Chapters 8 & 9: How to value bonds and stocks

Corporate Finance

Ross, Westerfield, and Jaffe

Outline

1. Bonds

2. Bond pricing

3. Bond concepts and reporting

4. Dividend discount model (DDM)

5. Stock market reporting

Notations, I

Bonds: debt securities with long-term maturities, typically longer than 1 year.

Coupon, typically C = C1 = C2 = … = CN: the stated interest payment made on a bond.

Par (face) value (FV): the principal value of a bond; by default, $1,000.

Maturity (N): specified date on which the principal is paid.

Notations, II

Coupon rate: annual coupon / FV. Yield to maturity (YTM): the rate required in the

market on a bond.– Market decides.– Time varying.– Related to default risk.– If coupons are paid out annually, i = YTM. If

coupons are paid out semiannually, i = YTM/2.

How to value bonds

Note that the cash flows of a typical bond consists of two cash flow streams: (1) an annuity of coupon payments, and (2) a final principal.

Recall that the PV of multiple cash flows is simply the sum of individual PV’s.

Thus, for a typical bond, PV = PVannuity + PVprincipal.

Bond pricing formula

PV = PVannuity + PVprincipal = C { [ 1 – 1 / (1 + i)N ] / i } + FV / (1 + i)N.

Of course, we can also use a financial calculator to do the job.

Note that computation itself is rather straightforward. The difficult part is to correctly figure out the values of parameters.

Bond example, I

Suppose that VTcredit Inc. is going to issue a bond. The maturity is 25 years. The average YTM on similar issues is 10%. A series of $120 as coupons is paid out annually. The face value is $1,000. What is the fair price of the bond?

Formula: PV = = C { [ 1 – 1 / (1 + i)N ] / i } + FV / (1 + i)N = $120 { [ 1 – 1 / (1 + 10%)25 ] / 10% } + $1,000 / (1 + 10%)25 = $1,181.54.

Calculator: 120 PMT; 1000 FV; 25 N; 10 I/Y; CPT PV. The answer is: PV = -1,181.5408.

Semiannual payments

Most corporate bonds pay coupons semiannually. The principle of calculation is the same.

But remember: the time frequency of i and N must be the same.

Bond example, II

Suppose that VTcredit Inc. is going to issue a bond. The maturity is 25 years. The average YTM on similar issues is 10%. A series of $60 as coupons is paid out semiannually. The face value is $1,000. What is the fair price of the bond?

Formula: PV = = C { [ 1 – 1 / (1 + i)N ] / i } + FV / (1 + i)N = $60 { [ 1 – 1 / (1 + 5%)50 ] / 5% } + $1,000 / (1 + 5%)50 = $1,182.56.

Calculator: 60 PMT; 1000 FV; 50 N; 5 I/Y; CPT PV. The answer is: PV = -1,182.5593.

Bond example, III

Suppose that Northern Inc. bonds have a $1,000 face value. Annual coupon is $100. The bonds mature in 20 years. YTM = 10%. What is the bond price?

Calculator: 100 PMT; 1000 FV; 20 N; 10 I/Y; CPT PV. The answer is: PV = -1,000.

Lesson: if payment rate equals to discount rate, the bond price is simply the FV.

YTM example

Northern Inc. issued 12-year bonds 2 years ago at a coupon rate of 8.4%. The bonds make semiannual payments. If these bonds currently sell for 110% of par value, what is the YTM?

Calculator: 42 PMT; 1000 FV; 20 N; -1100 PV; CPT I/Y. The answer is: I/Y = 3.4966.

YTM = 2 × 3.4966 = 6.9932 (%).

Bond price and YTM

Recall from Chapter 4: Holding time period constant – the higher the interest (discount) rate, the smaller the PV.

Because YTM is closely related to discount rate, one would expect that YTM is negatively related to bond price.

That is, market interest rates rise yield (YTM) increases bond price falls.

YTM and bond price

800

1000

1100

1200

1300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Discount Rate

Bo

nd

Va

lue

6 3/8

When the YTM < coupon, the bond trades at a premium.

When the YTM = coupon, the bond trades at par.

