Chapter 11

BOND PRICES AND YIELDS

Figuring out the Assured Returns

Outline

• Changes in Bond Market

• Bond Characteristics

• Bond Prices

• Bond Yields

• Risks in Bonds

• Rating of Bonds

• The Yield Curve

• Explaining the Term Structure

• Determinants of Interest Rates

• Analysis of Convertible Bonds

Changes In Debt MarketThen Now

• Plain vanilla bonds • Bonds with complex features

• Stable & administered • Volatile & market- determined interest rates interest rates

• Simplistic measures of • Precise measures of return & life return & life

• Rules of thumb • Analytical methods

• Few players • More players

• Passive approach • Relatively more active approach• Illiquid market • Liquid market ?

• Absence of a reference • Emergence of a reference rate rate

Bond Characteristics

• A bond is an IOU. It is described in terms of:

• Par value

• Coupon rate

• Maturity date

• Government bonds are also called government securities (G-secs) or gilt-edged securities. These are generally medium to long-term bonds issued by RBI on behalf of the government of

India and state governments.

• Corporate bonds or corporate debentures are debt instruments issued by companies

Types Of Bonds

• Straight bonds

• Zero coupon bonds.

• Floating rate bonds

• Bonds with embedded options

• Commodity-linked bonds

Bond Pricing (Valuation)

n C MP = +

t=1 (1+r)t (1+r)n

2n C/2 MP = +

t=1 (1+r/2)t (1+r/2)2n

Bond PricingA Rs.600 face value bond carries a coupon rate of 12 percent p.a. payable semi-annually. The bond is redeemable at par after 5 years. If investors require a return of 9% per half-year period, what will be the price of the bond.

10 36 600 P0 = +

t=1 (1.09)t (1.09)10

= 36 x 6.418 + 600 x 0.422 = Rs. 484.25

1 – F P = C +

r (1+r)n

36 x 6.418 + 600 x 0.422 = 484.25

1 (1+r)n

Price-Yield RelationshipPrice

Yield

Price changes with timeValue of Bond Premium Bond: rd = 11%

A

PAR VALUE BOND: rd = 13%

BDiscount Bond: rd = 15%

8 7 6 5 4 3 2 1 0

YEARS TO MATURITY

Bond Price Theorems1. Bond prices & yields move in opposite directions.

2. Bond prices are more sensitive to yield changes the longer their maturities.

3. The price sensitivity of bonds to yield changes increases at a decreasing rate with maturity.

4. High coupon bond prices are less sensitive to yield changes than low coupon bond prices.

5. With a change in yield of a given number of basis points, the associated percent gain is larger than the percent loss.

Bond Yields• Current Yield

Annual interest Price

• Yield To Maturity C C C M

P = + + …. + (1+r) (1+r)2 (1+r)n (1+r)n

8 90 1,000 800 = +

t=1 (1+r)t (1+r)8

AT r = 13% … RHS = 808 AT r = 14% … RHS = 768.1

808 - 800YTM = 13% + (14% - 13%) = 13.2%

808 - 768.1 C + (M - P) / n

YTM ≃ 0.4M + 0.6 P

• Yield to Call n* C M*

P = + t=1 (1+r)t (1+r)n

YTM

• Someone who invests in a coupon - paying bond will earn the YTM promised on the purchase date if and only if all of the following three conditions are fulfilled.

• The bond is held until it matures rather than being sold at a price which differs from its face value before its maturity

• The bond does not default

• All cash flows are re-invested at an interest rate equal to the promised YTM

Realised Yield To MaturityFuture Value of Benefits

(1+r*)5 = 2032 / 850 = 2.391 r* = 0.19 OR 19 PERCENT

0 1 2 3 4 5 INVESTMENT 850 ANNUAL INTEREST 150 150 150 150 150 RE-INVESTMENT

PERIOD (IN YEARS) 4 3 2 1 0 COMPOUND FACTOR

(AT 16 PERCENT) 1.81 1.56 1.35 1.16 1.00 FUTURE VALUE OF

INTERMEDIATE CASH FLOWS 271.5 234.0 202.5 174.0 150.0 MATURITY VALUE 1000

TOTAL FUTURE VALUE = 271.5 + 234.0 + 202.5 + 174.0 + 150.0 + 1000= 2032

Realised Yield to Maturity

The realised yield to maturity appears to be conceptually superior to

the conventional yield to maturity. However, its appeal seems to be

dubious because it is difficult to forecast the future reinvestment

rates, given the uncertainty characterising them. While the

conventional yields to maturity on coupon bonds are difficult to

interpret, realised yields to maturity are difficult to implement.

Stated YTM and Expected YTM

Corporate bonds are subject to default risk. So, you must distinguish

between the bond’s stated YTM and the bond’s expected YTM. The

stated or promised YTM will be realised only if the issuing firm meets

all the obligations on the bond issue. Thus, the stated YTM is the

maximum possible YTM on the bond. The expected YTM, however,

takes into account the possibility of a default.

