to also show the relationship that exists between a security's real and nominal returns, we must introduce some new variables. Let PC t denote the price of a consumption good at date t, and πt-t+1 denote the rate of inflation in consumption prices between dates t and t+1. Formally,

Ct

Cttt P

P 11 )1( ++− =+ π .

Now define the real price of an asset as the quantity of consumption goods for which the

asset would exchange. The real price of any asset X at date t is computed as Ct

xt

Pp

.

Similarly, the real value of any nominal payout xt made at the end of a time period is

computed as .1+Ct

t

Px

Observe that the consumption price at date t+1 is appropriate here

because the nominal payout xt is made an instant prior to date t+1. To compute a real rate of return, one simply uses real security prices and real payout in place of their nominal counterparts. As an illustration, the real, 1-year holding period

11

return between dates t and t+1 for security X as the value of Rx t-t+1 that satisfies the following:

Ctxt

Ctxtttxt Pp

PpxR 11

1)(

)1( +++−

+=+ = ]

)(1][

)([

1

1

CtCtxt

xtt

PPppx

+

++ = )1()1( 11 +−+− ++ tttxtr π .

The equation gives the exact relationship between real and nominal returns and the rate of price inflation. For normal (small) values of r and π a close approximation to the exact relationship is Rx t-t+1 ≈ rx t-t+1 - πt-t+1 . The real rate of return is approximately equal to the sum of the nominal return and the rate of price inflation over the one-year holding period. Students should recognize this approximation from discussions of "the real interest rate" in ECO100Y. As a second illustration, the real, 2-year rate of interest at date 0 is denoted by R20 and computed from the price and payoff of a 2-year T-Bill as follows:

(1+R20)2 = )1)(1()1()1()1()2(1$

21102

20202

2000

2−−− +++=++= πππ rr

PpP

CT

C .

As a third illustration, the real yield to maturity of security X is the value of Rytm that solves the following equation.

332

22110

00 )1()1()1( ytm

C

ytm

C

ytm

CCx R

PxRPx

RPx

Pp+

++

++

= .

Like its nominal counterpart, the real yield to maturity is not an especially useful concept for making investment decisions. In a world of perfect certainty Result 1 holds in real as well as nominal terms. That is, the real price of any security is the present discounted value of it future real payouts, discounted using real interest rates. Result 2 also holds in both real and nominal terms. That is, for any holding period, the real holding period returns on all assets must be identical. In the real world, future values for prices of consumption goods are unknown. We can, fox example, determine the real, 1-year holding period rate of return on any asset between dates 0 and 1 only at the latter date, after the value of the consumption price PC1 has been revealed. At date 0, the value of the real return is unknown and subject to risk. Consequently, all real world securities, including default-risk free Government of Canada bonds are "risky assets" insofar as their real rates of return are concerned.

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D. The Relationship between Short- and Long-Term Interest Rates Let us now investigate whether there exists any sort of relationship between short- and long-term interest rates, starting with the current 1-year and 2-year interest rates, r10 and r20, respectively. We know from Definition 2 that the two-year interest rates is computed as

(6) 0

220 )2(

1$)1(Tp

r =+ .

We also know from Result 1 that the current price of any asset is the present discounted value of its future payoffs. If an investor buys a 2-year T-bill at date 0, she need not wait until the T-Bill matures at date 2 to realize a payoff. Instead, she can resell the T-Bill at date 1 and realize an early payoff of p(1)T1 after only one year. Result 1 requires that the current price of the 2-year T-Bill equal the present discounted value of this early payoff, i.e.

(7) )1(

)1()2(10

10 r

pp TT +

= .

Substituting the RHS of Equation (7) for p(2)T0 in Equation (6) yields

(8) 1

102

20 )1(1$)1()1(

Tprr +=+ .

Result 1 also implies that the price of the 1-year T-Bill at date 1 is the appropriate present discounted value of its future payoff, i.e.

(9) )1(

1$)1(11

1 rp T +

= .

Then substituting Equation (9) into Equation (8) yields (10) )1)(1()1( 1110

220 rrr ++=+ .

A similar argument can be used to show that (11) )1)(1)(1()1( 121110

330 rrrr +++=+ .

