Chapter 5 Valuing Bonds Chapter 5 Topic Overview uBond Characteristics uReading Bond Quotes uAnnual...
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Transcript of Chapter 5 Valuing Bonds Chapter 5 Topic Overview uBond Characteristics uReading Bond Quotes uAnnual...
Chapter 5Chapter 5Chapter 5Chapter 5
Valuing BondsValuing Bonds
Chapter 5 Topic Overview
Bond Characteristics Reading Bond Quotes Annual and Semi-Annual Bond
Valuation Finding Returns on Bonds Bond Risk and Other Important
Bond Valuation Relationships
Bond Characteristics
Face (or Par) Value = stated face value that is the amount the issuer must repay, usually $1,000
Coupon Interest Rate Coupon (cpn) = Coupon Rate x Face Value Maturity Date = when the face value is
repaid. This makes a bond’s cash flows look like
this:
Characteristics of Bonds
Bonds pay fixed coupon (interest) payments at fixed intervals (usually every 6 months) and pay the face value at maturity.
00 1 1 2 . . .2 . . . nn
$I $I $I $I $I $I+$M$I $I $I $I $I $I+$M
The Financial Pages: Treasury Bonds
Maturity AskRate Mo/Yr Bid Asked Chg Yld6.5 Oct 06n 112:17 112:18 -2 2.23• Most values expressed as a %age of par
($1000).• xxx:## = xxx and ##/32nd % of parAsked = investor purchase price = 112 18/32% of
$1000 = $1,125.625Bid = investor selling price = $1,125.3125Rate = Annual coupon rate = 6.5% of par
$65/year: $32.50 semiannually Chg = change in price from previous day in 32nds
of % of parAsk Yld = 2.23% annual rate of return if
purchased and held until maturity in Oct 2006
Bonds
WARNINGWARNINGThe coupon rate IS NOT the discount rate used in the Present Value calculations.
The coupon rate merely tells us what cash flow the bond will produce.
Since the coupon rate is listed as a %, this misconception is quite common.
Bond Pricing
The price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.PV
cpn
r
cpn
r
cpn par
r t
( ) ( )
....( )
( )1 1 11 2
Bond Valuation• Discount the bond’s cash flows at the
investor’s required rate of return.– the coupon payment stream (an annuity).– the face (par) value payment (a single
sum).– PV = cpn (PVAF r, t) + par /(1+r)t
00 1 1 2 . . . 2 . . . nn
cpn cpn cpncpn cpn+parcpn+par
Bond Valuation Example #1
Duff’s Beer has $1,000 par value bonds outstanding that make annual coupon payments. These bonds have an 8% annual coupon rate and 12 years left to maturity. Bonds with similar risk have a required return of 10%, and Moe Szyslak thinks this required return is reasonable.
What’s the most that Moe is willing to pay for a Duff’s Beer bond?
0 1 2 3 . . . 12
1000 80 80 80 . . . 80
P/Y = 1 12 = N
10 = I/Y 1,000 = FV80 = PMT
CPT PV = -$863.73
Note: If the coupon rate < discount rate, the bond will sell for less than the par value: a discount.
Let’s Play with Example #1
Homer Simpson is interested in buying a Duff Beer bond but demands an 8 percent required return.
What is the most Homer would pay for this bond?
0 1 2 3 . . . 12
1000 80 80 80 . . . 80
P/Y = 1 12 = N
8 = I/Y 1,000 = FV80 = PMT
CPT PV = -$1,000
Note: If the coupon rate = discount rate, the bond will sell for its par value.
Let’s Play with Example #1 some more.
Barney (belch!) Barstool is interested in buying a Duff Beer bond and demands on a 6 percent required return.
What is the most Barney (belch!) would pay for this bond?
0 1 2 3 . . . 12
1000 80 80 80 . . . 80
P/Y = 1 12 = N
6 = I/Y 1,000 = FV80 = PMT
CPT PV = -$1,167.68
Note: If the coupon rate > discount rate, the bond will sell for more than the par value: a premium.
Bond Prices and Interest Rates have an inverse relationship!
Bond Values for 8% Annual Coupon Bonds
0200400600800
100012001400
0% 2% 4% 6% 8% 10% 12%
Required Return
($)M
ark
et V
alu
e
12-yr Bond
Bonds with Semiannual Coupons
Double the number of years, and divide required return and annual coupon by 2.
VVBB = = annual cpnannual cpn/2/2(PVAF(PVAFr/2,2tr/2,2t) + par ) + par /(1+r/2)2t
Semiannual Example
A $1000 par value bond with an annual coupon rate of 9% pays coupons semiannually with 15 years left to maturity. What is the most you would be willing to pay for this bond if your required return is 8% APR?
Semiannual coupon = 9%/2($1000) = Semiannual coupon = 9%/2($1000) = $45$45
15x2 = 30 remaining coupons15x2 = 30 remaining coupons
0 1 2 3 . . . 30
1000 45 45 45 . . . 45
P/Y = 1 15x2 =30 = N
8/2 = 4 = I/Y 1,000 = FV
90/2 = 45 = PMTCPT PV = -$1,086.46
Bond Yields
• Current Yield - Annual coupon payments divided by bond price.