Expected YTM and Stated YTM

An example may be given to illustrate the difference between the two measures of YTM. Alpha Corporation issued 12 percent coupon bonds 10 years ago. The bonds now have five years left until its maturity. Alpha is experiencing financial difficulties. Bondholders believe that Alpha will meet the remaining interest payments, but at the time of maturity bondholders will receive only 80 percent of par value. The bond is currently selling at Rs 850.

The following inputs would be used to calculate YTM:

Inputs Expected YTM Stated YTMCoupon payment Rs. 60 Rs 60Number of semiannual periods 10 periods 10 periodsFinal payment Rs. 800 Rs 1000Price Rs. 850 Rs 850

Using the approximate formula the YTM based on expected payments works out 6.63 percent, whereas the YTM based on promised payments works out to 8.24 percent.

YTM versus Holding Period Return

Don’t confuse the yield to maturity (YTM) of a bond with its holding period return. The YTM is the single discount rate at which the present value of payments received from the bond equals its price. It represents the average rate of return from the bond if it is held till maturity. In contrast, the holding period return is the income earned over a given holding period as a percentage of its price at the beginning of the period.

For example, if a 10 year Rs. 1000 par bond paying an annual coupon of Rs. 90 is bought for Rs. 1,000, its YTM is 9 percent. If the bond price increases to Rs. 1060 by year end, its YTM will fall below 9 percent (because it is selling at a premium), but its holding-period return for the year exceeds 9 percent:

90 + (1060 – 1000)Holding period return = = 0.15 or 15 percent

1000

Risks In Bond Investment• Interest rate risk Interest Bond (market risk) Rate Price

• Reinvestment Interest rate on Risk Interim cash flow

• Default risk Issuer may default (credit risk)

• Inflation risk Purchasing power risk

• Call risk Issuer may recall the bonds

• Exchange rate risk A non-rupee denominated bond

• Liquidity risk Marketability risk

• Event risk Issuer’s ability.. Change.. Unexpectedly(a) a natural accident or (b) .. corp. Restr’g

Debt Rating

• What is it ?

probability of timely payment of interest & principal by a borrower

• What it ‘is not’ ?

Not a recommendation

Not a general eval’n of the issuing organisation

Not a one-time evaluat’n credit risk . . valid entire life

• How is it done ?

Industry & bus analysis. Financial analysis

Quantitative rating models

• Value of ratings ?

• rating scenario ?

Functions Of Debt RatingDebt ratings (or debt rating firms) are supposed to :

• provide superior information

• Offer low-cost information

• Serve as a basis for a proper risk-return tradeoff.

• Impose healthy discipline on corporate borrowers.

• Lend greater credence to financial and other representations.

•Facilitate the formulation of public policy guidelines on institutional investment.

Credit Rating• Crisil’s Rating Symbols

AAA : Highest Safety

AA : High Safety

A : Adequate Safety

BBB : Low Safety

BB : Inadequate Safety

B : High Risk

C : Substantial Risk

D : In Default

• Key factors considered in credit rating

Industry & Business Analysis Financial Analysis

• Growth rate & rel’n with the economy • Earning power

• Industry risk characteristics • Business & financial risks

• Structure of industry & nature • Asset protection of competition

• Competitive position of the issuer • Cash flow adequacy

• Managerial capability of the issuer • Financial flexibility

• Quality of accounting

The Yield CurveThe yield curve., Or the term structure of interest rates, shows how YTM is related to term to maturity for bonds that are similar in all respects, expecting maturity.

YIELD CURVE

YIELD TO MATURITY(YTM)

14.0

13.0

12.0

1 2 3 4 5 TERM TO MATURITY (YRS)

FACEVALUE

INTERESTRATE

MATURITY(YRS)

CURRENTPRICE

YIELD TOMATURITY

100,000 0 1 88.968 12.40100,000 12.75 2 99,367 13.13

100,000 13.50 3 100,352 13.35

100,000 13.50 4 99,706 13.60

100,000 13.75 5 99,484 13.90

Types Of Yield CurveYTM A. Upward Sloping YTM B. Downward sloping

Term Term

YTM C. Flat YTM D. Humped

Term Term

Illustrative Data for Government Securities

Face Value Interest Rate Maturity Current Price Yield to (years) maturity

100,000 0 1 88,968 12.40

100,000 12.75 2 99,367 13.13

100,000 13.50 3 100,352 13.35

100,000 13.50 4 99,706 13.60

100,000 13.75 5 99,484 13.90

 

Forward Rates

To get forward interest rates, begin with the one-year treasury bill:88,968 = 100,000 / (1 + r1)

where r1 is the one-year spot rate i.e. the discount rate applicable to a riskless cash flow receivable a year hence. Solving for r1 gives r1 = 0.124. Next, consider the two-year government security and split its benefits into two parts, the interest of Rs. 12,750 receivable at the end of year 1 and Rs.112,750 (representing the interest and principal repayment) receivable at the end of year 2. The present value of the first part is: 

