7
Bonds Chapter and Their Valuation
Chapter 7 Topic Overview
x Bond Characteristics x Annual and Semi-Annual Bond Valuation x Reading Bond Quotes x Finding Returns on Bonds x Bond Risk and Other Important Bond Valuation
Relationships
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Bond Characteristics
Par (or Face) Value (M) = stated face value that
is the amount the issuer must repay, usually $1,000 Coupon Interest Rate Coupon (INT) = Coupon Rate x Face Value Maturity Date = when the face value is repaid. A legal contract called the bond indenture specifies these values. This makes a bonds cash flows look like this:
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Bond Valuation
Discount the bonds cash flows at the
investors required rate of return.
the coupon payment (INT) stream (an annuity). the par (M) value payment (a lump sum). V = INT (PVA r, n) + M /(1+r)n
INT 0 1
4
INT
INT+M
2
...
n
Bond Valuation Example #1
x Duffs Beer has $1,000 par value bonds
outstanding that make annual coupon payments. These bonds have a 7.5% annual coupon rate and 15 years left to maturity. Bonds with similar risk have a required return of 9%, and Moe Szyslak thinks this required return is reasonable. x Whats the most that Moe is willing to pay for a Duffs Beer bond?
5
? 0
75 1
r = 9%
75 2
75 3
... ...
1000 75 15
Lets Play with Example #1
xHomer Simpson is interested in
buying a Duff Beer bond but demands an 7.5 percent required return. xWhat is the most Homer would pay for this bond?
7
? 0
75 1
r = 7.5%
75 2
75 3
... ...
1000 75 15
Lets Play with Example #1 some more.
xBarney (belch!) Gumble is interested
in buying a Duff Beer bond and demands on a 6 percent required return. xWhat is the most Barney (belch!) would pay for this bond?
9
? 0
75 1
r = 6%
75 2
75 3
... ...
1000 75 15
Lesson from Example 1: Bond Prices and Interest Rates have an inverse relationship!
Bond Values for 7.5% Annual Coupon Bond
1400 1200 1000 800 600 400 200 0 0% 2% 4% 6% 8% 10% Required Return ($)M arket Value
15-yr Bond
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Another Example 1 Lesson: Bond Premiums and Discounts
What happens to bond values if required return
is not equal to the coupon rate?
The bond's price will differ from its par value
r > Coupon Interest Rate P0 < par value
=DISCOUNT =
PAR
r = Coupon Interest Rate
P0 = par value
r < Coupon Interest Rate
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P0 > par value
= PREMIUM
Bonds with Semiannual Coupons
x Double the number of years, and divide required
return and annual coupon by 2.
Vd = INT/2(PVAr/2,2n) + M /(1+r/2)2n
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Semiannual Example
x Kwickee-Mart has an $1000 par value bond with
an annual coupon rate of 6% that pays coupons semiannually with 20 years left to maturity. What is the most you would be willing to pay for this bond if your required return is 7% APR? x Semiannual coupon = 6%/2($1000) = $30 x 20x2 = 40 remaining coupons
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30 0 1
30 2
30 3
... ...
1000 30 40
Bond Yields
Current Yield - Annual coupon payments divided
by bond price. Yield To Maturity - Interest rate for which the present value of the bonds payments equal the price. Also known as the markets required rate of return.
Yield To Maturity = total expected return = current
yield + expected capital gains yield (change in price)
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Yield Definitions
Annual coupon payment Current yi (CY) = eld Current price Change price in Capital gainsyield(CGY)= Beginningprice Expected Expected Expected totalreturn= YTM = CY + CGY
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Bond Yields
Calculating Yield to Maturity (YTM=r) If you are given the price of a bond (PV) and the coupon rate, the yield to maturity can be found by solving for r.
cpn cpn (cpn + par ) PV = + +....+ 1 2 (1 + r ) (1 + r ) (1 + r ) t
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Yield to Maturity Example
x Burns Power $1000 face value bond with a 5%
coupon rate paid annually with 10 years left to maturity sells for $890. x What is this bonds yield to maturity?
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-890 0
50 1
50 2
50 3
... ...
1000 50 10
What is bonds YTM?
What is the bonds current yield?
U.S. Treasury Bond Quotations
RATE MATURITY MO/YR Feb 20 BID ASKED CHG ASK YLD 4.67
Government Bonds & Notes
8.50 136:30 136:31 +42
Rate
Coupon rate of 8.5% Bid price: the price traders receive if they sell a bond to the dealer. Quoted in increments of 32nds of a %age of par Ask price: the price traders pay to the dealer to buy a bond Bid-ask spread: difference between ask and bid prices.