When the YTM > coupon, the bond trades at a discount.

Bond markets

Trading is inactive for corporate bonds; typically, trading occurs in OTC (over-the-counter; not exchange); Illiquid.

Trading is active for U.S. government debt instruments. This sector has many international participants, e.g., Chinese Central Bank.

Bond rating

Junk bonds: rating at BB or below by S&P; rating at Ba or below by Moody’s.

Investment-quality bonds: rating at BBB or above by S&P; rating at Baa or above by Moody’s.

P. 251

Bond reporting

WSJ used to publish corporate bond quotation for the 40 most active corporate bonds; but not anymore.

Bond trading information can be found at FINRA (Financial Industry Regulatory Authority) http://cxa.marketwatch.com/finra/MarketData/Default.aspx

A FINRA bond quotation

Issue:   IBM.GT   IBM 8.375   11/01/2019 Rating -------------------Last Sale ---------------------------- Moody's/S&P/Fitch Date Price Yield A1 / A+ / AA- 05/02/2007   125.00   5.569263 This TRACE quotation was retrieved on 05/15/2007. This issue pays coupons semiannually: 05/01 and 11/01. I/Y = YTM / 2 = 2.7846. Coupon rate = 8.375%. So, the semiannual payment is

$41.875. Calculator: 25 N; 2.7846 I/Y; 1000 FV; 41.875 PMT; CPT PV.

The answer is: PV = -1,250.2572, which is 125.02572% of par.

Bond features, I

Indenture: The written agreement between the corporation and the lender detailing the terms of the debt issue.

Collateral: the asset pledged on a debt. Debenture (> 10 years) and note (< 10

years): an unsecured debt. Sinking fund: an account managed by the

bond trustee for early bond redemption.

Bond features, II

Call bond: a bond that allows the firm to repurchase or “call” part or all of the bond issue at stated prices over a specific period.

Put bond: a bond that allows the holder to force the issuer to buy the bond back at a stated price.

Protective covenant: a part of the indenture limiting certain actions that might be taken during the term of the loan, usually to protect the lender’s interest.

Treasury notes (2-10 years) and bonds (>10 years): long-term debt securities issued by the U.S. federal government.

Bond features, III

Municipal bonds (munis): debt securities issued by state and local governments.

Zero coupon bond: a bond that makes no coupon payments, thus initially priced at a deep discount.

Floating-rate bond: a bond whose coupon payments are adjustable.

Convertible bond: a bond that can be swapped for a fixed number of shares of stock anytime before maturity at he holder’s option.

Inflation and interest rates

Inflation erodes purchasing power Nominal rate (R): the percentage change in

the number of dollars you have. Real rate (r): the percentage change in how

much you can buy with your dollars. Fisher effect/equation: (1 + R) = (1 + r) × (1 +

h), where h is the inflation rate.

Nominal rate

Suppose that the real risk-free rate is 2.5% and the inflation rate is expected to be 4.7%. What rate would you expect to see on a 1-year T-bill?

(1 + R) = (1 + r) × (1 + h) (1 + R) = (1 + 2.5%) × (1 + 4.7%) R =

7.32%.

Dividends as cash flows

For bond pricing, we just discounted coupons and par (2 types of cash flows) to arrive at the PV.

For stock pricing, this course focuses on one particular type of cash flows—cash dividends.

Discounting dividends makes sense because dividends are the only types of cash flows that investors can actually receive from the firm.

When dividends are used as cash flows for discounting, we need to use cost of equity, i.e., the required rate demanded by shareholders, as the applicable discount rate.

But the problems are:

Discounting all (possibly infinite terms) dividends to arrive at PV (fundamental value of the stock) is theoretically correct.

However, the problems are: (1) cash flows are uncertain; we usually plug in a series of estimates as if they were certain, (2) the required rate of return is unknown and frequently time unstable, (3) the life of the security is actually unknown.

How about those no-dividend firms?

About 50% of U.S. publicly traded firms do not pay dividends.

However, the valuation concept is the same, except that some of the early dividend payments are zero. After all, there should be expectations that at some point the firm will start paying dividends. Otherwise, the investment will have no value.

