Valuation of stocks and bonds
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Valuation of Stocks and Bonds. 3. 1 Bonds and Bonds Valuation. What is Bond ?. A long-term debt instrument in which a borrower agrees to make payments of principal and interest, on specific dates, to the holders of the bond.

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Valuation of stocks and bonds

Valuation of Stocks and Bonds

3.1 Bonds and Bonds Valuation

What is bond

What is Bond ?

  • A long-term debt instrument in which a borrower agrees to make payments of principal and interest, on specific dates, to the holders of the bond.

  • A bond is a security that obligates the issuer to make specified interest and principal payments to the holder on specified dates.

    • Coupon rate

    • Face value (or par)

    • Maturity (or term)

  • Bonds are sometimes called fixedincome securities.

  • Bond is normally an interest-only loan, meaning that the borrower will pay the interest every period, but none of the principal will be repaid until the end of the loan.

Key features of a bond

Key Features of a Bond

  • Par value or face value – face amount of the bond, which is paid at maturity (assume $1,000).

  • Bond that sells for its par value is called a par value bond

  • Coupon interest rate – stated interest rate (generally fixed) paid by the issuer. Multiply by par to get dollar payment of interest.

  • Because the coupon is constant and paid every year, the type of bond we are describing is sometimes called a level coupon bond.

  • Maturity date – years until the bond must be repaid.

  • Issue date – when the bond was issued.

  • Yield to maturity (YTM) - rate of return earned on a bond held until maturity (also called the “promised yield”).

Characteristics of bonds

$I $I $I $I $I $I+$M

. . .





Characteristics of Bonds

  • Bonds pay fixed coupon (interest) payments at fixed intervals (usually every 6 months) and pay the par value at maturity.

example: AT&T 9s of 2018

par value = $1000

coupon = 9% of par value per year.

= $90 per year ($45 every 6 months).

maturity = 20 years.

issued by AT&T.

Example at t 9s of 2018

$45 $45 $45 $45 $45 $45+$1000

. . .





Example: AT&T 9s of 2018

par value = $1000

coupon = 9% of par value per year.

= $90 per year ($45 every 6 months).

maturity = 20 years.

issued by AT&T.

Types of bonds

Types of Bonds

  • Pure Discount or Zero-Coupon Bonds (Zeroes)

    • Pay no coupons prior to maturity.

    • Pay the bond’s face value at maturity.

    • Priced at a deep discount.

  • Coupon Bonds

    • Pay a stated coupon at periodic intervals prior to maturity.

    • Pay the bond’s face value at maturity.

  • Perpetual Bonds (Consols)

    • No maturity date.

    • Pay a stated coupon at periodic intervals.

Types of bonds1

Types of Bonds

  • Self-Amortizing Bonds

    • Pay a regular fixed amount each payment period over the life of the bond.

    • Principal repaid over time rather than at maturity.

  • Debentures –

    • unsecured bonds.

  • Subordinated debentures –

    • unsecured “junior” debt.

  • Mortgage bonds –

    • secured bonds.

  • Junk bonds –

    • speculative or below-investment grade bonds; rated BB and below.

Types of bonds2

Types of Bonds

  • Eurobonds - bonds denominated in one currency and sold in another country. (Borrowing overseas).

  • example - suppose Disney decides to sell $1,000 bonds in France. These are U.S. denominated bonds trading in a foreign country. Why do this?

    • If borrowing rates are lower in France,

    • To avoid SEC regulations.

Other types features of bonds

Other types (features) of bonds

  • Convertible bond – may be exchanged for common stock of the firm, at the holder’s option.

  • Warrant – long-term option to buy a stated number of shares of common stock at a specified price.

  • Putable bond – allows holder to sell the bond back to the company prior to maturity.

  • Income bond – pays interest only when income is earned by the firm.

  • Indexed bond – interest rate paid is based upon the rate of inflation.

Bond markets

Bond markets

  • Bond market is bigger in value than stock market.

  • Largest securities market in the world is not NYSE but U.S. Treasury Market.

  • Primarily traded in the over-the-counter (OTC) market.

  • This means that there’s no particular place where buying and selling occur.

  • Instead dealers around the country (and around the world) stand ready to buy and sell and they are connected electronically.