Current long-term interest rates are geometric averages of the current and future 1-year interest rates. We can verify that Equations (10) and (11) are satisfied in or numerical example. We were given the values r10 = 0.04, r20 = 0.0435, and r30 = 0.0526 in the Table relating to the three T-Bills on page 3 above. Following Definition 3, we deduced (on page 9 above)

13

the values r11 = 0.047 and r12 = 0.071. Verifying Equation (10): we see that (1.0435)2 is indeed equal to (1.04)(1.047) = 1.0888. Verifying Equation (11): (1.0526)3 = 1.1662, which is indeed equal to (1.04)(1.047)(1.071). The results here generalize to any long-term interest rate, which prompts the following. Result 3 For all values of N > 1, the current N-period rate of interest is related to current and future 1-year rates of interest according to )1..().........1)(1)(1()1( 111211100 −++++=+ N

NN rrrrr .

Because this equation must hold for all maturities N, it follows that the 1-year rate of interest that will prevail at any date T ≥ 0 may be computed from

TT

TT

T rr

r)1()1(

)1(0

110

1 ++

=++

+ .

(The notation here is awkward. 10+Tr denotes the (T+1)-year interest rate at date 0). ________________________________________________________________________ This result can not literally apply to interest rates in the (risky) real world because we can only observe the values of current interest rates while the values of future rates are unknown. However, the second equation can be used to compute real-world values of what are called forward interest rates. (See Bodie, Ch. 12, pp 436-39.). I will use some actual data to illustrate this. The following were the values of 1-, 2-, 3-, and 4-year interest rates on zero coupon Government of Canada bonds at the end of March, 2003 (which represents date 0 here). r10 = 0.03493 r20 = 0.038145 r30 = 0.040518 r40 = 0.042284 From the perspective of date 0, an investor could not know what 1-year interest rates would prevail at the ends of March, 2004, 2005, and 2006 (dates 1, 2, and 3, respectively). However, the investor could employ the second equation in Result 3 to compute forward interest rates for each of those future dates. Foe example, the one- year forward rate for the end of March, 2004 is denoted f1 and is computed from

04137.1)03493.1()038145.1(

)1()1(

)1(2

10

220

1 ==++

=+r

rf ,

which yields a value f1 = 0.04137 (or 4.137 %). This tells the prospective investor at date 0 that the 1-year interest rate prevailing at the end of March, 2004 will have to be exactly 4.134 % in order for 1- and 2-year T-Bills to end up with the same 1-year holding period returns between the two dates. In other words, an amount W0 invested in 2-year T-

14

Bills at the end of Mach, 2003 that are to be sold at the end of March, 2004 will yield exactly the same payoff as an amount Wo invested in1-year T-Bills if and only if the 1-year interest rate at the end of March, 2004 turns out to be 4.137 %. If the 1-year rate that prevails at the end of March, 2004 turns out to be greater (less) than 4.137 %, the 2-year T-Bill will have a smaller (greater) payoff that the 1-year T-Bill. The 1-year forward interest rates for the ends of March, 2005 and 2006 are denoted f2 and f3, and computed from

04528.1)038145.1()040518.1(

)1()1(

)1( 2

3

220

330

2 ==++

=+rr

f ; f2 = 0.04528 (or 4.528 %)

04760.1)040518.1()042284.1(

)1()1(

)1( 3

4

330

440

3 ==++

=+rr

f ; f3 = 0.04760 (or 4.76 %)

The forward rate f3 tells the prospective investor that the 1-year interest rate prevailing at the end of March, 2005 will have to be exactly 4.528 % in order for 2- and 3-year T-Bills purchased now to end up with the same 1-year holding period returns between the ends of March 2004 and 2005. Similarly, the forward rate f4 tells the prospective investor that the 1-year interest rate prevailing at the end of March, 2006 will have to be 4.76 % in order for 3- and 4-year T-Bills purchased now to end up with the same 1-year holding period returns between the ends of March 2005 and 2006. Many market analysts use forward interest rates as measures of market participants' expectations of future short-term interest rates. We have just observed that at the end of March, 2003 the situation was that r10 < f1 < f2 < f3 . Analysts at the time interpreted this to mean that investors were anticipating that short-term interest rates would rise over the next three years. You might be curious as to what actually happened. Short-term rates fell between the end of March 2003 and 2004. The 1-year rate of interest prevailing on the latter date was 1.9806 %. Following that, short term rates began to rise. The 1-year interest rate was 2.8756 % at the end of March, 2005 and climbed to 3.9561 % at the March, 2006. It should be obvious that the concept of a forward interest rate is not restricted to 1-year durations. For example, we could calculate the 2-year forward rate of interest prevailing at date 0 via the following

)1()1(

)1(10

3302

2 rr

f yr ++

=+ .

