security valuation bonds updated

Security Valuation By Sudarshan Kadariya 1 Bond, Equity and Preferred Stock

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Page 1: Security valuation bonds updated


Security Valuation

BySudarshan Kadariya

Bond, Equity and Preferred Stock

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Basis for all valuation approachesThe use of valuation models in

investment decisions (i.e., in decisions on which assets are under valued and which are over valued) are based upon

a perception that markets are inefficient and make mistakes in assessing value

an assumption about how and when these inefficiencies will get corrected

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In an efficient market, the market price is the best estimate of value. The purpose of any valuation model is then the justification of this value.

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In general, the value of an asset is the equilibrium price that a buyer and a seller is ready to transact.

Note that if either the buyer or seller is not both willing and able, then an offer does not establish the value of the asset.

What is Value?

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Among the several types of value, the following three are useful for valuation purpose:

oBook Value - The asset’s accounting value after depreciation

oMarket Value - The price of an asset in the market determined competitively

o Intrinsic Value - The present value of the expected future cash flows discounted at the required rate of return

Types of value

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There are two primary determinants of the intrinsic value of an asset as:

o The size and timing of the expected future cash flows

o The individual’s required rate of return (this is determined by a number of other factors such as risk/return preferences, returns on competing investments, expected inflation, etc.)

Note that the intrinsic value of an asset can be, and often is, different for each individual so that the markets exist. (why??)

Determinants of Intrinsic Value

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BondsBonds are long-term fixed income securities. It is

a trade-able instrument which is also termed as debt securities. Both the bonds (?) and debentures (?) are the debt securities. Most commonly, bonds pay interest periodically (usually semiannually) and then return the principal at maturity.

In India, debt securities issued by the government and public sector are generally referred as bonds whereas the debt securities that are issued by the private sector joint stock companies are called debentures. The terms - bonds and debentures are often used interchangeably.

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Par or Face ValueThe amount of money that is paid to the bondholders at

maturity. For most bonds this amount is $1,000. It also generally represents the amount of money borrowed by the bond issuer.

Coupon RateThe coupon rate, which is generally fixed, determines the

periodic coupon or interest payments. It is expressed as a percentage of the bond's face value. It also represents the interest cost of the bond to the issuer. For example: if the coupon rate is 12%, $120 is payable to bondholder.

Definitions of useful terms

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Spot interest rateZero coupon bond is a special type of bond which

does not pay annual interests. This type of bonds is also called pure discount or deep discount bond. The return received from a zero coupon bond expressed on an annualized basis is the spot interest rate. In other words, spot interest rate is the annual rate of return on a bond that has only one cash inflow to the investor. For instance, if a bond with face value $1000 issued at a discount for $797.19 for 2 years. The spot rate is 12%, an annual interest rate.

Coupon PaymentsThe coupon payments represent the periodic interest

payments from the bond issuer to the bondholder. For example: coupon payments is $1000 @12% = $120. Since most bonds pay interest semiannually, generally one half of the annual coupon is paid to the bondholders every six months.

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Maturity DateThe maturity date represents the date on which the bond

matures, i.e., the date on which the face value is repaid. The last coupon payment is also paid on the maturity date.

Current YieldThe current yield is the annual interest receivable on a

bond to its current market price. For instance, if the market price is $800, coupon rate is 12% and the face value is $1000 then the current yield equals $120/$800 * 100 = 15% (Discount or Premium ??)

(If the current yield > coupon rate = bond is selling at discount,(If the current yield< coupon rate = bond is selling at premium)

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Bonds are valued using time value of money concepts.

Their coupon, or interest, payments are treated like an equal cash flow stream (annuity).

Their face value is treated like a lump sum.

How many types of cash flows that provides to a bond investments or the bondholder?

Periodic interest payments Repayment of the face value

Bond Valuation

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Maturity period: 10 years bond Date of purchase: January 1, 2003 Face value: $1000 Coupon rate: 10% Market rate of return: 12%. Required: Calculate the price of this bond today.Draw a timeline


$100 $100$100






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1. First of all calculate the present value of the coupon stream

PV = $100/(1+.12)1 + $100/(1+.12)2 + $100/(1+.12)3 + $100/(1+.12)4 + $100/(1+.12)5 + $100/(1+.12)6 + $100/(1+.12)7 + $100/(1+.12)8 + $100/(1+.12)9+ $100/(1+.12)10

Or, we can find the PV of an annuityPVA = $100 * {[1-(1+.12)-10]/.12} (PVIFA12%,10 years = 5.650)

PV = $565.02

2. Find the PV of the face valuePV = CFt / (1+r)t

PV = $1000/ (1+.12)10 (PVIF 12%, 10 years = 0.3219)

PV = $321.97

3. Add the two values to get the total PVTherefore, the price of bond today is $565.02 + $321.97 = $886.99

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If you purchased a bond for $800 5-years ago and sold the bond today for $1200. The bond paid an annual 10% coupon. What is the realized rate of return?


