Laurence Booth Sean Cleary. LEARNING OBJECTIVES Bond Valuation and Interest Rates 6 6.1 Describe the...

Laurence Booth

Sean Cleary

LEARNING OBJECTIVES

Bond Valuation and Interest Rates66.1 Describe the basic structure and the various features of different

types of bonds.6.2 Explain how to value a bond given an appropriate discount rate.6.3 Determine the discount rate or yield for a given market value of a

bond.6.4 List and describe the factors, both domestic and global, that affect

interest rates.6.5 List and describe the characteristics and pricing of other debt

instruments.6.6 Explain how interest rate parity works.

6.1 THE BASIC STRUCTURE OF BONDS• In the broadest sense, a bond is any debt instrument that

promises a fixed income stream to the holder• Bonds usually have the following characteristics:– A fixed face or par value, paid to the holder at maturity– A fixed coupon, which specifies the interest payable over

the life of the bond– A fixed maturity date

• Fixed income securities are often classified according to maturity:– Bills or paper have maturities less than one year– Notes have maturities between one and seven years– Bonds have maturities greater than seven years

• Bonds may be either bearer bonds or registered bonds

Booth • Cleary – 3rd Edition 3© John Wiley & Sons Canada, Ltd.

• The market price of a bond is the present value of the payments promised by the bond

• The bond indenture is the contract between issuer and holder, which specifies:– Details regarding payment terms– Collateral– Positive or negative covenants– Par value or face value (usually in increments of $1,000)– Bond pricing, usually shown as the price per $100 of par

value which is equal to a percentage of the bond’s face value


6.1 THE BASIC STRUCTURE OF BONDS

• Term to maturity is the time remaining to the maturity date

• Coupon rate is the annual percentage interest paid on the bond’s face value– Multiply the coupon rate by the bond’s face value

to calculate, and divide by two if the coupon is paid semi-annually

– Example: A $1,000 bond with an 8% coupon rate will have an $80 annual coupon or a $40 semi-annual coupon



• Security and Protective Provisions– Mortgage bonds are secured by real assets– Debentures are either unsecured or secured, with

a floating charge over the firm’s assets– Collateral trust bonds are secured by pledged

financial assets, such as common stock, other bonds or Treasury bills

– Equipment trust certificates are secured by pledged equipment, such as railway rolling stock



• Covenants are another type of protective provision– Positive covenants specify actions the firm agrees

to do, such as supply periodic financial statements and maintain certain ratios

– Negative covenants specify actions the firm agrees to avoid, such as restrictions on the size of its debt or acquiring or disposing of assets



Additional Bond Features• Call features allow the issuer to redeem or pay off the

bond prior to maturity, usually at a premium• Retractable bonds allow the holder to extend bonds

back to the issuer before maturity• Extendible bonds allow the holder to extend the bond’s

maturity• Sinking funds are funds set aside by the issuer to ensure

the firm is able to redeem the bond at maturity• Convertible bonds can be converted into common stock

at a pre-determined conversion price



6.2 BOND VALUATION

• Equation 6-1 shows the price of a bond is a function of:– Par or face value , F– Term to maturity, n– Interest or coupon payment, I– The investor’s required rate of return (also known as

discount rate or yield to maturity), kb


6.2 BOND VALUATION Annual Coupons

• Example: The market price of a 5% annual coupon-paying Eurobond with 10 years to maturity, a face value of $1,000 and a yield-to-maturity of 6% is $926.40.

• TI BAII+ Calculator: (2nd) (CLRTVM) 1000 (FV) 50 (PMT) 10 (N) 6 (I/Y) (CPT) (PV) = 926.40


• As Figure 6-3 shows, interest rates are inversely related to bond prices– Bond prices increase when interest rates decrease– Bond prices decrease when interest rates increase


6.2 BOND VALUATION Factors Affecting Bond Prices

• The relationship between the coupon rate and the bond’s yield to maturity (YTM) determines if the bond will sell at a premium, a discount, or at par


Coupon vs YTM Price vs Face Value Pricing

Coupon Rate < YTM Price < Face Value Discount

Coupon Rate = YTM Price = Face Value At Par

Coupon Rate > YTM Price > Face Value Premium


6.2 BOND VALUATION Semi-Annual Coupons• Example: Find the price of a 4% semi-annual coupon Government of Canada

bond has five years to maturity, $1,000 in par value and a YTM of 6%.• Since coupons are paid semi-annually, we must make three changes:• Divide the coupon payment by 2• Multiply the number of years by 2 (since we have twice as many semi-annual

periods• Divide the yield to maturity by 2

• TI BAII+ Calculator: (2nd) (CLRTVM) 1000 (FV) 20 (PMT) 10 (N) 3 (I/Y) (CPT) (PV) = 914.70


