Bond Prices and Yields

11.1 Bond Characteristics

Face or par value Coupon rate

◦ Zero coupon bond Compounding and payments

◦ Accrued Interest Indenture

T Note maturities range up to 10 years T bond maturities range from 10 – 30 years Bid and ask price

◦ Quoted in points and as a percent of par Accrued interest

◦ Quoted price does not include interest accrued

Accrued interest◦ Interest that accrues between coupon payment

datesAccrued interest=(annual coupon payment)/2

*days since last coupon payment/days separating coupon payments, if semiannually paid

Example:◦ Coupon rate 8%, 40 days have passed since the

last coupon payment, if semiannually paidThe accrued interest on the bond

=8%*1000*0.5*40/182=8.79

Most bonds are traded over the counter Registered (record) Bearer bonds (without record) Call provisions (price?) Convertible provision (conversion value) Put provision (putable bonds) (price?) Floating rate bonds (yield spread fixed) Preferred Stock (cumulate, not tax

deductible, offsetting tax adv , 30% of dividend taxed)

Coupon, maturity, price, yield to maturity, rating

Call provisions (price?)◦ Allowing the issuer to repurchase the bond at a

specified call price before the maturity date◦ Refunding: retire high-coupon debt and issue new

bonds at a lower coupon rate◦ Call option valuable to the firm, higher coupon

and promised yields to maturity than non-callable bonds

Put provision (puttable bonds) (price?) ◦ Give the option to the bondholder to extend the

bond’s life, when the coupon rate exceeds current market yield

Floating rate bonds (yield spread fixed) ◦ Interest payments tied to some measure of

current market rate. Yield spread fixed if financial health kept.

Convertible provision (conversion value)◦ Give bondholders an option to exchange each

bond for a specified number of shares of common stock

◦ Conversion ratio, Market conversion value Preferred Stock (cumulate, not tax

deductible, offsetting tax adv) International bonds

◦ Foreign bonds: issued by a borrower from a country other than the one in which the bond is sold

◦ Eurobonds: bonds issued in the currency of one country but sold in other national markets.

International Governments and Corporations◦ Foreign bonds

Foreign Issuer , Denominated in local currency Yankee bonds (in USA) Samurai bonds (in JAPAN) Bulldog bonds (in UK)

◦ Eurobonds Foreign issuer, denominated in foreign currency

Inverse Floaters (suffer doubly and benefit doubly)◦ The coupon rate falls when market rate rises

Asset-backed bonds◦ Income from a specified group of assets is used to

service the debt

Indexed Bonds◦Payments are tied to a general price

index or the price of a particular commodity

◦Example: Treasury inflation protected securities (TIPS), par value tied to general level of prices, coupon payments and final repayment of par value increase in direct proportion to the Consumer Price Index.

◦risk-free real rate

11.2 BOND PRICING

Value a security, discount its expected cash flows by the appropriate discount rate

Bond value= present value of coupons + present value of par value

1 (1 )(1 )

T

TtTt

t

BParValueCP

rr

PB = Price of the bondCt = interest or coupon paymentsT = number of periods to maturityr = semi-annual discount rate or the semi-annual

yield to maturity

Price= coupon *annuity factor (r, T) + Par value * PV factor (r, T)

annuity factor (r, T)=

PV factor (r, T)=

1 1

11

Tr r

1

1T

r

77.148,1

)03.1(

11000

)03.1(

140 20

20

1

P

P

B

ttB

Coupon = 4%*1,000 = 40 (Semiannual)

Discount Rate = 3% (Semiannual)

Maturity = 10 years or 20 periods

Par Value = 1,000

Prices and Yields (required rates of return) have an inverse relationship

When yields get very high the value of the bond will be very low

When yields approach zero, the value of the bond approaches the sum of the cash flows

Coupon rate=8%, semiannual, face value=1000

Maturity : 1-yr , 10-yr, 20-yr, 30-yr Market interest rate, 4%--12%

TIME TO MATURITY

MARKET INTEREST RATE

4% 6% 8% 10% 12%

1￥ -

1,038.83 ￥ -1,019.13 ￥ -1,000.00 ￥ -981.41 ￥ -963.33

10￥ -

1,327.03 ￥ -1,148.77 ￥ -1,000.00 ￥ -875.38 ￥ -770.60

20￥ -

1,547.11 ￥ -1,231.15 ￥ -1,000.00 ￥ -828.41 ￥ -699.07

30￥ -

1,695.22 ￥ -1,276.76 ￥ -1,000.00 ￥ -810.71 ￥ -676.77

Issued at par value Secondary market, price move in

accordance with market forces, fluctuate inversely with the market interest rate

Interest rate fluctuations, main source of risk in fixed-income market

Maturity, key factor of sensitivity Longer maturity, greater sensitivity

Invoice price= flat price + accrued interest

11.3 BOND YIELDS

Measure of rate of return that accounts for both current income and the price increase over the life

