Present Value and Bond Valuation Professor Thomas Chemmanur

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Present Value and Bond Valuation

Professor Thomas Chemmanur

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Review of Present Value Computation

All of you must be somewhat familiar with computing present values from previous classes.

Computing present value is at the core of all valuation problems in finance. To ensure you are comfortable with present value computation, I will therefore briefly review this material.

i. Future value of a single amount

where F is the future value of an investment P after n periods, at a rate of return r per period. The term (1 + r)n is tabulated, and is often denoted by FVIFn,r.

(1) (1 )nF P r

(2) (1 )n

FPr

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ii. Present value of a single amount.where P is the present value of a single amount F to be received n periods from now, if the opportunity cost of capital is r per period. The term [1/(1 + r)n] is tabulated, and is often denoted by PVIFn,r.

iii. Future Value of an annuity. An annuity is a stream of cash flows

where the cash flow in each period is the same. Consider the following three period annuity;

t = 1 t =2 t = 3t = 0Date:

Cash Flow: A AA

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Note: Throughout this course, we will use this distinction between dates and periods.

Let us first compute the future value (i.e., the value at the end of the last period: i.e., at t=3) of this three period annuity. Using the formula (1) on each individual cash flow, and adding value at t = 3 (remember, we can add values at the same date):

F = A + A(1 + r) + A(1 + r)2 Notice that the index on the last term is 3 -

1 = 2. Generalizing to the n period case, the future value of an n period annuity is,

F = A + A(1 + r) + A(1 + r)2 + A(1 + r)3 +.....+ A(1 + r)n - 1. = A [ 1 + (1 + r) + (1 +r)2 + (1 + r)3 +....+ (1 + r)n-1 ]

55

Review of Present Value Computation The series inside the square brackets is a

geometric series, with common ratio (1 + r). Making use of the formula for the sum of a geometric series, the term in the square brackets can be shown to be equal to Thus,

The term in square brackets is tabulated, and is often denoted by FVIFAn,r.

iv. Present value of an annuity.Using the same three period cash flow stream above as an example, the present value of the three period annuity (i.e., value at t=0) is,

(1 ) 1[ ]nr

r

(1 ) 1(3) [ ]nrF A

r

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Generalizing to the n period case as before,

The series in the square brackets is a geometric series with common ratio 1/(1 + r). Using the formula for the sum of a geometric series, this series sums to

2 3(4) (1 ) (1 ) (1 )

A A APr r r

2 3(5) .....(1 ) (1 ) (1 ) (1 )n

A A A APr r r r

2 1

1 1 1 1(6) ( ) [1 ..... ]1 (1 ) (1 ) (1 )nP A

r r r r

11(1 )(1 )

nrrr

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This gives,

The term in square brackets is tabulated, and is denoted by PVIFAn,r.

If the number of periods, n, goes to infinity, (7) can be shown to reduce to

which is the present value of a perpetual annuity or perpetuity.

11A 1(1 )(7) = 1r (1 )

n

nrP A

r r

(8) APr

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v. Compounding and discounting more than once in a given period.Sometimes the period over which the data is given may not be the period over which compounding is done. Eg. Let R be the interest rate per year, T be the number of years.

a) Future value of an amount P compounded half-yearly after T years at R is given by converting this data to correspond with the compounding period: n = number of effective periods = 2T in this example. r = interest rate per effective period = R/2. Then the future value is given by plugging in these values into (1). Generalizing to the case where compounding is done m times a year, r = R/m, and n = mT.Using these in (1),

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b) Similarly, the present value of an amount F, T years in the future, if discounting is done m times a year at a rate of return of R per year is,

vi. Continuous compounding and discounting.Consider the case when m goes to infinity, i.e., compounding is done continuously. Taking the limit of (9) as ,

(9) [1 / ]mTF P R m

(10) [1 / ]mT

FPR m

m

/

(11)

lim (1 / ) 2.718... e is the base of the natural logarithms.

RT

m Rm

F Pe

where e R m

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The present value of an amount F to be received T years from now, where discounting is done continuously at the rate of return R per year is easily obtained from (11) as,(12) RTP Fe

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Appendix: Sum of a geometric series.

Consider a series of terms of the form 1, k, k2, k3,....kn-1.

Each successive term is k times the previous one; k is therefore called the common ratio. The sum of all the above terms can be shown to be equal to,

S = 1 + k + k2 + k3 +...+ kn-1 = [(kn - 1)/(k-1)]. Whenever we see a series of the above form, we can

sum it using the above formula, by setting k equal to the common ratio of that particular series.