Bid prices Ask prices (percentage of par value)
Ask Yield 21
Yield to maturity on the ask price
Verifying the T-bonds YTM (Ask Yld)
Par Value = $1000, semi-annual coupons Ask Price = 136.969%($1000) = $1,369.69 INT/2 = 8.5%($1000)/2 = $42.50 N = 2020-2007 = 13, 2N = 26
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Causes of Bond Price Changes
Since a bonds cash flows are fixed: 2. Changes in interest rates, and 3. Passage of time. cause changes in a bonds price.
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Bond Value Changes Over Time
x Returning to the Duffs Beer original example #1,
where r = 9%, N = 15, C (PMT) = $75, par (FV) = $1000, & PV = $879.09. x What is bond value one year later when N = 14 and r is still = 9%?
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Bond values over time
At maturity, the value of any bond must equal its
par value. If rd 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.
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Changes in Bond Value over Time
What would happen to the value of these three
bonds is bond if required its rate of return remained at 10%:
VB
1,184 1,000 816
10
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13% coupon rate 10% coupon rate.
7% coupon rate 5 0 Years to Maturity
What about this?
What if Barney buys Duff Bond at $1145.68 with
n =15 and sells at r = 7% with n = 14. What is his return?
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Interest Rate Risk
Measures Bond Price Sensitivity to changes in
interest rates. In general, long-term bonds have more interest rate risk than short-term bonds. Also, for bonds with same time to maturity, lower coupon bonds have more interest rate risk than higher coupon bonds.
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Interest Rate Risk Example
Recall from our earlier example (#1), the 15-year,
7.5% annual coupon bond has the following values at kd = 6%, 7.5%, & 9%. Lets compare with a 2-yr, 7.5% annual coupon bond. 15-year bond 2-year bond
r=6%: PV = $1,145.68 r=7.5%: PV = $1,000 r=9%: PV = $879.09 PV = $1,027.50 PV = $1,000 PV = $973.61
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Interest Rate Risk: Bond Price Sensitivity Graph
Bond Values for 7.5% Annual Coupon Bonds
2400 2150 1900 1650 1400 1150 900 650 400 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%
2-yr Bond 15-yr Bond 30-yr Bond
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Reinvestment rate risk
Reinvestment rate risk is the concern that
rd 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.
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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 1year 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. annual income.
If you choose the 10-year bond strategy:
You can lock in a 10% interest rate, and $50,000
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Conclusions about interest rate and reinvestment rate risk
Short-term AND/OR High coupon bonds Interest rate risk Reinvestment rate risk Low High Long-term AND/OR Low coupon bonds High Low
CONCLUSION: Nothing is riskless!
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Default Risk
Credit risk Default premium Investment grade Speculative grade (Junk bonds)
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Default Risk
Moody' s Standard & Poor's Safety Aaa Aa A Baa Ba B Caa Ca C
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AAA AA A BBB BB B CCC CC C
The strongest rating; ability to repay interest and principal is very strong. Very strong likelihood that interest and principal will be repaid Strong ability to repay, but some vulnerability to changes in circumstances Adequate capacity to repay; more vulnerability to changes in economic circumstances Considerable uncertainty about ability to repay. Likelihood of interest and principal payments over sustained periods is questionable. Bonds in the Caa/CCC and Ca/CC classes may already be in default or in danger of imminent default C-rated bonds offer little prospect for interest or principal on the debt ever to be repaid.
Callable Bonds & the 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.
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A 10-year, 10% semiannual coupon bond selling for $1,135.90 can be called in 4 years for $1,050, what is its yield to call (YTC)?
The bonds yield to maturity can be determined to be
8%. Solving for the YTC is identical to solving for YTM, except the time to call is used for N and the call premium is FV.
INPUTS OUTPUT
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8 N I/YR
3.568
- 1135.90
50 PMT
1050 FV
PV
Yield to call
3.568% represents the periodic semiannual yield
to call. YTCNOM = rNOM = 3.568% x 2 = 7.137% is the rate that a broker would quote. The effective yield to call can be calculated
YTCEFF = (1.03568)2 1 = 7.26%
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If you bought these callable bonds, would you be more likely to earn the YTM or YTC?
The coupon rate = 10% compared to
YTC = 7.137%. The firm could raise money by selling new bonds which pay 7.137%. Could replace bonds paying $100 per year with bonds paying only $71.37 per year. Investors should expect a call, and to earn the YTC of 7.137%, rather than the YTM of 8%.
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When is a call more likely to occur?
In general, if a bond sells at a premium, then (1)
coupon rate > rd, so (2) a call is more likely.
So, expect to earn:
YTC on premium bonds. YTM on par & discount bonds.
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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.
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How are sinking funds executed?
Call x% of the issue at par, for sinking fund
purposes.
Likely to be used if rd is below the coupon rate and
the bond sells at a premium.
Buy bonds in the open market.
Likely to be used if rd is above the coupon rate and
the bond sells at a discount.
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Other types (features) of bonds
Convertible bond may be exchanged for common stock of
the firm, at the holders 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 interest is earned by the firm. Indexed bond interest rate paid is based upon the rate of inflation.
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