  • Most bonds are owned by and traded among large financial institutions.

  • Full information on bond trades in the OTC market is not published, but a representative group of bonds is listed and traded on the bond division of the NYSE.

Bond markets contd

Bond markets (contd…)

  • Because Bond market is almost entirely OTC, it has little or no transparency.

  • It is near to impossible to get the information on price and quantity of transactions because transactions are privately negotiated between parties, and there is little of no centralized reporting of transactions.

Bond issuers

Bond Issuers

  • Federal Government and its Agencies

  • Local Municipalities

  • Corporations

U s government bonds

U.S. Government Bonds

  • Treasury Bills

    • No coupons (zero coupon security)

    • Face value paid at maturity

    • Maturities up to one year

  • Treasury Notes

    • Coupons paid semiannually

    • Face value paid at maturity

    • Maturities from 2-10 years

  • Treasury Bonds

    • Coupons paid semiannually

    • Face value paid at maturity

    • Maturities over 10 years

    • The 30-year bond is called the long bond.

Agencies bonds

Agencies Bonds

  • Mortgage-Backed Bonds

    • Bonds issued by U.S. Government agencies that are backed by a pool of home mortgages.

    • Self-amortizing bonds.

    • Maturities up to 20 years.

U s government bonds1

U.S. Government Bonds

  • No default risk. Considered to be riskfree.

  • Exempt from state and local taxes.

  • Sold regularly through a network of primary dealers.

  • Traded regularly in the over-the-counter (OTC) market.

Municipal bonds munis

Municipal Bonds (Munis)

  • Maturities from one month to 40 years.

  • Exempt from federal, state, and local taxes.

  • Riskier than U.S. Government bonds.

  • Rated much like corporate issues.

  • They are almost always callable.

Corporate bonds

Corporate Bonds

  • Secured Bonds (Asset-Backed)

    • Secured by real property

    • Ownership of the property reverts to the bondholders upon default.

  • Debentures

    • General creditors

    • Have priority over stockholders, but are subordinate to secured debt.

Common features of corporate bonds

Common Features of Corporate Bonds

  • Senior versus subordinated bonds

  • Convertible bonds

  • Callable bonds

  • Putable bonds

  • Sinking funds



  • Some bonds may be converted to common stock.

  • Can be swapped for a fixed number of shares of stock anytime before maturity at the holder’s option.

  • Is this a benefit to the investor?

    Yes !

Effect of a call provision

Effect of a call provision

  • Allows issuer to refund the bond issue if rates decline (helps the issuer, but hurts the investor).

  • Borrowers are willing to pay more, and lenders require more, for callable bonds.

  • Most bonds have a deferred call and a declining call premium.

What is a sinking fund

What is a sinking fund?

  • Provision to pay off a loan over its life rather than all at maturity.

  • Similar to amortization on a term loan.

  • Reduces risk to investor, shortens average maturity.

  • But not good for investors if rates decline after issuance.

Security valuation

Security Valuation

  • In general,

    The intrinsic value of an asset = the present value of the stream of expected cash flows discounted at an appropriate required rate of return.

  • Can the intrinsic value of an asset differ from the market value?


The value of financial assets











The value of financial assets

Bond valuation

Bond Valuation

  • Determining the value of a bond requires:

    • An estimate of expected cash flows

    • An estimate of the required return

  • Assumptions:

    • The coupon interest rate is fixed for the term of the bond

    • The coupon payments are made annually and the next coupon payment is receivable exactly a year from now

    • The bond will be redeemed at par on maturity.

    • The bond is non-callable.

Bond valuation contd

Bond Valuation (contd…)

Example 1

Example 1

Consider a 10 year, 12 % coupon bond with a par value of Rs 1000. Let the required yield on this bond is 13%.

The cash flows for this bond are as follows:

  • 10 annual coupon payment of Rs 120

  • Rs 1000 principal repayment 10 years from now

Bond values with semi annual interest

Bond Values with Semi-Annual Interest

Example 2

What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the required yield to maturity is 10% compounded semiannually?