Further elaboration of the relationship that exists between forward interest rates and realized future values of actual interest rates is a topic that is examined in Financial / Monetary Economics under the label: The Term Structure of Interest Rates. This is a

15

topic we will not pursue in this course. The curious student might wish to consult Bodie, Ch. 12, Sections 12.2 – end for an introduction to this topic. E. Valuation of Equities (Materials introduced in this section do not appear directly in Bodie but are presented in CWS, Ch. 2). The holders of common stock in a corporation are the owners of the firm. Suppose a corporation has N shares of equity (or, common stock) outstanding. Holding 1 share gives its owner claim to one-Nth of all the corporation's future earnings. We will assume for simplicity that the corporation pays 100 percent of its earnings out as cash dividends. We will also assume that the corporation is an ongoing enterprise that will operate into the indefinite future. Let D0, D1, D2, ………….. denote the stream of current and future cash dividends that the corporation will pay out to each share of equity. In a world of perfect certainty Results 1, 2 and 3 apply, and so the current price of one share is e0, determined as the present discounted value of current and future cash dividends:

(12) e0 = .....................)1)(1)(1()1)(1()1( 121110

3

1110

1

10

0 ++++

+++

++ rrr

Drr

Dr

D .

If the corporation is to last forever, the number of terms on the RHS of Equation (12) approaches infinity. Let us assume that the corporation will indeed last forever and consider a couple of special cases. Special Case A: Annual cash dividends are constant over time at the value D. The 1-year rate of interest is also constant over time at the value r. Then Equaition (12) becomes

(13) e0 = i

i rD )

11(

1∑∞

= +.

The value of the share price is the product of the constant D and the sum of an infinite,

declining geometric series. [Each term in the summation is )1

1(r+

times the preceding

term, which defines a geometric series. The fact that )1

1(r+

< 1 means that each term is

smaller than its predecessor, so the geometric series is declining as we progress from one term to the next.] Though it will not be proved here, it is relatively easy to deduce that the sum of an infinite, declining geometric series collapses to something quite simple. In particular, it can be shown that

i

i r)

11(

1∑∞

= + =

r1 .

Substituting this into Equation (13), yields the very simple result

16

(14) e0 = rD .

Suppose that D = $1.50 and r = 0.04 (or 4 %). Then the current share price is $37.50. Further, the share price will remain constant over time at this value. Equation (14) also applies to a financial asset known as a consol, which is a coupon bond with an infinite term to maturity. Consider a consol with par/face value of $1000 and a coupon rate of 6 %. Such a security will pay 6 % of $1,000, or $60 per year forever. If we assume that the coupon payments are made at the end of each year and that the 1-year rate of interest is constant over time at a value of, say, r = 0.05, the Equation (14) may be used to determine that the current price of the consol is $1,200.00 and that thia price will remain constant over the indefinite future. Special Case B: Annual cash dividends grow over time at the rate g per year. The 1-year rate of interest is also constant over time at the value r. Let D0 denote the value of the cash dividend to be paid at the end of the first year. Then the dividend to be paid at the end of the second year is D1 = (1+g)D0, the dividend to be paid at the end of the third year is D2 = (1+g)2D0, and so on. For this case valuation Equation (13) may be written as

(15) e0 = i

i rg

gD

)11(

)1( 1

0 ∑∞

= ++

+.

Provided that g < r, the summation above is another infinite, declining geometric series. We will make the assumption that g is indeed smaller than r. It can be shown that

i

i rg )

11(

1∑∞

= ++ =

grg

−+1 .

Substituting this result into Equation (15) yields

(16) e0 = gr

D−

0 .

Updating Equation (16) to the next time period implies that the share price at date 1 will be

e1 = gr

D−

1 = grDg

−+ 0)1(

= (1+g) e0.