PV = S [CFt / (1+r)t] t=0

$800 = [$100/(1+r) + $100/(1+r)2 + $100/(1+r)3 + $100/(1+r)4 + $100/(1+r)5] + [$1200/(1+r)5]

To solve, you need to use a “trail and error” approach.

The realized rate of return on this bond is 19.3%.

931.8276 (15%) & 781.3143 (20%)

Realized rate of return

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The following timeline illustrates a typical bond’s cash flows:

0 1 2 3 4 5

100 100 100 100 1001,000

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The value of the bond is simply the present value of the annuity-type cash flow and the lump sum:

V Pmtk






d d


1 11


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Assume that you are interested in purchasing a bond with 5 years to maturity and a 10% coupon rate. If your required return is 12%, what is the highest price that you would be willing to pay?

0 1 2 3 4 5

100 100 100 100 1001,000



1 11 012


1 000

1 012927 90







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Interest rate risk factsBond prices are sensitive to the market

interest rate

If interest rates rise, the market value of bonds fall in order to compete with newly issued bonds with higher coupon rates.

Sensitivity to the interest rate change become more severe for longer term bonds

Percentage change (rise) in price is not symmetric with percentage decline.

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Some Notes About Bond Valuation

The value of a bond depends on several factors such as time to maturity, coupon rate, and required return

We can note several facts about the relationship between bond prices and these variables (ceteris paribus):

o Higher required returns lead to lower bond prices, and vice-versa

o Higher coupon rates lead to higher bond prices, and vice versa

o Longer terms to maturity lead to lower bond prices, and vice-versa

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Common stock represents an ownership interest in a corporation, but to the typical investor, a share of common stock is simply a piece of paper characterized by two features

◦ It entitles its owner to dividends, but only if the company has earnings out of which dividend can be paid, and only if the management chooses to pay dividends rather than retaining and reinvesting all the earnings.

◦ Stock can be sold at some future data, hopefully at a price greater than the purchase price. If the stock is actually sold at a price above its purchase price, the investor will receive a capital gain.

Stock price Volatile, rapidly changes Price swings are even larger for smaller companies For large companies, it is relatively stable

Common stock & Stock price

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Common Stock Valuation

Just like with bonds, the first step in valuing common stocks is to determine the cash flows

For a stock, there are two: Dividend payments The future selling price

Again, find the present values of these cash flows and adding them together will give us the value

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Terms used in stock valuationsD0 : Most recent dividend, which has already been paid CertainDt : Dividend the stockholder expects to receive at the end of year t UncertainD1 : First dividend expected, and will be paid at the end of this year UncertainD2 : The dividend expected at the end of second year UncertainP0 : Actual market price of stock CertainPˆt : Expected price of stock at the end of year t ("P hat t") Uncertain

Pˆ0 :

The intrinsic or theoretical value of the stock today as seen by the particular investor doing the analysis. It could differ among investors depending on how optimistic they are regarding the company


g :Expected growth rate in dividends as predicted by a investor. Different investors may use different g's to evaluate a firm's stock.


ks :Minimum acceptable, or required rate of return, on the stock, considering both its riskiness and the return available on other investments.


kˆs :Expected rate of return which an investor who buys the stock expects to receive. It could be above or below ks, but one would buy the stock only if kˆs is equal or greater than ks.


k¯s : Actual, or realized, after the fact rate of return, pronounced "k bar s." Certain

D1/P0 : Expected dividend yield on the stock during the coming year. Uncertain(Pˆt –P0)/P0 The expected capital gain yield on the stock during year t Uncertain

The expected total return = exp.div. yield + exp. Capital gain yield

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Common Stock Valuation: An Example

Assume that you are considering the purchase of a stock which will pay dividends of $2 next year, and $2.16 the following year. After receiving the second dividend, you plan on selling the stock for $33.33. What is the intrinsic value of this stock if your required return is 15%?


2 00

1 15

216 3333

1 1528571 2



. .