Example 2

0 6 12 18 24 ... 120Months

45 45 45 45 1045

Bond valuation contd1

Bond Valuation (contd…)

  • To determine the value of a bond at a particular point in time, we need to know:

    • Number of periods remaining until maturity

    • The Face Value of the Bond

    • The Coupon (Interest)

    • The market interest rate for bond with similar features. (Yield To Maturity – YTM)

  • Interest rate required in the market on particular bond type is called the bond’s YTM or simply YIELD of the bond.

Bond yields and prices the case of coupon bonds

Bond Yields and PricesThe case of coupon bonds

  • Suppose you purchase the U.S. Treasury bond described earlier (Example 2) and immediately thereafter interest rates fall so that the new yield to maturity on the bond is 8% compounded semiannually. What is the bond’s new market price?

  • Suppose the interest rises, so that the new yield is 12% compounded semiannually. What is the market price now?

  • Suppose the interest equals the coupon rate of 9%. What do you observe?

Bonds yields and prices

  • New Semiannual yield = 8%/ 2 = 4%

  • What is the price of the bond if the yield to maturity is 8% compounded semiannually?

  • Similarly:

    • If r=12%: P =$ 827.95

    • If r= 9%: P =$ 1,000.00

Bonds Yields and Prices



  • S’pose our firm decides to issue 20-year bonds with a par value of $1,000 and annual coupon payments. The return on other bonds of similar risk is currently 12%, so we decide to offer a 12% coupon interest rate.

  • What would be a fair price for these bonds?

Valuation of stocks and bonds


120 120 120 . . . 120

0 1 2 3 . . .20

Period/Yr = 1

N = 20

r% per year = 12

FV = 1,000

Coupon = 120


P = $1,000

Note: If the coupon rate = yield, the bond will sell for par value.

Exercise contd

Exercise (contd…)

  • Suppose interest rates fall immediately after we issue the bonds. The required return on bonds of similar risk drops to 10% i.e. Yield falls to

    10 %

  • What would happen to the bond price?

Valuation of stocks and bonds

Period/Yr = 1

N = 20

r% per Year = 10

Coupon = 120

FV = 1000


P = $1,170.27

Note: If the coupon rate > yield, the bond will sell for a premium.

Exercise contd1

Exercise (contd…)

  • Suppose interest rates rise immediately after we issue the bonds. The required return on bonds of similar risk rises to 14%.

  • What would happen to the bond price?

Valuation of stocks and bonds

Period/Yr = 1

N = 20

r% per year = 14

Coupon = 120

FV = 1000


P = $867.54

Note: If the coupon rate < yield, the bond will sell for a discount.

Relationship between bond prices and yields

Relationship Between Bond Prices and Yields

  • Bond prices are inversely related to interest rates (or yields).

  • A bond sells at par only if its coupon rate equals the required yield.

  • A bond sells at a premium if its coupon rate is above the required yield.

  • A bond sells at a discount if its coupon rate is below the required yield.

Volatility of coupon bonds

Volatility of Coupon Bonds

  • Consider two bonds with 10% annual coupons with maturities of 5 years and 10 years.

  • The yield is 8%

  • What are the responses to a 1% yield change?

  • The sensitivity of a coupon bond increases with the maturity

Bond prices and yields

Bond Prices and Yields

Longer term bonds are more

sensitive to changes in (yields)

Interest rates than shorter term bonds.

Bond Price





10% 12% 14%

Bond yields and prices the problem

Bond Yields and PricesThe problem

  • Consider the following two bonds:

    • Both have a maturity of 5 years

    • Both have yield of 8%

    • First has 6% coupon, other has 10% coupon, compounded annually.

  • Then, what are the price sensitivities of these bonds to a 1% increase (decrease) in bond yields?

  • Lesser coupon rate bonds are more sensitive to change in yield.

Interest rate price risk

Interest Rate (Price) Risk

  • The risk that arises for bond owners from fluctuating interest rates is called interest rate risk.

  • How much interest rate risk a bond has, depends on how sensitive its price is to interest rate changes.

  • This sensitivity directly depends on two things:

    • The time to maturity

    • The coupon rate.

  • All other things being equal, the longer the time to maturity, the greater the interest rate risk.

  • All other things being equal, the lower the coupon rate, the greater the interest rate risk.

What is interest rate or price risk

What is interest rate (or price) risk?