Successive updating implies e2 = (1+g) e1, and so on. The price of a share will grow over time at a rate of g per year. For example, suppose the first year dividend is D0 = $2.00, the growth rate of dividends is g = 0.02 (or, 2 % per year), and the 1-year rate of interest is r = 0.05. Then the share

17

price at date 0 is e0 = $66.67. The share price at date 2 will be $68.00, and the price will continue to rise at a 2 % rate into the indefinite future. Let us calculate the 1-year holding period rate of return between dates 0 and 1 for this security. Since Result 3 holds in this setting, we know that re0-1 will have to be equal to the 1-year rate of interest, which is 5 %, but let us verify that.

Applying Definition 3, we can write re0-1 = 0

01

0

0 )(e

eeeD −

+ .

Observe that the holding period return is the sum of two components. The first of these is

0

0

eD

, which is called the dividend yield. Here the dividend yield has a value of 0.03 (or 3

%). The second component is 0

01 )(e

ee −, which represents the proportionate capital

gain/loss on the security. In our example the share price is rising and the proportionate capital gain/loss is +0.02 (or a capital gain of 2 %). The total holding period return is 0.05 (or 5 %), as we knew it had to be. Would a rational investor ever hold a security which has a price that declines over time? The answer is, yes, providing the security has the same holding period return as other securities. We can illustrate that with another numerical example for Special Case B. Suppose the first year dividend is D0 = $5.00, the growth rate of dividends is g = -0.01 (or, minus 1 % per year), and the 1-year rate of interest is r = 0.04. [Note that our derivation of Equation (16) assumes only that g < r; it does not require that g be a positive number]. Then, from Equation (16), the share price at date 0 is e0 = $100. The share price at date 2 will be $99.00, and the price will continue to decline at a rate of 1% per year into the indefinite future. For this security the 1-year dividend yield is 0.05 and the proportionate capital gain/loss is -0.01 (a capital loss of 1%). The capital loss is offset by a high dividend yield, so the total 1-year holding period return is 0.04, which is equal to the 1-year rate of interest as Result 3 requires. In the (risky) real world the value of a corporation's equity is often presumed to be to the present discounted value of the expected value of future dividends, discounted by a risk-adjusted discount rate. Whether this is appropriate and, if so, how the risk-adjusted discount rate is determined are topics to be investigated in detail in the remainder of ECO2503. F. Business Investment in Physical Capital (Materials introduced in this section do not appear directly in Bodie but are presented in CWS, Ch. 2). The objective of business managers is to make decisions that will maximize the value of the enterprise to its owners. In the case of a corporation, decisions by the CEO and other

18

executives should be aimed at maximizing the price of its common shares. This objective applies both to our world of perfect certainty and also to the real world. A decision that managers frequently have to make is to decide whether undertaking a new investment in physical capital will be to the benefit of the firm's owners. The following example provides a simple illustration of the appropriate criteria to apply in making such a decision. Suppose a corporation is considering the purchase of a new piece of machinery at date 0 for a cost of C0 dollars. If the machine is installed, the firm's future net earnings will increase by amounts y0 , y1, y2, and y3 over the next 4 years. After that the machine will be scrapped and will have no effect on earnings further in the future. Should the corporation purchase this machine? It should if and only if the purchase will cause the current share price e0 to rise. We can calculate the impacts of the purchase of the machine on the current share price by examining its impacts on future cash dividends. We will continue to assume that the corporation pays out all net earnings as cash dividends. For simplicity we will assume that the1-year interest rate over the next four years is constant at a value r. We will also assume that there are currently N shares of equity outstanding. Then the change in the current share price from under taking the investment may be determined using Equation (12):

(17) ∆e0 = ])1()1()1()1(

[14

43

32

10

rD

rD

rD

rD

N +∆

++∆

++∆

++∆

,

where ∆Dt denotes the change in cash dividends in year t that will occur if the investment is undertaken, for t = 0, 1, 2, 3. We will evaluate ∆e0 under each of two alternative assumptions about how the firm decides to pay for the cost of acquiring the machine. Assumption 1: The firm pays for the acquisition of the machine by borrowing C0 dollars at date 0. The firm repays this borrowing, plus accrued interest, at the end of the current year by reducing its dividend payout D0 . Under this assumption the acquisition of the machine will have the following impacts on the firm's cash dividends. ∆D0 = y0 – (1+r)C0 ∆D1 = y1 ∆D2 = y2 ∆D3 = y3 Assumption 2: The firm pays for the acquisition of the machine by borrowing C0 dollars at date 0. The firm repays this borrowing, plus all accrued interest, at the end of the fourth year by reducing its dividend payout D3. Under this assumption the acquisition of the machine will have the following impacts on the firm's cash dividends. ∆D0 = y0 ∆D1 = y1 ∆D2 = y2 ∆D3 = y3 – (1+r)4C0

