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Lecture 5. How to Value Bonds and Stocks. Valuing Bonds. A bond is a certificate (contract) showing that a borrower owes a specified sum that will be repaid on a number of specified dates , along with a schedule of interest payments. How to value Bonds.

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Lecture 5

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Lecture 5

How to Value Bonds and Stocks

Valuing Bonds

A bond is a certificate (contract) showing that a borrower owes a specified sum that will be repaid on a number of specified dates, along with a schedule of interest payments

How to value Bonds

  • Pure discount bonds (zero coupon bonds)

  • Level coupon bonds

  • US government bonds

  • Consoles

Pure discount bonds(zero coupon bonds)

A pure discount bond makes one payment (the face value) at a specified date (the maturity date). The face value is also called principal or denomination

A pure discount bond paying F in T years, when the annual interest rate r in each 1,…,T year will have a value

A discount bond of value PV paying F in T years has spot return

(T -year spot rate)

Level coupon bonds

Most bonds issued by governments or corporations pay couponsC in addition to a face value F at maturity T

The value of a level coupon bond with face value F, coupon C and a maturity of T years will be, where r is the annual interest rate

US government bonds

A US government bond called “13 of November 1999” will have

- a face value of $1000

- an annual coupon of 13% of the face value $ 130

- coupons paid in May and in November $ 65

until November 1999when the bond is redeemed for $1000


- it is November 1995,

- the stated annual market rate is 10% , and hence the semi annual rate is 5% .

US government bonds (continued)

The cash flows from the bond would be

The value of this bond is


Consoles are bonds with no maturity date.

The value of a console with the coupon C at the interest rate r will be

Relationship between Bond values and Interest rates

  • Value of a bond depends inversely on interest rate r.

    • Coupons reflect interest rates at issue time.

    • Coupon rate is the market interest rate at the issue time.

    • If r falls below the coupon rate, the bond sells at premium.

    • If r rises above the coupon rate, the bond sells at discount.

  • Determining Yields from Bond Prices:

    • The yield to maturity is the interest rate that

    • equates the PV of the payments on the bond to the current bond price.


Suppose that

the current spot rate on a one year discount bond is 8%

the current annual spot rate on a two year zero coupon bond is 10%

I.e., market interest rate for year 1 is r1 = 8% , for year 2 r2 = 10% .

The value of a 5% coupon 2 year bond with annual payments is

The yield to maturityy on this bond solves

The value of a 12% coupon 2 year bond with annual payments is

The yield to maturityy on this bond solves

Example (continued)

Therefore,higher coupon bonds have lower yield to maturity.

An individual investing $1 in a 2 year zero coupon bond will receive

Notice that

Term Structure of Interest Rates

The term structure of interest rates relates the annual spot rates (yields to maturity) on zero-coupon government bonds to their terms to maturity.

Recall our earlier example where the one year spot rater1 = 8% and the annual spot rate (or annual yield to maturity) on a two year zero coupon bond is r2 = 10% .

We can breakdown the 2 year spot rater2 into one year spot rater1 and forward ratef2 for next year. More formally,

An investor in the 2 year bond effectively invest in a 1 year bond at r1 and “locks in” an investment for 1 year at f2. Forward rates for later years can be calculated as :

where fn is a forward rate over n-th year and rn is a n-year spot rate.

Term Structure of Interest Rates (continued)

Estimating the Price of a Bond at a future date

One year spot ratefrom year1 to year2 is unknown at date 0.

The price of Bond B at date 1 is unknown at date 0. Thus we consider expected value of Bond B at date 1, which is given by

Proceeds from the investment I at date 1 is

Estimating the Price of a Bond (continued)

Now consider the following investment strategies at date 0.

I : Buy a 1 year bond at date 0

II : Buy a 2 year bond at date 0 and sell it at date 1

Proceeds from the investment II (expected) at date 1

Estimating the Price of a Bond (continued)

If f2 (=12.04%)= expected spot rate over year2, then I and II give the same proceed at date 1.

So, the investors should be indifferent.

If f2 > expected spot rate over year 2, then the proceed from II is greater than I.