• Yield to maturity, time to maturity and size of coupon affect the price volatility of a bond

• Yield to Maturity– Bond prices increase when the YTM decreases– Bond prices decrease when the YTM increases– The price-YTM relationship is convex to the origin (see Figure 6-2)– Because of convexity, the price rise due to a fall in YTM is greater than

the price decline due to a rise in YTM given an identical change in YTM– For a given change in YTM, bond prices will change more when

interest rates are low than when they are highBooth • Cleary – 3rd Edition 14© John Wiley & Sons Canada, Ltd.


• Time to Maturity– Long-maturity bonds have greater price volatility than short-maturity bonds– The longer the bond, the longer the period for which the cash flows are

fixed• Size of Coupon

– Low coupon bonds have greater price volatility than high coupon bonds– High coupon bonds act like a stabilizing device, since a greater proportion of

the bond’s total cash flows occur closer to today and are therefore their present value is less affected by a change in YTM

• Interest Rate Risk and Duration– The sensitivity of a bond’s price to change in interest rates is a measure of

the bond’s interest rate risk. Interest rate risk is affected by: yield to maturity, term to maturity and size of coupon

– These impact of interest rate risk is measured using duration



• Duration measures of interest rate risk as a change in price for a given change in interest rates

• The higher a bond’s duration, the more sensitive its price is to changes in interest rates

• A bond’s duration will be higher if its:– YTM is lower– Term to maturity is longer– Coupon is lower

• Bond Quotations– Example: This quotation shows a 5.000% CIBC bond maturing on

September 10, 2010. Its YTM is 2.75% and its price is 106.97% of face value.



• The quoted price is the price reported by the media• The cash price is the price paid by an investor, and includes both the

quoted price plus any interest that has accrued since the last coupon payment date

• Example: Suppose you want to purchase a $1,000 bond with a 5% coupon, paid semi-annually. Today is July 15th and the last coupon was paid on June 30th. If the quoted price is $902, how much is the cash price?– The cash price will be equal to the quoted price ($902) plus 15 days of

accrued interest (between June 30th and July 15th)


05.904$05.2$902$

365

1505.0000,1$902$

Interest Accrued Price Quoted PriceCash

6.2 BOND VALUATION Cash Prices versus Quoted Prices

6.3 BOND YIELDSYield to Maturity (YTM/kb)

• The yield to maturity (YTM) is the discount rate used for bond valuation

• YTM is the yield an investor would earn if:– She purchases the bond at the current market price– She holds the bond to maturity– She reinvests all of the coupons paid by the bond at the YTM

• YTM is, therefore, the bond’s internal rate of return (IRR)• YTM is, also, the discount rate that causes the present value

of the bond’s future cash flows to equal its current price


• There is no closed-form algebraic solution to the following equation for YTM, so we must use a computer or calculator to solve YTM

• Example: What is the YTM on a 6% semi-annual coupon bond with 20 years to maturity that is selling for $1,030?

• The result is 2.87% each semi-annual period, or 5.74% per year. Remember to express YTMs as annual rates, not semi-annual rates.



• Example: What is the YTM on a 6% semi-annual coupon bond with 20 years to maturity that is selling for $1,030?

• Excel: = RATE(nper,pmt,pv,fv,type,guess) = RATE(40,30,-1030,1000,,) = 2.87% (multiply by 2 to annualize)

• TI BAII+ Calculator: (2nd) (CLRTVM) 30 (PMT) -1030 (PV) 1000 (FV) 40 (N) (CPT) (I/Y) = 2.87, then multiply by 2 to annualize– Always enter the price as a negative number since it is a cash

outflow to purchase the bond, and enter the coupon and principal payments as positive numbers since these are cash inflows



• If a bond has a call feature, the issuer can call (or force the investor to sell the bond back to it) before the maturity stated in the bond indenture

• Callable bonds are initially protected from call for a period of a few years (e.g., five, seven or ten), after which the issuer may call the bond

• Replace the maturity date with the first call date, the face value with the call premium (CP), and the time to maturity (n) with the number of periods until expected call (c) to calculate the yield to call (YTC), as in Equation 6-2:


6.3 BOND YIELDSYield to Call (YTC/kc)

• Example: What is the YTC on a 6% semi-annual coupon bond with 20 years to maturity that is selling for $1,030? This bond is callable in five years with a call price of $1,050.