Total return: current income and price change

Average rate of return (bought now and held until maturity)

YTM is the discount rate that makes the present value of a bond’s payments equal to its price

Solve the bond formula for r

1 (1 )(1 )

T

TtTt

t

BParValueCP

rr

10 yr Maturity Coupon Rate = 7%

Price = $950

Solve for r = semiannual rate r = 3.8635%

20

1

35 1000950(1 )(1 )

Ttt rr

8% coupon, 30-year bond selling at $1,276.76:

60

601

$40 $1,000$1,276.76

(1 ) (1 )tt r r

r = 3%, semiannual rate

Bond Equivalent Yield (semiannual yield doubled, YTM)7.72% = 3.86% x 2; 6%=3%*2

Effective Annual Yield (accounts for compound interest)(1.0386)2 - 1 = 7.88%; (1.03)2 - 1 = 6.09%

Current Yield (annual coupon payment divided by bond price)Annual Interest / Market Price$70 / $950 = 7.37 %; $80 / $1276.76 = 6.27 %

Coupon rate 7% =70/1000; 8% =80/1000

YTM◦ internal rate of return◦ Compound rate of return over the life

(assumption, all bond coupons can be reinvested at the yield)

◦ Proxy for average return Premium bonds (selling above par)

◦ Coupon rate > current yield > YTM Discount bonds (selling below par)

◦ Coupon rate < current yield < YTM

if the bond is callable Example , Callable at $1100 The call provision allows the issuer to

repurchase the bond at call price Yield to Call

◦ Call price replaces par◦ Time until call replaces time until maturity

8% coupon, 30-year bond selling at $1150, callable in 10 years at call price of 1100

yield to call yield to maturity

coupon payment 40 40

number of semiannual periods 20 60

final payment 1100 1000

price 1150 1150

Reinvestment Assumptions◦ All coupon are reinvested at an interest rate equal

to YTM With a reinvestment rate equal to YTM,

the realized compound yield equal to YTM

Conventional YTM not equal realized compound return, if reinvestment rates can change over time

Example: 2 year, selling at par (1000), coupon rate 10%, annual pay

If interest rate earned on the first coupon is less than 10%., the final value of the investment will be less than 1210, realized compound return will be less than 10%

Example:◦ if reinvest interest rate 8%

21000 1 1208

9.91%

r

r

Horizon analysis◦ Forecasting the realized compound yield over

various holding periods or investment horizons◦ Forecast of total return depends on forecasts of

both the selling price of the bond and the rate to reinvest coupon income

Horizon analysis (Example)◦ 30-year, 7.5% annual payment coupon bond, sell

for 980, YTM is 7.67%, plan to hold for 20 years.◦ Forecast that YTM will be 8% when it is sold,

reinvestment rate on the coupons will be 6%◦ At the end of your investment horizon, the bond

will have 10 years remaining, the forecast sales price (when YTM is 8%) will be 966.45.

◦ 20 coupon payments will grow with compound interest (6%)to 2758.92

◦ 980 investment will grow in 20 years to 966.45+2758.92=3725.37

20980 1 3725.37

6.90%

r

r

10 10

2020

20

1 1 100075 1 966.45

8% 1.08 1.08

1 175 1 1.06 2758.92

6% 1.06

$980(1 ) 966.45 2758.92 3725.37

6.9%

r

r

When interest rates change, bond investors are subject to two sources of offsetting risk◦ Rate rise, bond prices fall, reduce the valued of

the portfolio◦ Reinvested coupon income will compound more

rapidly at those higher rates. The reinvestment rate risk will offset the impact of price risk

11.4 BOND PRICES OVER TIME

HPR = [ I + ( P1 - P0 )] / P0

whereI = interest paymentP1 = price in one period

P0 = purchase price

Requires actual calculation of reinvestment income

Solve for the Internal Rate of Return using the following:◦ Future Value: sales price + future value of

coupons◦ Investment: purchase price

Par bond◦ Coupon rate equals market interest rate (fair

compensation, no further capital gain) Discount Bond

◦ Coupon rate is lower than market rate (need to earn price appreciation, built-in capital gain)