19

Substituting the values for the ∆D's for either assumption into Equation (17) yields the identical result

(18) ∆e0 = ])1()1()1()1(

[104

33

22

10 Cr

yr

yr

yr

yN

−+

++

++

++

.

It is readily apparent from this that the impact of the investment on the share price depends directly upon the sign of the term in square brackets in Equation (18) and is independent of how the firm chooses to finance the initial cost of the investment. [There are financing options other than the two we have looked at but they will all yield Equation (18)]. The term in square brackets defines what is called the net present value (NPV) of the investment. The NPV is the present discounted value of the investment's net payoffs, less the initial cost of the investment. In this example

NPV = 043

32

210

)1()1()1()1(C

ry

ry

ry

ry

−+

++

++

++

.

The investment will make shareholders better off and should be undertaken only if its NPV > 0. If the NPV < 0, undertaking the investment will cause the share price to fall and the investment should not be undertaken. In the rare event that NPV = 0, the investment will have no impact on the share price and the firm should be indifferent between taking it or not. The preceding statements apply to all prospective investment projects being considered by a firm. The general rule is to undertake all investments that have positive NPV's and to avoid all investments that have negative NPV's. Further, in perfectly certain world without taxes, an investment project's NPV is independent of how the initial cost of the project is financed. As an exercise, the student should compute the NPV and determine whether the investment project with the following characteristics should be undertaken when the 1-year rate of interest is constant at the value r = 0.045. C0 = $1,000 y0 = $150 y1 = $375 y2 = $500 y3 = $350 As a brief digression, we can ask: Do firms in the real world use NPV criteria in deciding whether to undertake investment projects? They most certainly do, but with some necessary modifications to the concept of NPV defined above for a world of certainty. In the real world, while the initial cost of an investment project may be known with certainty, its future payoffs are usually subject to some uncertainty. A businessman must determine the expected values of the payoffs and use these in computing the project's expected NPV. The businessman will also use a "risk-adjusted" rate of return instead of the prevailing interest rate to discount these expected future payoffs. In other words, the businessman will add a risk premium to the prevailing (risk-free) interest rate and use the

20

result as the discount rate. This kind of risk-adjusted discount rate is often referred to as "the cost of capital". This topic is investigated in Financial Economics II and will not be further examined in ECO2503H. (However, I will say that economic theory has not yet determined that making investment decisions using expected NPV yields optimal results). Returning to our world of certainty, observe that investing C0 in a piece of machinery at date 0 in order to receive future net earnings of y0 , y1, y2, and y3 over the next 4 years is equivalent to purchasing a security that has price C0 at date 0 and future dollar payouts of y0 , y1, y2, and y3. Among other things, this equivalence means that we can apply Definition 1 and compute a yield to maturity for the investment. It is common practice to refer to the yield to maturity on an investment in physical capital as the investment's internal rate of return (IRR). We saw in an earlier illustration that yield to maturity is not a useful concept to employ in making a decision about whether or not to invest in a particular security. The same naturally applies to IRR. An investment's internal rate of return does not provide reliably useful information regarding whether or not the investment should be undertaken. (This point is forcefully demonstrated in CWS, Ch. 2). G. Consumption & Investment in a World of Perfect Certainty (Materials introduced in this section do not appear in Bodie but are discussed in CWS, Ch. 1). We are now in a position to investigate how households make decisions regarding consumption spending and asset accumulation. For simplicity, the analysis is restricted to certain world consisting of only two time periods. The first time period begins at date 0, the second time period begins a year later at date 1. (At the end of the second time period the world ceases to exist). It is currently date 0 and we consider the actions of a household with current nominal wealth $W0, which is the dollar value of all assets held by the household (net of liabilities) at date 0, plus the present discounted value of any nominal labour income to be earned by the household during the two time periods. The current price of a consumption good is PC0 and the price at date 1 will be PC1. The household must decide how much of $W0 to spend on current consumption and how much to invest in assets to be carried over and spent on future consumption at date 1. (The household will spend everything available on consumption at date 1 because there is no point in buying assets that can't be redeemed until after the world ends at the end of the second year). Because this is a world of perfect certainty, Result 2 applies and all assets have the same holding period returns between dates 0 and 1. Consequently, the household does not have to decide which assets to purchase. One is just as good as another and they will all yield the prevailing 1-year, nominal, rate of interest r. Let C0 and C1 denote the real quantities of consumption in each of the two time periods, respectively. At date 0 the household must choose values for C0 and C1 that satify the following budget constraint: (19) PC1 C1 = (1+r) [ $W0 – PC0 C0 ] . Another way to write this budget constraint is