Interest rate risk is the concern that rising r will cause the value of a bond to fall.

% change 1 yr r 10yr% change

+4.8%$1,048 5% $1,386 +38.6%

$1,00010% $1,000

-4.4% $95615% $749 -25.1%

The 10-year bond is more sensitive to interest rate changes, and hence has more interest rate risk.

Interest rate risk contd

Interest Rate Risk (contd…)

  • Reason that longer-term bonds have greater interest rate sensitivity:

    • A large portion of a bond’s value comes from the discounting of face value at maturity,

    • The PV of this amount isn’t greatly affected by a small change in interest rates if the amount is to be received in smaller years to maturity,

    • Even a small change in the interest rate, however, once it is compounded for greater years to maturity, can have a significant effect on the present value.

  • Interest rate risk, increases at a decreasing rate.

    • Diff of interest rate risk betn 1 yr bond and 10 yr bond is greater, but this diff is not that greater between 20 yr bond and 30 yrs bond

Interest rate risk contd1

Interest Rate Risk (contd…)

  • Reasons that the bonds with lower coupons have greater interest rate risk:

    • Value of the bond depends on the PV of coupons and the PV of the face value.

    • Value of one with the lower coupon is proportionately more dependent on the discounted value of face value.

    • The bond with higher coupon has a larger cash flow early in its life, so its value is less sensitive to the changes in the discount rate.

What is reinvestment rate risk

What is reinvestment rate risk?

  • Reinvestment rate risk is the concern that r will fall, and future CFs will have to be reinvested at lower rates, hence reducing income.

    EXAMPLE: Suppose you just won

    $500,000 playing the lottery. You

    intend to invest the money and

    live off the interest.

Reinvestment rate risk example

Reinvestment rate risk example

  • You may invest in either a 10-year bond or a series of ten 1-year bonds. Both 10-year and 1-year bonds currently yield 10%.

  • If you choose the 1-year bond strategy:

    • After Year 1, you receive $50,000 in income and have $500,000 to reinvest. But, if 1-year rates fall to 3%, your annual income would fall to $15,000.

  • If you choose the 10-year bond strategy:

    • You can lock in a 10% interest rate, and $50,000 annual income.

Conclusions about interest rate and reinvestment rate risk

Conclusions about interest rate and reinvestment rate risk





CONCLUSION: Nothing is riskless!

Bond values over time

Bond values over time

  • At maturity, the value of any bond must equal its par value (assuming that there’s no risk of default)

  • A bond that is redeemable for Rs 1,000 (which is its par value) after 5 years when it matures, will have a price of Rs 1,000 at maturity, no matter what the current price is.

  • If r (yield) remains constant:

    • The value of a premium bond would decrease over time, until it reached $1,000.

    • The value of a discount bond would increase over time, until it reached $1,000.

    • A value of a par bond stays at $1,000.

The price path of a bond







r = 7%.

r= 10%.

r = 13%.


to Maturity

3025 20 15 10 5 0

The price path of a bond

  • What would happen to the value of this bond if its required rate of return remained at 10%, or at 13%, or at 7% until maturity?

Yield to maturity ytm

Yield To Maturity (YTM)

  • The average annual rate of return investors expect to receive on a bond if they hold it to maturity.


  • YTM of a bond is the interest rate that makes the present value of the cash flows receivable from owning the bond equal to the price of the bond.

P = $A (PVIFA r, n) + $M (PVIF r, n)

Just solve for r = YTM !!!

Valuation of stocks and bonds

YTM Example

Suppose we paid $898.90 for a $1,000 par 10% coupon bond with 8 years to maturity and semi-annual coupon payments.

What is our yield to maturity?



Period/YR = 2

N = 16

PV = 898.90

Coupon per period = 50

FV = 1000


r %= 12%

898.90 = 50 (PVIFA r, 16 ) + 1000 (PVIF r, 16 )

An approximation

An Approximation

Use the same formula with annual coupon and n as no. of years even if the coupon payment is semiannual. (To find out approx YTM)



Consider a Rs 1,000 par value bond, carrying a coupon rate of 9%, maturing after 8 years. The bond is currently selling for Rs 800. What is YTM on this bond ? The YTM is the value of r in the following equation:

Hit and Trial Method:

  • Try r = 12%, RHS = Rs 851.0

  • Try r = 14%, RHS = Rs 768.1

  • Try r = 13%, RHS = Rs 808.0

    Thus, value of r lies between 13% and 14%.