◦ Bond price will increase to par over its maturity◦ Yield to maturity exceeds coupon rate

Discount Bond Example◦ Market rate when issuing, 7% , annual coupon

rate 7%. when the market rate is 8%◦ 3 years left, Bond price= 70* annuity factor (8%,

3) + 1000 * PV factor (8%,3)=974.23◦ 2 years left, Bond price= 70* annuity factor (8%,

2) + 1000 * PV factor (8%, 2)=982.17◦ HPR over this year=(982.17-974.23+70)/974.23=8%

Premium Bond◦ Coupon rate exceeds market rate (investors need

to bid up the price above their par value, capital losses offset the large coupon payment, fair rate)

◦ Coupon rate exceeds yield to maturity◦ Bond price will decline to par over its maturity

YTM , measure of the average rate of return if the bond is held to maturity

If YTM is unchanged over the period, the HPR equals YTM

If YTM fluctuate, so will HPR. Unanticipated changes in market rates will

result in unanticipated changes in bond returns and HPR can be better or worse than YTM which it initially sells.

YTM depends on coupon, current price, par value. Observable today

HPR is a rate of return over a particular investment period and depends on the market price of the bond at the end of the holding period

No coupons and provides all returns in form of price appreciation

Provide only one cash flow on maturity date US. Treasury bills STRIPS (Treasury strips) :break down the

cash flows of a Treasury coupon bond to be paid by the bond into a series of independent securities, each security is a claim to one of the payments of the original bond

Prices of zeros over time◦ On maturity dates, sell par ◦ Before maturity, sell at discounts from par,

approach par value

11.5 DEFAULT RISK AND BOND PRICING

Rating companies◦ Moody’s Investor Service◦ Standard & Poor’s◦ Fitch

Rating Categories◦ Investment grade◦ Speculative grade

Coverage ratiosEBIT/INTEREST, ratio of earnings to all fixed cash

obligations Leverage ratios D/E Liquidity ratios current asset/current liabilities Profitability ratios ROA, ROE Cash flow to debt

Z-score◦ Edward Altman◦ Used discriminant analysis to predict bankruptcy,

get a score based on financial financial ratio

Indenture, contract between the issuer and the bondholder (Protective covenants)◦ Set of restrictions that protect the rights of the

bondholders◦ Provisions relating to collateral, sinking funds,

dividend policy, further borrowing

Sinking funds- Help ensure the commitment of the par value payment at the end of the bond’s life, the firm agrees to establish a sinking fund to spread the payment burden over several years◦ The firm may repurchase a fraction of the

outstanding bonds◦ May purchase a fraction of the outstanding at a

special call price

Subordination of future debt◦ Restrict the amount of additional borrowing.

Additional debt might be required to be subordinated in priority to existing debt

Dividend restrictions Collateral

Promised yield and expected yield◦ The promised or stated yield will be realized

only if the firm meets the obligations of the bond issue, maximum possible YTM

◦ When a bond more subject to default risk, price will fall, its promised YTM will rise, default premium will rise.

Default premium is the difference between the promised yield on a corporate bond and the yield of an otherwise-identical government bond that is riskless in terms of default.

Risk structure of interest rates◦ Greater default risk, higher default premium

Default premiums◦ Yields compared to ratings◦ Yield spreads over business cycles

Major mechanism to reallocate credit risk in the fixed-income markets

First establish a legally distinct entity to buy and later resell a portfolio of bonds or other loans (Structured Investment Vehicle, SIV, often used to create the CDO

Raise funds by issuing short-term commercial paper, using the proceeds to buy corporate bonds or other forms of debt

Loans are pooled together and then split into a series of classes (tranche)

Each tranche is given a different level of seniority in terms of its claims on the underlying loan pool, and each can be sold as a stand-alone security

Proceeds of the loans in the pool are distributed to pay interest to each tranche in order of seniority.

Priority structure implies the different exposure to credit risk.

Mortgage-backed CDOs were an Investment disaster in 2007

CDOs formed by pooling sub-prime mortgage loans made to individuals whose credit standing did not allow them to qualify for conventional mortgages. When home prices stalled in 2007, and interest rates on these typically adjustable-rate loans reset to market levels, mortgage delinquencies and home foreclosures soared, investors lost billions of dollars