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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