    Use linear interpolation to find exact value of r = 13.2%

Using approximate formula

Using approximate formula

The YTM calculation considers the current coupon income as well as the capital gain or loss the investor will realize by holding the bond to maturity. In addition, it takes into account the timing of the cash flows.

Another approximate formula

Another Approximate Formula

Use the same formula with annual coupon and n as no. of years even if the coupon payment is semiannual. (To find out approx YTM)

For callable bond

For Callable Bond

  • C = annual coupon payment

  • n = term to call

  • Call price can be on premium sometimes.

  • Eg. Call to premium of 9% means, bond will be called at the value 9% greater than the face value

Bond yields

Bond Yields

An example current and capital gains yield

An example: Current and capital gains yield

Find the current yield and the capital gains yield for a 10-year, 9% annual coupon bond that sells for $887, and has a face value of $1,000.

Current yield = $90 / $887

= 0.1015

= 10.15%

Calculating capital gains yield

Calculating capital gains yield

Find YTM = 10.91 %

YTM = Current yield + Capital gains yield


= 10.91% - 10.15%

= 0.76%

Could also find the expected price one year from now and divide the change in price by the beginning price, which gives the same answer.

Zero coupon bonds

$508 $1000

0 10

Zero Coupon Bonds

No coupon interest payments. The bond holder’s return is determined entirely by the price discount.

Suppose you pay $508 for a bond that has 10 years left to maturity. What is your yield to maturity?

PV = FV (PVIF r, n )

508 = 1000 (PVIF r, 10 )

Zero example contd

Zero Example (contd…)

Period /Yr = 1

N = 10

P = 508

FV = 1000


r% per year = 7%

Valuing zero coupon bonds



















Valuing Zero Coupon Bonds

What is the current market price of a U.S. Treasury strip that matures in exactly 5 years and has a face value of $1,000. The yield to maturity is r=7.5%.

What is the yield to maturity on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $591.11?

Bond yields and prices the case of zero coupon bonds

Bond Yields and PricesThe case of zero coupon bonds

  • Consider three zero-coupon bonds with maturity of 1 yr, 3 yrs and 5 yrs, all with

    • face value of F=100

    • yield to maturity of r=10%, compounded annually.

      We obtain the following table:

The impact of price responses

The Impact of Price Responses

  • Suppose the yield would drop suddenly to 9%, or increase to 10%. How would prices respond?

  • Bond prices move up if the yield drops, decrease if yield rises

  • Prices respond more strongly for higher maturities

Default risk

Default risk

  • If an issuer defaults, investors receive less than the promised return. Therefore, the expected return on corporate and municipal bonds is less than the promised return.

  • Influenced by the issuer’s financial strength and the terms of the bond contract.

Evaluating default risk bond ratings

Evaluating default risk:Bond ratings

  • Bond ratings are designed to reflect the probability of a bond issue going into default.

  • Bond ratings are an assessment of the creditworthiness of the bond issuer.

  • Bond ratings don’t address the issue of interest rate risk

Bond ratings

Bond Ratings

Factors affecting default risk and bond ratings

Factors affecting default risk and bond ratings

  • Financial performance

    • Debt ratio

    • TIE ratio

    • Current ratio

  • Bond contract provisions

    • Secured vs. Unsecured debt

    • Senior vs. subordinated debt

    • Guarantee and sinking fund provisions

    • Debt maturity

Other factors affecting default risk

Other factors affecting default risk

  • Earnings stability

  • Regulatory environment

  • Potential antitrust or product liabilities

  • Pension liabilities

  • Potential labor problems

  • Accounting policies

Want to know more about it ?


The bond indenture deed of trust

The Bond Indenture (Deed of Trust)

  • Indenture is the written agreement between the corporation (the borrower) and its creditors detailing the terms of the debt issue.

  • Usually a trustee (a bank, perhaps) is appointed by the corporation to represent the bondholders.

  • Trust company’s jobs:

    • Making sure that the terms of the indenture are obeyed.