• Yield To Maturity - Interest rate for which the present value of the bond’s payments equal the price.
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.PV
cpn
r
cpn
r
cpn par
r t
( ) ( )
....( )
( )1 1 11 2
Yield to Maturity Example
$1000 face value bond with a 10% coupon rate paid annually with 20 years left to maturity sells for $1091.29.
What is this bond’s yield to maturity?
0 1 2 3 . . . 20
1000-1091.29 100 100 100 . . . 100
P/Y = 1 -1091.29 = PV
20 = N1,000 = FV100 = PMT
CPT I/Y = 9% = YTM
Bond Yields
Rate of Return - Earnings per period per dollar invested.
Rate of return =total income
investment
Rate of return =Coupon income + price change
investment
Let’s try this together.• Imagine a year later, the discount
(required) rate for the bond from the YTM example fell to 8%.
• What is the bond’s expected price?• What is the rate of return, if we sell the
bond at this time assuming we bought the bond a year earlier at 1091.29?
• PMT =100, FV = 1000
YTM for semiannual coupon bonds: back to
our T-bond Maturity Ask
Rate Mo/Yr Bid Asked Chg Yld6.5 Oct 06n 112:17 112:18 -2 2.23• $1000 par value, today’s price = $1125.625 = PV• Semiannual coupon = $1000(6.5%/2) = $32.50• Assume 2006-2003 = 3 years to maturity x 2 = 6
semiannual payments left.• -1,125.625 = PV, 32.50 = PMT, $1000 = FV, 6 = N, CPT I/Y
= 1.1% semiannually• Annual YTM = 2(1.1%) = 2.2% APR
Bond Value Changes Over Time
Returning to the original example #1, where k = 10%, N = 12, cpn (PMT) = $80, par (FV) = $1000, & PV = $863.73.
What is bond value one year later when N = 11 and r is still = 10%?
80 = PMT, 1000 = FV, 11 = N, 10 = I/Y, CPT PV = 870.10
PV = $80(PVAF10%,11) + $1000/(1.10)11 = $870.10
What is the bond’s return over this year?
Rate of Return = (Annual Coupon + Price Change)/Beg. Price
Annual Coupon = $80 Beg. Price = $863.73, End Price =
$870.10 Price Change = $870.10 - $863.73 =
$6.37 Rate of Return = ($80 +
$6.37)/$863.73 = 10%
Bond Prices over time approach par value as maturity date approaches assuming same YTM
Bond Values Over Tim e
$870.10$ 8 6 3 .7 3
$ 1 ,1 6 7 .6 8
$800.00
$900.00
$1,000.00
$1,100.00
$1,200.00
12 10 8 6 4 2 0
Tim e to Maturity
Bo
nd
Va
lue
k = 1 0 %k = 8 %k = 6 %
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.
Interest Rate Risk Example
• Recall from our earlier example (#1), the 12-year, 8% annual coupon bond has the following values at kd = 6%, 8%, & 10%. Let’s compare with a 2-yr, 8% annual coupon bond.
• 12-year bond 2-year bondr=6%: PV = $1,167.68 PV = $1,036.67r=8%: PV = $1,000 PV = $1,000r=10%: PV = $863.73 PV = $965.29
Bond Price Sensitivity Graph
Bond Values for 8% Annual Coupon Bonds
400650900
1150140016501900215024002650
2% 4% 6% 8% 10% 12% 14%
2-yr Bond
12-yr Bond
30-yr Bond
Default Risk• Credit risk• Default premium• Investment grade• Junk bonds
Default RiskStandard
Moody' s & Poor's Safety
Aaa AAA The strongest rating; ability to repay interest and principalis very strong.
Aa AA Very strong likelihood that interest and principal will berepaid
A A Strong ability to repay, but some vulnerability to changes incircumstances
Baa BBB Adequate capacity to repay; more vulnerability to changesin economic circumstances
Ba BB Considerable uncertainty about ability to repay.B B Likelihood of interest and principal payments over
sustained periods is questionable.Caa CCC Bonds in the Caa/CCC and Ca/CC classes may already beCa CC in default or in danger of imminent defaultC C C-rated bonds offer little prospect for interest or principal
on the debt ever to be repaid.
Other Types of Bonds
Zero Coupon Bonds: no coupon payments, just par value.
Convertible Bonds: can be converted into (fixed # of) shares of stock.
Floating Rate (Indexed) Bonds: coupon payments and/or par value indexed to inflation. TIPs: Indexed US Treasury coupon bond, fixed coupon rate,
face value indexed. Callable Bonds: Company can buy back the
bonds before maturity for a call price. More likely as interest rates fall. Yield to Call: calculate like yield to maturity but use time to
earliest call date as N, and call price as FV.