21

)1(

$ 100 r

PPW C

C ++= ,

which says that the present discounted value of lifetime consumption expenditures must equal the current value of nominal wealth. A third – and in this case more useful -- way to write the budget constraint is to express it in real terms as (19)' C1 = (1+R) [ W0 – C0 ] ,

where R ( = 1)1(

01

−+

CC PPr ) is the current value of the real, 1-year interest rate and

W0 ( = 0

0$

CPW

) is the real value of current wealth.

Economists treat consumption at different points as if they were different goods, which facilitates the application of standard microeconomic techniques. We can say that at date 0 the household has a lifetime utility function defined over C0 and C1 and denoted by U(C0 , C1). A standard representation of this kind of lifetime function is the following: (20) U(C0 , C1) = u( C0 ) + β u( C1 ). In Equation (20), u( C0 ) describes the utility to be derived by the household at date 0 from consumption of the quantity C0 during the first time period. The function u(·) has the properties of a standard utility function; specifically it exhibits positive but diminishing marginal utility, i.e. u' > 0 and u'' < 0. The term u( C1 ) describes the utility to be derived by the household at date1 from consumption of the quantity C1 during the second time period. The parameter β is just a numerical constant that is used to convert utility to be derived at 1 into lifetime utility at date 0. Empirical evidence regarding consumer behavior suggests that β is a number that is slightly smaller than 1. This is usually interpreted to mean that consumers have a "time preference" in favour of current versus future consumption. That is, if offered a free hamburger either today or a year from today, most consumers would to have it today. With the specification of the lifetime utility function, we can describe the household's actions as choosing the (C0 , C1) pair that maximizes lifetime utility, subject to the budget constraint. This is a straightforward optimization problem of the sort students encountered repeatedly in intermediate microeconomics. We can readily solve it by substituting budget constraint (19)' for C1 in Equation (20) and expressing lifetime utility as u( C0 ) + β u((1+R) [ W0 – C0 ] ). Taking the derivative of this function with respect to C0 and setting it equal to 0 gives the following first-order condition (something Economists also call an Euler equation):

22

(21) u'( C0 ) – β (1+R) u'((1+R) [ W0 – C0 ] ) = 0. The optimal value for C0 solves Equation (21) and this solution can be inserted into the budget constraint to determine optimal C1. Let us illustrate this by assuming that the utility function has the specific form u(C) = ln(C). {The student should verify that u' > 0 and u'' < 0 for this logarithmic utility function). Then Euler equation (21) becomes

0))(1(

)1(1

000

=−+

+−

CWRR

Cβ .

This equation, together with budget constraint (19)' yields optimal solution values

00 )1

1( WCβ+

= and 01 )1

)(1( WRCβ

β+

+= .