• Excel: = RATE(nper,pmt,pv,fv,type,guess) = RATE(10,30,-1030,1050,,) = 3.08 % (multiply by 2 to annualize) = 6.16%

• TI BAII+ Calculator: (2nd) (CLRTVM) 30 (PMT) -1030 (PV) 1050 (FV) 10 (N) (CPT) (I/Y) = 3.08 then multiply by 2 to annualize = 6.16%– Always use the call premium as the face value and the

expected call date as the time to maturity to find the yield to earliest call


6.3 BOND YIELDSYield to Call (YTC/kc)

• A bond’s current yield (CY) is the yield on the bond’s current market price provided by the annual coupon

• Since current yield does not consider potential capital gains or losses, it’s not a true measure of return to the bondholder

• Example: What is the current yield of a 5.5% coupon bond with a current market price of $1,050?

• Using Equation 6-3:


%24.5050,1$

55$Interest Annual

BCY

6.3 BOND YIELDSCurrent Yield

6.4 INTEREST RATE DETERMINANTSBase Interest Rates

• Interest is the “price” of borrowed money• Interest is measured in basis points, or 1/100th of 1% (e.g., 250

basis points is 2.5%)• Interest rates rise when the demand for loanable funds rises• Interest rates fall when the demand for loanable funds falls• The Risk-Free Interest Rate

– The risk-free rate is an abstract concept, and usually the yield on short-term government treasury bills is used as a proxy for practical purposes

– The risk-free rate is comprised of two components• The real rate, which is compensation for deferring consumption• The expected inflation rate, which is compensation for the expected

loss of purchasing power over the term of the short-term T-bill


• Figure 6-3 shows there is, generally, a relationship where interest rates respond to the rate of inflation


6.4 INTEREST RATE DETERMINANTSBase Interest Rates

Figure 6-4 Interest Rates and Inflation

• Canadian interest rates are heavily influenced by changes in interest rates in other countries

• Macroeconomic factors, both domestic and global, also play an important role

• Interest rate parity theory (IRP) states that foreign exchange forward rates will be established that equalize the yield an investor can earn, whether investing domestically or in a foreign jurisdiction

• Example: A country with both high inflation and high interest rates will have a depreciating currency


6.4 INTEREST RATE DETERMINANTSGlobal Influences on Interest Rates

• The term structure of interest rates is the set of rates (YTM) for all maturities of a given risk-class of debt securities (e.g., Government of Canada bonds) at a given point in time

• When plotted on a graph the result is called a yield curve• There are three typical shapes for the yield curve: upward sloping or

normal (1994), downward sloping or inverted (1990) and flat (1998)


6.4 INTEREST RATE DETERMINANTSThe Term Structure of Interest Rates

Figure 6-5 Historical Yield Curves 1990, 1994, 1998, 2004, and 2012

Three Theories:• Liquidity preference theory posits that investors must be paid

a liquidity premium in order to be compensated for the interest rate risk inherent in holding less liquid, longer-term debt

• Expectations theory suggests that longer-term interest rates are the result of expectations of future short-term interest rates. Or, in other words, the interest rates of various maturities are dependent on each other

• Market segmentation theory suggests that different markets exist for securities of different maturities that therefore the two ends of the yield curve can have different factors affecting them


6.4 INTEREST RATE DETERMINANTSThe Term Structure of Interest Rates

• More risky bonds (i.e., BBB-rated corporate bonds) will have their own yield curve and it will plot at higher YTM at every maturity than government bonds because of the additional default risk that BBBs carry

• The yield spread is the difference between the YTM on a BBB-rated corporate bond and a Government of Canada bond of the same maturity and it represents the default risk premium investors demand for investing in the more risky corporate bond

• Yield spreads widen during recessions and narrow during times of economic expansion


6.4 INTEREST RATE DETERMINANTSRisk Premiums

• Equation 6-5 can be used to determine the YTM on a corporate bond:

– RF is the risk-free rate– The maturity yield differential is the extra yield

required for taking on a longer maturity– The spread is the additional yield required for default

risk


Spread aldifferenti yieldMaturity RFkb


• Debt ratings – rating agencies, such as the Dominion Bond Rating Service (DBRS), Standard & Poors (S&P), and Moody’s assign all publicly traded bonds a risk rating