2.00 2.1633.33



1 )1(


)1(t tt


CS ks




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In valuing the common stock, we have made two assumptions:o We know the dividends that will be paid in the

future (Di)o We know the price that we will be able to sell the

stock in the future (Pi)

Both of these assumptions are unrealistic, especially knowledge of the future selling price

Furthermore, suppose that when we intend to hold the stock for twenty years, the calculations would be very tedious!

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Further Assumptions

We cannot value common stock without making some simplifying assumptions

If we make the following assumptions, we can derive a simple model for common stock valuation:

Assumption 1: The holding period is infinite (i.e., you will never sell the stock)

Assumption 2: The dividends will grow at a constant rate forever

Note that the second assumption allows us to predict every future dividend, as long as we know the most recent dividend

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The Dividend Discount Model (DDM)

With these assumptions, we can derive a model which is known as the Dividend Discount Model, or the Gordon Model

This model gives us the present value of an infinite stream of dividends that are growing at a constant rate:


D g

k g


k gCS


0 11

Where,Ke = Kcs = Cost of equityg = growth rateDo = Recent dividend rateD1 = One period dividend

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The DDM: An ExampleRecall the previous example in which the

dividends were growing at 8% per year, and your required return was 15%

The value of the stock must be:


185 1 08

15 08

2 00

015 0828 57

. .

. .


. ..

(Note that this is exactly the same value that we got earlier)

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The DDM Example (cont.) In the earlier example, how did we know that the

stock would be selling for $33.33 in two years?

Note that the period 3 dividend must be 8% larger than the period 2 dividend, so:


2 16 1 08

15 08

2 33

015 0833 33

. .

. .


. ..

(Therefore, the selling price is equal to 33.33, considering 8% growth rate per year)

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Preferred Stock – Hybrid stock Preferred stock, like as bonds imposes a fixed

charge & the failure to make this fixed charge will not lead to bankruptcy.

Preferred stock represents an ownership claim on the firm that is superior to common stock in the event of liquidation.

Typically, preferred stock pays a fixed dividend periodically, and

the preferred stockholders are usually not entitled to vote as are the common shareholders.

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Preferred Stock Valuation Most preferred stocks are perpetuities, and the

value of a share of perpetual preferred stock is found as the dividend divided by the required rate of return.

Preferred stock is very much like common stock, except that the dividends are constant (i.e., the growth rate is 0%)

Therefore, we can use the DDM with a 0% growth rate to find the value:







0 1 0


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Preferred Stock: An Example

Suppose that you are interested in purchasing shares of a preferred stock which pays a $5 dividend every year. If your required return is 7%, what is the intrinsic value of this stock?

VP 5

0 077143


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Bond valuation is less glamorous than stock valuation. Why?

Two reasons:

First, the returns from investing from bonds are less imperative and fixed.

Second, the bond prices fluctuate less than the equity prices

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Bond Yields

Yield to Maturity:

Same as market rate of return at maturity

Yield to Call:

Market rate of return at call which is issuer’s option

When coupon>market interest

Current Yield:

Annual interest payment divided by bond’s current price

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Yield to Maturity: The Approximation Approach


The yield to maturity (YTM) is the average annual rate of return that a bondholder will earn. YTM is generally the same as the market rate of interest, kd which can be solve as “trial and error” or by interpolation.

For example, a 14-year, 10 percent coupon with par value $1000 is offered at a market price $1494.93. What rate of interest we earn if we bought the bond and held it to maturity?

(Answer is 5%)

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The Approximation FormulaF = Face Value = Par Value = $1,000P = Bond PriceC = the semi annual coupon interestN = number of semi-annual periods left to maturity

1YTM) annual-semi (1YTM

YTM annual-semi 2YTM2


Maturity toYield annual-Semi




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Example Find the yield-to-maturity of a 5 year 6%

coupon bond that is currently priced at $850. (assume the coupon interest is paid semi-annually.)

Therefore there is coupon interest of $30 paid semi-annually i.e. ($1000 x 6% = $60 p.a. & $30 s.a.)

There are 10 semi-annual periods left until maturity (i.e. nx2 = 5x2 = 10 periods)

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%97.91)0486.1(1YTM) annual-semi (1YTMreturn of rate Realized

9.3%0.0927320.0486YTM annual-semi 2YTM








Maturity toYield annual-Semi




The approximation approach only gives us an approximate answerThe approximation approach only gives us an approximate answer

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