As an exercise, the student should determine the optimal (numerical) values for the case

where the period utility function u(C) = C1

− , initial real wealth is W0 = 100, R = 0.025,

and β = 0.96. The analysis of this section generalizes to any number of future time periods. Provided that there exist securities that permit the household to transfer consumption between date 0 and any future date T (by buying/selling securities at date 0 and receiving/making their payouts at date T), the household is able to contract for all future consumption and asset holdings at date 0. No future decisions need be made. This requires simply that there exist 1-year bonds that connect each successive pair of dates. When this requirement is satisfied, it is said that we have complete securities markets. Then Result 2 tells us that any additional securities are superfluous and the household is never required to make a decision about which assets to hold because all must yield the same 1-year holding period returns. (The corresponding concept of complete securities markets in the real world is a very important issue that we will examine later in the course). H. But What Determines the Values of Interest Rates? Everything in the entire foregoing has been deduced /derived under the presumption that prevailing values for current and all future 1-year interest rates are givens. That begs the question of what determines the values of these interest rates. To answer that question we need to impose upon our analysis some kind of complete macroeconomic model that describes the environment in which households and other investors are operating – such as the standard IS-LM model of intermediate macroeconomics. The model would impose market-clearing restrictions that ensure that the quantities of securities supplied and demanded at date 0 be equal. Then, for example, the equilibrium value for r10 would be whatever value is required to equate the supply and demand for securities with a 1-year

23

maturity. (A brief discussion of this appears in Bodie 6th ed., Ch. 5, pp.135-139 or Bodie 5th ed. Ch. 5, pp145-149). Addendum: Duration Real world analysts often use the concept of a bond's duration in assessing how the risk of a change in interest rates might affect the bond's price. Duration is a measure of the effective maturity of a bond. Consider a coupon bond selling for current price P, a face value of F, and a maturity of M years. For simplicity we will continue to assume that coupon payment are made at the end of each year. Let the dollar value of the annual coupon be denoted by C. Recall that we can express the relationship between P and the bond's other features via the following, where rytm is the bond's yield to maturity.

(22) .)1(

.......)1()1()1( 32 M

ytmytmytmytm rFC

rC

rC

rCP

++

+++

++

++

=

Observe that the current price P may be considered to be the present discounted value of the individual payments to be received at the end of each year over its M-year maturity. Let w1 denote the proportion of the current price attributable to the payment to be received at the end of the first year, let w2 denote the proportion of the current price attributable to the payment to be received at the end of the second year and so on through wM.

P

rC

w ytm )1(1

+= ,

Pr

C

w ytm2

2

)1( += ,…,

Pr

C

wt

ytmt

)1( += ,…,

Pr

FC

wM

ytmM

)1()(

++

= .

Definition 4 A bond's duration is the average maturity of its cash payouts, computed as follows: MMwwwwwD +++++= ........4321 4321 . Because a coupon bond makes a series of payments prior to maturity, its duration will always be less that its maturity. But for a zero-coupon bond, D is always equal to M. As a numerical example, consider a coupon bond with P = $936.6427, M = 4, F = $1,000, C = $80, and rytm = 0.10 . The student should verify that the values for the wt's are w1 = 0.077647, w2 = 0.070588, w3 = 0.064171, w4 = 0.787552. (These values sum to 1.0. Why?) Then bond's duration is D = 3.5615 years.

24

[Note: Virtually all coupon bonds make semi-annual coupon payments. To compute duration for this kind of payment stream one must measure time in 6 month intervals. That is, t = 1 is 6 months from the present, t = 2 is 1 year from the present, and so on until the bond matures 2M six-month intervals in the future. One must also measure the yield to maturity as a 6-month rate. If the annual yield to maturity is rytm, then the six month yield is determined from (1+rytm)1/2. The resulting value for D will also be in terms of 6-month intervals. For some examples using bonds with semi-annual coupons see Bodie, pp.462-469.] The usefulness of duration is that it can readily be used to compute the sensitivity of a bond's price to changes in interest rates. It can be shown that if a bond's yield to maturity were to change by an amount dr, the proportionate change in the price of the bond is given by

(23) drrD

PdP

ytm )1( +−

= .

(The term )1( ytmr

D+

is sometimes referred to as modified duration and denoted by the

symbol D*). Using the previous numerical example, if medium-term interest rates were to rise by 50 basis points (dr = 0.005), then the bond's yield to maturity could be expected to rise by the same amount. The bond's price would change by the amount

.01619.0)005.0()10.1(

5615.3−=

−=

PdP

In other words, the bond's price would fall by 1.619 percent. ________________________________________________________________________