Figure 6-6 Debt-Rating Categories

for Standard & Poor’s, Dominion

Bond Rating Service, and

Moody’s

Risk, Liquidity, and Bond Features:• The greater the default risk, the higher the required

YTM• The less liquid the bond, the higher the required YTM• Call features generally increase the required YTM• Extendable bonds generally have lower required

YTMs• Retractable bonds generally have lower required

YTMs


6.4 INTEREST RATE DETERMINANTS

6.5 OTHER TYPES OF BONDS/DEBT INSTRUMENTS Treasury Bills

• Treasury bills are short-term obligations of the government with an initial term to maturity of one year or less

• Issued at a discount to face value with face value being paid at maturity• The difference between the discounted issue price and the face value is

treated as interest income• Equation 6-7 can be used to value a Treasury bill:

where, P = the market price of the T-billF = the face value of the T-billkBEY = the bond equivalent yield

n = the number of days until maturity


• Example: What is the price of a $1 million Canadian treasury bill with 80 days until maturity and a bond-equivalent yield of 4.5%?

• Example: What is the yield on a $100,000 Treasury bill with 180 days until maturity and a market price of $98,200?


32.233,990$

365

80045.01

000,000,1$

3651

bill-T

n

k

FP

BEY

%72.3180

365

200,98$

200,98$000,100$365

nP

PFkBEY

6.5 OTHER TYPES OF BONDS/DEBT INSTRUMENTS Treasury Bills

6.5 OTHER TYPES OF BONDS/DEBT INSTRUMENTS Zero Coupon Bonds

• Zero coupon bonds are bonds issued at a discount which pay no coupons and mature at par or face value

• Since no coupons are paid, there is no reinvestment rate risk• Equation 6-9 shows the price of a zero-coupon bond is the

present value of the face value of the bond:

• Example: What is the market price of a $50,000 zero coupon bond with 25 years to maturity that is currently yielding 6%?


93.649,11$)06.1(

000,50$

)1( 25

nbk

FB

6.5 OTHER TYPES OF BONDS/DEBT INSTRUMENTSFloating Rate, Real Return, and Canada Savings Bonds• Floating rate bonds have coupon rates that float with some

reference rate, such as the yield on Treasury bills– Since the coupon rate floats, or is variable, the market price will

typically be close to the bond’s face value• Real return bonds are issued by the Government of Canada to

protect investors against unexpected inflation– Each period, the face value of the real return bond is grossed up by

the inflation rate. The coupon is then paid on the grossed up face value

• Canada Savings Bonds (CSBs) are issued by the Government of Canada as either regular interest bonds (interest paid annually) or compound interest bonds (interest compounds over the life of the bond) – There is no secondary market for Canada Savings Bonds; instead, they

are redeemable at any chartered bank in Canada at their face valueBooth • Cleary – 3rd Edition 36© John Wiley & Sons Canada, Ltd.

6.6 INTEREST RATE PARITY

• The interest rate parity (IRP) theory demonstrates why differences in interest rates between countries should be offset by forward exchange rates

• If no arbitrage opportunities exist, Equation 6A-1 should hold:

where,F = current forward exchange rate (domestic units for foreign units)S = current spot exchange ratekdomestic = domestic interest rate

kforeign = foreign interest rate


foreign

domestic

k

k

S

F

1

1

• Example: Assume U.S interest rates are presently 1.25% on one-year American T-Bills, that the U.S. dollar is quoted at USD$1 = CAD$1.0500, and that the interest rate on one-year Canadian T-Bills is 1.00%. Find the one-year forward exchange rate.


0474.1$0125.1

01.10500.1

1

1

foreign

domestic

k

kSF


• Example continued: Can an investor benefit from higher U.S. interest rates without assuming risk? Suppose there are two Canadian investors, both with $1,000 to invest. Investor A invests domestically while investor B invests in the U.S. and eliminates foreign exchange risk with a forward contract.

• Investor A’s ending wealth: $1,000(1.01) = $1,010• Investor B’s ending wealth:

• Notice that if $1.0474 is the prevailing forward rate, both investors earn the same amount


010,1$05.1

000,1$0125.10474.1


© John Wiley & Sons Canada, Ltd.

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Booth • Cleary – 3rd Edition 40

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http://www.wiley.com/go/boothcanada

Copyright © 2013 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (the Canadian copyright licensing agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these files or programs or from the use of the information contained herein.

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