Estimating the Price of a Bond (continued)

Under Expectation hypothesis :

(investors are assumed to be risk-neutral)

f2 =expected spot rate over year2

Under Liquidity Preference hypothesis :

(investors are assumed to be risk-averse : in order to induce risk averse investors to hold the riskier two year bonds, the market sets the forward rate f2 over the second year to be above the spot rate expected over year2.)

f2 >expected spot rate over year2

If the required return on the stock is r, the price of the stock will be

How to value Stocks

Consider a shareholder who intends to hold a stock for 1 year, earn a dividendD1 and sell the stock for an expected priceP1.

Fundamental equation of yield

dividend + expected capital gain = opportunity cost

Note that P1 is unknown now, and consequently we need to use its expected value, which can be computed if we know expected values of the dividend in 2 periods D2 and the price of the stock in period 2, P2.

How to value Stocks (continued)

Substituting P1 into the first Fundamental yield equation gives

The current price of the stock P0 can be obtained by repeating the above process.

How to value Stocks (continued)

All future dividends Di affect the price P0 even if the investor’s investment horizon is only one year.

Some Special Cases

Zero Growth :the share price of a stock that pays fixed dividendD in perpetuity should be

For example, preferred stocks

Some Special Cases (continued)

Constant Growth : if the dividends are expected to grow at the constant rateg, then

WW is expected to pay per-share dividend of $3 next year, growing at 8% forever. What is the price of the WW stock if the required return is 12% ?

PV of the expected dividends from 9 year on

Some Special Cases (continued)

Differential Growth :

A stock has just paid a dividend of $1, which is expected to grow at 20% for 5 years, 15% for 3 years, and then 8% for all future periods. Suppose the discount rate is 10% .

PV of the expected dividends for the first 8 years= 11.61

Current stock price= 11.61 + 95.33 = 106.94

Consider a firm with a fixed retention ratio

Such a firm would have

and this in turn gives

Estimating the dividend growth rateg

Now, notice that

Then we have

Estimating the dividend growth rateg(continued)

Earning next year = earning this year + increase in earning

increase in earning = retained earning *

expected gross return on retained earning at t

use the historical gross return on equity

to approximate the expected gross return at t

Estimating the dividend growth rateg(continued)

growth rate of earnings (dividends)

= retention ratio * return on retained earnings

If the firm pays all these earnings out as dividends to shareholders, then at all dates,

earnings per share = EPS = d = dividends per share

The share value at date 0, P0 should be EPS/r.

Now suppose the dividend at date 1 is retained and invested in an investment project. The share value should increase by the NPV of the “growth opportunity”(NPVGO) induced by the investment project.

Growth Opportunities

Consider a company with a constant stream of earnings in perpetuity.

Growth Opportunities (continued)

Example :Sam shipping with 100,000shares outstanding expects to earn$1,000,000 per year in perpetuity, if it distributes all its earnings to shareholders. Suppose the appropriate discount rater = 10% . Then

The firm finds an investment opportunity that will cost$1 million at date 1, but will increase earnings in every subsequent period by $210,000. If the firm decides to retain the earning at date 1 and invest in the project, what is the share price?

  • NPV of the investment opportunity at date 1

    • -1,000,000 + 210,000/0.1 = 1,100,000

  • NPVat date 0

    • 1,100,000/1.1 = 1,000,000 or 10 per share

Growth Opportunities (continued)

The share price with the investment project

P0 = EPS/r + NPVGO = 100 + 10 = 110

The above share prices can be obtained from calculating PV’s of the future earnings with or without the investment opportunity.

Growth Opportunities (continued)

Price-earning ratio(PER)

P0 / EPS = 1 / r + NPVGO / EPS

PER depends positively on the growth opportunities.

Hence, the stocks of firms retaining earnings to invest in growth opportunities do have higher PER.

PER depends negatively on the discount rater.

Firms with risky earnings will therefore have lower PER.

Reported accounting earnings are used.

Conservative accounting rules leads to higher PER’s.

For instances, Japanese firms PER’s

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