Consider now the case where the number of terms in the series,. If k > 1, each successive term is larger than the previous one, and . However, in the special case where k < 1, we can show that the sum is given by, S = 1/(1 - k).

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Bond Valuation

A bond represents borrowing by firms from investors. F Face Value of the bond (sometimes known as "par

value"); this is the amount the company will pay back at the maturity of the bond.

Ct Coupon or interest payment at date t. Usually coupon amounts are the same at each date, in which case we will simply use C for coupon.

A coupon is often expressed as a percentage c of the face value. This percentage is called "coupon rate". Then, C = c•F, where c is expressed in decimals. Notice that the coupon rate is specified as part of the bond (it is not something that changes according to economic conditions).

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Bond Valuation

n The number of periods to maturity. r The rate of return per period at which the bond cash flows

are to be discounted, determined by financial market conditions. For the present, we will assume that cashflows at all dates are to be discounted at the same rate. r changes as the general level of interest rates in the economy changes.

The price that you are willing to pay for the bond is simply the present value of all the cash flows that the bond entitles you to; Thus,

31 22 3 .....

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

n

C C FC CPr r r r

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Bond Valuation

Usually coupon payments are made semi-annually; this means that the length of a period is a half-year, and r should be in terms of return per half-year; n should also be in half-years.

If the price of the bond is less than the face value, we say that the bond is selling at a "discount"; if they are the same, the bond is selling at par; if the price is higher than the face value, it is selling at a "premium".

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Example

If Teletron Electronics has a bond outstanding that pays a $75 coupon semi-annually, what is its price? The annual interest rate = 10%; 15 years to maturity.

r = 10/2 = 5% n = 15*2 = 30 PERIODSP = 75 (PVIFA 5%, 30) + 1000(PVIF 5%, 30) = 75 (15.3725) + 1000(0.2314) = $1384.34

There are several rules about bond pricing that can be kept in mind.

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Some Rules of Bond Valuation

• (i) WHEN c = r , THE BOND SELLS AT PAR• (ii) WHEN c > r, THE BOND SELLS AT A PREMIUM

(FOR n > 0); WHEN c < r, THE BOND SELLS AT A DISCOUNT (FOR n > 0);

• (iii) WHEN r , PRICE FALLS; AND WHEN r , PRICE INCREASES.

• (iv) THE LONGER THE MATURITY, THE MORE SENSITIVE THE BOND PRICE TO INTEREST RATE CHANGES.

• (v) AS n 0 (THE BOND APPROACHES MATURITY), P F (FACE VALUE); AT n = 0, P = F.

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Summary of Bond Valuation Rules

r < c

r = c

r > c

F

0

Bond Price, p

Time to Maturity, n

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Yield to Maturity

The yield to maturity of a bond is the rate of return we would earn if we bought the bond at a price P and held it till it matures.

That is, it is that rate of return r that solves the bond pricing equation for a known P. Clearly YTM will be different for different values of P. (Usually, yield to maturity is computed and quoted in the financial press assuming semi-annual compounding).

The assumption here is that you are able to re-invest all coupons you receive also at the rate of return r.

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What determines the yield of bonds? 1. Real Interest Rate 2. Inflation Rate 3. Time to Maturity

Real Interest Rate ( rc )Is defined as the interest rate that would prevail in a world without inflation. It depends on:

(a) Investor Preferences (current vs. future consumption) (b) Production Opportunities

Inflation Rate (p) Inflation exists in most economies, measured by:

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What determines the yield of bonds?

p = (Ratio of PRICES) – 1 E.g. Consumer Price Index

Relationship between real and nominal (money) interest rates. (Fisher’s Theory)

(1 + r) = (1 + rc) (1 + p)

PRICE OF A BASKET OF GOODS NEXT YEAR(1 + ) = PRICE OF SAME BASKET THIS YEAR

p

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Term Structure of Interest Rates

Often, the discounting rate to be applied to a one year loan is different from that for a two year loan, etc.

The relationship between the length of a loan (or, to talk in terms of a bond, the time to maturity) and the rate of return you can earn on a loan (the yield to maturity in the case of a bond) is called the term-structure of interest rates or the yield curve.

To make this relationship clear, we will think in terms of 'spot rates.'