25

Summary of the Implications of Asset Pricing Under Certainty for Fixed Income Securities 1. Yield-to-Maturity Under Perfect Certainty The yield-to-maturity can be computed for any fixed income security. Two securities with identical payment streams and identical maturities must have identical yields-to-maturity. Under Uncertainty (the Real World) The yield-to-maturity can be computed for any fixed income security. Two securities with identical payment streams and identical maturities need not have identical yields-to-maturity if they have differing degrees of default risk. In general, the higher the default risk, the higher will be the yield-to-maturity. (e.g. a Baa rated bond will generally have a greater yield-to-maturity that a comparable Aaa rate bond). 2. Duration Under Perfect Certainty Duration can be computed for any fixed-income security. Under Uncertainty (the Real World) Duration can be computed for any fixed-income security. 3. Present Discounted Values Under Perfect Certainty The current price of any fixed income security must equal the present discounted value of its future payment stream using interest rates as discount rates. Under Uncertainty (the Real World) Only the current price of any (default risk free) Federal Government Bond will equal the present discounted value of its future payment stream using interest rates on Federal securities as discount rates.

26

4. Holding Period Yields Under Perfect Certainty For any holding period, the yield on any fixed income security may be computed on an ex ante basis. For example, the 1-year holding period yield on a coupon bond maturing in more that 1-year in the future is the value r that solves the following equation:

0

1)1(P

PCr

+=+ ,

where C is the coupon payment to be made at the end of the year, P0 is the current price of the bond and P1 is the price at which the bond will sell 1 year from now. For any holding period, all securities must have identical yields. Under Uncertainty (the Real World) Since future security prices are unknown, it is not possible to compute the ex ante holding period yield for any fixed income security with a maturity that extends beyond the holding period. However, ex post (realized) holding period yields can be computed for any security. For example, the ex post 1-year holding period yield on a coupon bond maturing in more that 1-year in the future may be computed at Date 1 after its selling price has been observed. This ex post yield is the value r' that solves the following equation:

0

1)1(P

PCr

′+=′+ ,

where C is the coupon payment made at the end of the year, P0 is the current price of the bond and P1' is the price at which the bond actually sold at Date 1. On an ex ante basis, the expected holding period return can be computed for any security by using the expected value of its future selling price. For example, the expected 1-year holding period yield on a coupon bond maturing in more that 1-year in the future may be computed at Date 0 as the value E[r] that solves the following equation:

0

1 ][])[1(

PPEC

rE+

=+ .

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5. Real Rates of Return Under Perfect Certainty The real rate of return can be computed ex ante for any security for any holding period. Under Uncertainty (the Real World) Realized real rates of return can computed for any security but only on an ex post basis. For example, the ex post real rate of return over a 1-year holding period on a coupon bond maturing in more that 1-year in the future may be computed at Date 1 after its selling price has been observed and after the price of goods has been observed.. This ex post yield is the value R' that solves the following equation:

))(()1(1

0

0

1

C

C

PP

PPC

R′′+

=′+ ,

where C, P0, P1' are as previously defined and PC0, PC1 denote the observed values for consumption prices at Dates 0 and 1, respectively. Also one may compute the expected real rate of return for any fixed income security and any holding period. For example, the expected real return on a 1-year Treasury Bill with nominal yield of r10 is the value of E[R10] that satisfies the following:

].[)1(])[1(1

01010

C

C

PP

ErRE +=+

E[R10] is often referred to as "the real rate of interest". A frequently used approximation for E[R10] is the following ][][ 101010 −−= πErRE , where ][ 10−πE denotes the expected rate of price inflation between Dates 0 and 1. 6. Future and Forward Rates of Interest Under Perfect Certainty

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All future interest rates are known with certainty but can be expressed in terms of currently observed interest rates. For example, the 1-year rate of interest that will prevail at Date 1 (denoted by r11) is related to the current 1- and 2-year interest rates via

.)1()1(

)1(10

220

11 rr

r++

=+

Under Uncertainty (the Real World) Future interest rates are unknown but observed values of current interest rates can be used to compute forward interest rates. For example, the forward rate of interest for a 1-year Treasury Bill is denoted by f1 and computed as follows:

.)1()1(

)1(10

220

1 rr

f++

=+

Forward rates may be interpreted as "predictions" of future interest rates implicit in the current yield curve. In the preceding example, f1 is a prediction of what the 1-year TBill rate might be at Date 1. Forward rates can be useful in forming certain kinds of investment strategies. ________________________________________________________________________