    • Managing the sinking fund

    • Representing the bondholders in default

The bond indenture contd

The Bond Indenture (contd…)

  • Bond indenture is a legal document and can run several hundred pages.

  • Includes:

    • The basic terms of the bonds

    • The total amount of bonds issued

    • A description of property used as security

    • The repayment arrangements

    • The call provisions

    • Details of protective covenants

Terms of bond

Terms of Bond

  • Face Value / Par Value / Principal Value

  • Corporate bonds are usually in registered form.

  • This means that the company has a registrar who will record the ownership of each bond and changes on ownership.

  • For eg, it might read as:

    Interest is payable semiannually on July 1 and January 1 of each year to the person in whose name the bond is registered at the close of business on June 15 or December 15 respectively.

  • A corporate bond may be registered and have attached “coupon”

  • Alternatively, the bond could be in bearer form.

    • Difficult to recover if they are lost or stolen

    • Company cannot notify bondholders of important events.



  • Debt securities are classified according to the collateral and mortgages used to protect the bondholder.

  • Collateral – general term that frequently means securities (for eg. Bonds and Stocks) that are pledged as security for payment of debt. However, the term collateral is commonly used to refer to any asset pledged on a debt.

  • Mortgage securities are secured by a mortgage on the real property of the borrower.

  • The property involved is usually real estate.

Security contd

Security (contd…)

  • The legal document that describes the mortgage is called a mortgage trust indenture or trust deed.

  • A blanket mortgage pledges all the real property owned by the company.

  • A debenture is an unsecured bond, for which no specific pledge of property is made.

  • Debenture holders only have a claim on property that remains after mortgages and collateral trusts are taken into account.

  • Note is an unsecured debt usually with a maturity under 10 years.



  • Seniority indicates preference in position over other lenders and debts are sometimes labeled as senioror junior to indicate seniority.

  • Some debt is subordinated

  • The subordinated lenders will be paid off only after the specified creditors have been compensated.

  • However, debt cannot be subordinated to equity.



  • Can be repaid at maturity.

  • The may be repaid in part or in entirely before maturity.

  • Early repayment in some form is more typical and is often handled through a sinking fund.

  • A sinking fund is an account managed by the bond trustee for the purpose of repaying the bonds.

  • The company makes annual payments to the trustee, who then uses the funds to retire a portion of debt.

Repayment contd

Repayment (contd…)

  • The trustee does this by either buying up some of the bonds in the market or calling in a fraction of outstanding bonds (Call Provision)

  • There are many different kinds of sinking fund arrangements:

    • Some sinking fund start about 10 years after the initial issuance.

    • Some sinking fund establish equal payments over the life of the bond.

    • Some high quality bond issues establish payments to the sinking fund that are not sufficient to redeem the entire issue. As a consequence, there is the possibility of a large “balloon payment” at maturity.

The call provision

The Call Provision

  • Allows the company to repurchase or “call” part or all of the bond issue at the stated price over a specific period.

  • Corporate bonds are usually callable.

  • Generally the call price is above the bond’s stated value (par value)

  • The difference between the call price and the stated value is the call provision.

  • The amount of the call premium usually becomes smaller over time.

  • Deferred call provision – Call protected bond – prohibited form calling the bonds for the first few years.

Protective covenants

Protective Covenants

  • The part of indenture or loan agreement that limits certain actions a company might otherwise wish to take during the term of the loan.

  • Classified into two types:

    • Negative Covenants

    • Positive Covenants (affirmative)

  • A negative covenant is a “thou shalt not” type of covenant.

  • A positive covenant is “thou shalt” type of covenant.

Examples of negative covenants

Examples of Negative Covenants

  • The firm must limit the amount of dividends it pays according to some formula.

  • The firm cannot pledge any assets to other lenders.

  • The firm cannot merge with another firm.

  • The firm cannot sell or lease any major assets without approval of the lender.

  • The firm cannot issue additional long-term debt.

Examples of positive covenants

Examples of Positive Covenants

  • The company must maintain its working capital at or above some specified minimum level.

  • The company must periodically furnish audited financial statements to the lender.

  • The firm must maintain any collateral or security in good condition.

End of section

End of section

3.1 Bonds and Bonds Valuation

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