A spot rate is the rate of return on a loan or a bond which has only one cash flow to the investor: the investor makes a loan of an amount P and gets back the amount F at the maturity of the loan. Remember that this is exactly the same as investing in a bond with a zero coupon rate, which has a current price P and face value F.

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Thus the one-year spot rate is the yield to maturity on zero coupon bonds (sometimes referred to as 'pure discount' bonds) of one year maturity. Let us denote the one year spot rate by r1. Then, the price of a one-year maturity zero-coupon bond with face value F is given by:

Similarly, the two-year spot rate r2 is the rate of return on a two-year loan, or equivalently, the yield to maturity on a two year zero-coupon bond. Then, the price of a two-year zero coupon bond is,

011(1 )

FPr

02 2(1 )FP

r

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Term Structure of Interest Rates We can write down similar relationships between the prices of a three

year pure discount bond, a four year zero coupon bond etc., and the corresponding spot rates r3, r4, r5 etc.

Now, if r1 < r2 < r3 ...etc., the term structure is uniformly upward sloping, which is very often the case.

On the other hand, if r1 > r2 > r3... the term structure is uniformly downward sloping.

The term structure is 'flat' if r1 = r2 = r3 ...etc. The term-structure can also take other shapes as well.

With all the spot rates, we can price any riskless bond with cash flows of any magnitude and at arbitrary points in time.

This is because we can show that all cash flows of similar riskiness occurring at the same point in time should be discounted at the same rate.

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Thus, consider a bond with coupons C1, C2, C3, ...Cn occurring at time periods t=1, 2, 3, ...n. In addition, let the face value of the bond be F. The price of the bond is then given by,

1 22

1 2

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

nn

n

C FC CPr r r

TYPICAL YIELD CURVES

0

YTM

Time to Maturity, n

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Example

Compute the spot rates given the following zero coupon bond prices : P01 = $900; P02 = $820;

P03 = $725 (Assume $1000 face values).

011

1000 1000P = 900 = r = - 1(1 + r ) 900

02 222

1000 1000P = 820 = r = - 1(1 + r ) 820

303 33

3

1000 1000P = 725 = r = - 1(1 + r ) 725

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Decomposing a Zero Coupon Bond

Decomposing a zero coupon bond into a portfolio of zero coupon bonds:

1. If the yield curve is upward sloping, the yield on a coupon bond will be lower than on a zero coupon bond (why?)

2. If the yield curve is downward sloping, the yield on a coupon bond will be higher than on a zero-coupon bond.

3. The key is to realize that the yield on coupon bonds are weighted averages of the corresponding spot rates.

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The Law of One Price

What happens if the bond price is different from the price computed by the above formula? We can show that in that case, 'the law of one price' is violated, and there are opportunities for making arbitrage profits which can be taken advantage of by investors, and the price of the bond will be driven back to the above price.

law of one price: securities (or portfolios of securities) which have the same riskiness, and which entitle the holder to the same stream of cash flows should have the same current price.

Remember that whenever there are arbitrage opportunities, the security markets cannot be in equilibrium: thus the equilibrium price of the above bond should be given by the above formula.

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Problem

Is there an arbitrage opportunity? BOND CASH FLOW AT DATE: PRICE

1 2 A 80 1080 982B 1100 - 880C 120 1120 1010

Let us try to replicate bond C with bonds A and B:nA (80) + nB (1100) = 120 (1)

nA (1080) + nB (0) = 1120 (2)

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Problem

From (2)

Substituting in (1),

So, by the law of one price,PC = nA (PA) + nB (PB) = 1.037 (982) + 0.03367 (880) = 1047.96 1010

Therefore, an arbitrage opportunity exists. Buy bond C and sell and equivalent portfolio of bonds A and B.

B120 - 1.037(80)n = = 0.03367

1100

A1120n = = 1.0371080

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Valuing Perpetual Bonds

If a bond pays a coupon C each period for ever (no face value), its market value (at a discounting rate r) is, P = C/r, since the coupon stream forms a perpetual annuity.

Also called a consol bond Example:

C = $50/yr r = 12% = 0.12P = 50/0.12 = $416.67

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Valuing Preferred Stock

“Preferred stock” usually promise a fixed payment forever (however, unlike corporate bonds, the company can miss payments without going bankrupt).

Thus, they can be treated as a perpetuity: P = C/r. Here C denotes the "preferred dividend".

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Present Value and Bond Valuation Professor Thomas Chemmanur