12,750 12,750 =

(1+r1) 1.124

r1 11,343.4

Forward Rates

To get the present value of the second year’s cash flow of Rs.112,750, discount it twice at r1 (the discount rate for year 1) and r2 (the discount rate for year 2):  112,750 112,750

=(1+r1) (1+r2) (1.124) (1+r2)

 r2 is called the ‘forward rate’ for year two, that is, the current estimate of the next year’s one-year spot interest rate. Since r1, the market price of the bond, and the cash flow associated with the bond are known the following equation can be set up: 

12,750 112,75099,367 = +

(1.124) (1+r2)

Forward RatesSolving this equation gives r2 = 0.1289To get the forward rate for year 3(r3), set up the equation for the value of the three-year bond:

13,500 13,500 113,500100,352 = + +

(1+r1) (1+r1) (1+r2) (1+r1) (1+r2) (1+r3) 

13,500 13,500 113,500100,352 = + +

(1.124) (1.124) (1.1396) (1.124) (1.1396) (1+r3) Solving this equation gives r3 = 0.1389. This is the forward rate for year three.Continuing in a similar vein, set up the equation for the value of the four-year bond: 

13,500 13,500 13,500 113,50099,706 = + + +

(1+r1) (1+r1) (1+r2) (1+r1) (1+r2) (1+r3) (1+r1) (1+r2) (1+r3) (1+r4)  13,500 13,500 13,500 113,500 = + + + (1.124) (1.124) (1.1396) (1.124)(1.1396)(1.1389) (1.124)(1.1396)(1.1389)(1+r4) 

Solving this equation for r4, leads to r4 = 0.1458. Exhibit 11.7 plots the one-year spot rate and forward rates r2, r3, r4. Notice that while the current spot rate and forward rates are known, the future spot rates are not known - they will be revealed as the future unfolds.

Forward Rates

Forward rate

15.0--

14.0--

13.0--

12.4--

1 2 3 4

Year

Explaining The Term Structure

• Expectations theory

• Liquidity preference theory

• Preferred habitat theory

• Market segmentation theory

Expectations Theory

Shape …. Yield curve … depends on .. expectations … those who participate … market

(1 + tRn) = [ (1 + tR1) (1 + t+1R1) … (1 + t+n-1Rn)]1/n

Yield Curve Explanation

Ascending Short-term rates rise in future

Descending Short-term rates … fall in future

Humped Short-term rates … rise …. fall

Flat Short-term rates . . unchanged in future

Liquidity Preference TheoryForward rates should incorporate interest rate expectations as well as a risk (or liquidity) premium

(1 + tRn) = [ (1 + tR1) (1 + t+1R1+L2) … (1 + t+n-1Rn +Ln)]1/n

An upward-sloping yield curve suggests that future interest rates will rise (or will be flat) or even fall if the liquidity premium increases fast enough to compensate for the decline in the future interest rates.

Preferred Habitat Theory

Investors prefer to match the maturity of investment to their investment objective

Borrowers . . too

If mismatch … inducement to shift

Market segmentation theory

Extreme form of preferred habitat theory

Determinants Of Interest Rates• Inflation Rate

• Real Growth Rate

• Time Preference

Short-Term Risk-Free Rate

Maturity Premium

•Future Expectations

•Liquidity Preference

•Preferred Habitat

Default Premium

•Business Risk

•Financial Risk

•Collateral

Special Features

•Call/Put Feature

•Conversion Feature

•Other Features

Interest Rate

Valuation Of Convertible BondsValue of the Straight Conversion Option

Convertible = Max Bond Value + Value

Bond Value ,

Valuation of Convertible Bonds

A. Straight Debt Value B. Conversion Value C. Value of Convertible Bond

Firm Value Firm Value Firm Value

STRAIGHT DEBT

VALUE

CONVER SION VALUE

VALUE

OF

CONVERTIBLE BONDS

Summing Up

• The debt market in India has registered an impressive growth

particularly since mid-1990s and has been accompanied

by increasing complexity in instruments, interest rates,

methods of analysis, and so on .

• The value of a non callable, nonconvertible bond is:

n C M P = + t =1 (1+ r)t (1 + r)n

• The commonly employed yield measures are : current yield,

yield to maturity (YTM), yield to call, and realised yield

to maturity.

• The YTM of a bond is the discount rate that makes the

present value of the cash flows receivable from owning the

bond equal to the price of the bond.

• Bonds are subject to diverse risks, such as interest rate risk,

inflation risk, real interest rate risk, default risk, call risk,

and liquidity risk.

• Default risk or credit risk is reflected in credit rating of debt

instruments.

• The term structure of interest rates, popularly called the

yield curve, shows how yield is related to maturity.

• Three principal explanations have been offered to explain

the term structure of interest rates : expectations theory,

liquidity preference theory, and preferred habitat theory.

• The interest rate is determined by four factors : short-

term risk-free interest rate, maturity premium, default

premium, and special features.

• Convertible bonds may be viewed as a debenture –

warrant package.