Of course, it is often difficult to forecast when these firms will pay dividends and how much will they pay.

Infinite terms?

The fundamental value of a stock is the PV of its future dividends.

Because a firm can possibly live forever, thus we are discounting an infinite number of cash flows.

To make the calculation doable, we need to make some assumptions.

Zero-growth DDM

The easiest assumption one can make is to assume that there is no growth in dividends, i.e., D1 = D2 = … = D.

Because this is a perpetuity, the pricing is rather straightforward: PV = D / i.

Suppose that GE paid $2 dividend per share last year. Investors expect no growth in GE’s future dividends. The applicable discount rate is 10%.

PV = $2 / 10% = $20.

Constant-growth DDM

Another easy assumption one can make is to assume that there is a constant growth rate, g, in dividends, i.e., D1 = D0 × (1 + g), D2 = D0 × (1 + g)2, etc.

That is, this is a growing perpetuity. Recall from Chapter 4, the PV of a growing

perpetuity is: PV = D1 / (i – g).

Constant-growth DDM example

Vermont Financial Inc. paid a dividend of $1 last year. The constant growth rate of dividends is 5%, and the required rate of return is 10%.

PV = [D0 × (1 + g)] / (i – g) = [$1 × (1 + 5%)] / (10% – 5%) = $21.

Multiple-stage DDM model

This model allows different growth rates for different stages.

Typically, it takes care of recent, supernormal growth.

There are formulas for PV. However, they do not look neat.

Let us use Excel to visualize the discounting process.

Multiple-stage example, I

HP has a cost of equity at 14%. HP just paid an annual dividend of $2.

The expected dividend growth rate between year 1-3 is 25%. The expected dividend growth rate between year 4-5 is 15%. The expected dividend growth rate for year 6 and afterwards is 5%.

Multiple-stage example, II

Year Dividend Growth rate Discount rate PV5 PV0 2 0.141 2.5 0.25 2.19298252 3.125 0.25 2.4045863 3.90625 0.25 2.63660754 4.492188 0.15 2.65973565 5.166016 0.15 60.2701823 33.9855116 5.424316 0.05 43.879422

Quality of inputs

The above stock pricing examples seem quite simple. In reality, stock valuation is extremely difficult. The difficulty does not arise from the calculation itself, but from the uncertainty associated with model inputs (estimates).

Professional investors compete on the quality of inputs.

Estimating g

There are potentially many ways to estimate g.

One possible way is to use the industry’s long-term average growth rate.

Another way: g = retention ratio × ROE (return on equity).

Retention ratio = (NI – dividends) / NI. ROE = NI / equity.

P/E ratio, I

The ratio of the current market price per share to the expected next-12-month earnings per share (or the current annual earnings per share).

When the expected next-12-month EPS is used, this P/E ratio is also called “forward P/E ratio.”

The level of P/E ratio is related to growth potential. A stock with higher expected growth potential tends to have a higher P/E ratio.

P/E ratio, II

Embedded in the seemingly high prices of those high PE stocks are expectations that in the aggregate are unlikely to be met. You should expect a high strikeout rate. Example: During 1996-2000, only 1/3 of all high PE, high-tech stocks have higher returns than the S&P 500; but once they beat the S&P 500, they tend to have extremely good performance.

An effective, widely-used communication tool because of its simplicity.

Stock market reporting (WSJ)

STOCK SYM CLOSE NET CHG

GenMotor GM 30.62 1.16 Yesterday’s closing price: $30.62. Yesterday’s closing price is higher than the

closing price of the previous trading day by $1.16.

Review: Let us work on this

Q2, P. 267. Microhard has issued a bond with the following characteristics: (1) par: $1000, (2) time to maturity: 15 years, (3) coupon rate: 7%; (4) semiannual payments. What is the price of the bond if the YTM is (a) 7%, and (b) 9%?

End-of-chapter

Chapter 8: Concept questions: 1-8, 12-14, 16-17; Questions and problems: 1-10, 13-17, and 21-23.

Chapter 9: Concept questions: 1-10; Questions and problems: 1-9, 12-14, and19.