bonds 2016
TRANSCRIPT
BondsPrices & Yields
Learning Objectives
• Types of bonds and bond parameters • Bond yield to maturity• Bond price
• Computed from discounted expected cash flow• Applications
• Compute bond yield from a known price • Compute bond price from a known yield• Plot the price v. yield • Compute bond price when yield is not known• Plot price v. time to maturity • Compute mortgage payments
• Bond price quotes2
Bonds
• Corporations and government entities can raise capital by selling bonds
• Long term liability (accounting)• Debt capital (finance)
• The bond has • Principal, par, or face value: F• Price: P• Yield: y (also the discount rate) • Maturity date, time to maturity, term, or tenor: T
• Date at which the bond principal, F, is returned to investors• In the case of a coupon bond (as opposed to a zero coupon
bond)• Coupon rate: c (annual, simple, nominal rate) • Annual payment frequency: m; or period Dt
• In the U.S. semiannual coupons is typical: m = 2 or Dt = .5
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Zero Coupon Bonds
• ZCBs do not pay a coupon• No intermediate cash flow
• The return and rate of return (yield) is due to the bond bought at a discount to face value
• U.S. Treasury bills (T – bills) are zero coupon bonds
• Time-to-maturity at issue is 4, 13, 26, 52 weeks • Face value $100 to $5,000,000
• A ZCB yield is the interest rate, (and the discount rate) denoted z
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F
P
t=0
t=T
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F
P
t=0
t=T
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Bonds: MSFT
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June 30 2011 2010
Bonds: MSFT
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Bonds: MSFT
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Bonds: MSFT
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Coupon Bond
Coupon Bond
P = current price
C = coupon payment
F = face or par value
t=0.0 t=Dt t=2∙Dt t=M∙Dt=T
i=0 i=1 i=2 i=M
t0=0.0 t1=Dt t2=2Dt tM= M∙Dt =T
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Coupon Payment • Bond coupon cash flows, C, are defined by a
nominal, simple coupon rate, c, and a compounding frequency per year, m, or coupon period measured in years, Dt
• The total cash flow at time ti, CFi, is defined as:
Coupon Bond Yield
• Yield to maturity is the actual yield achieved for a coupon bond if
• The bond is held to maturity, and • Each coupon payment is reinvested at a rate of return, y,
through time T• The risk that coupons cannot be reinvented at a rate greater than
or equal to y due to market conditions is called “reinvestment risk”
• The yield to maturity is the investor’s expected return on investment and is thus the issuer’s rate cost
• It’s the issuer’s cost of debt, kD, for the bond• The yield reflects both the time value of money and the
credit worthiness of the borrower• The expected variance in the cash flow is reflected in the yield
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Bond Price
• The discount rate y is the yield to maturity or simply the yield on a coupon bond
• It’s an internal rate of return that sets the discounted cash flow on the right hand side to the market price of the bond, P, on the left hand side
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y is the nominal annual yield to maturity in this formula with integer periods
ybar is the effective annual yield to maturity in this formula with discrete real time periods
Current interest and yield summary
• Bloomberg• Yahoo• Bankrate
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Yield Curve
• A graph can be developed that plots various (annualized) yields-to-maturity against times-to-maturity for bonds from the same issuing entity or from issuing entities with the same risk
y
0 T19
Yield Curve
• The most useful version of the yield curve plots zero coupon bond yields against times-to-maturity
• A zero coupon yield curve depicts pure interest rates with no ambiguity in the risk associated with the yield
• Thus we can use ZCB yield curve to plot the interest rate with respect to time-to-maturity, or holding period
• Animated yield curve
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Yield Curve
• Yields on zero coupon U.S. Treasury debt are risk free
• The additional yield for non-U.S. Treasury debt is due largely to credit risk and is called a risk premium or a credit spread
• Moody’s and Standard & Poor’s evaluate bond credit risk via analysis of the issuer’s financial position and assign risk levels such as AA that translate into credit risk premiums
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Reinvestment Risk• Consider a $1000 bond with a coupon rate of 10% paid annually
for 10 years. Initially, the yield is 11%, the price is $941.11, and the yield curve is flat. Prior to the payment of the next coupon, we consider three scenarios1. the yield curve shifts parallel down to 9%2. the yield curve remains flat at 11%3. the yield curve shifts parallel up to 12%What are the actual yields?
$1,000 F10.00% c nominal Year CF DF DCF 9% 11% 12%11.00% y nominal 1 100$ 0.9009 90.09$ 217.19$ 255.80$ 277.31$
2 100$ 0.8116 81.16$ 199.26$ 230.45$ 247.60$ 3 100$ 0.7312 73.12$ 182.80$ 207.62$ 221.07$ 4 100$ 0.6587 65.87$ 167.71$ 187.04$ 197.38$ 5 100$ 0.5935 59.35$ 153.86$ 168.51$ 176.23$ 6 100$ 0.5346 53.46$ 141.16$ 151.81$ 157.35$ 7 100$ 0.4817 48.17$ 129.50$ 136.76$ 140.49$ 8 100$ 0.4339 43.39$ 118.81$ 123.21$ 125.44$ 9 100$ 0.3909 39.09$ 109.00$ 111.00$ 112.00$
10 1,100$ 0.3522 387.40$ 1,100.00$ 1,100.00$ 1,100.00$ Sum 941.11$ 2,519.29$ 2,672.20$ 2,754.87$
Yield To Maturity 10.35% 11.00% 11.34%
Bond Price Calculation Future Value of Coupon Reinvestment
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• For a fractional initial coupon period: t1 < ∆t
Fractional Initial Time Period
For a bond with semi-annual coupons, assume that the next coupon payment is in 3 months. The coupon payments occur at t0=0.0, t1=0.25, t2=0.75, t3=1.25, t4 = 1.75, …
i=0 i=1 i=2 i=M
t0=0.0 t1 t2=t1+Dt tM= T
C = coupon payment F = face or par value
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Zero Coupon Bonds Again
• A bond dealer can split a coupon bond into ZCBs• one for the principal and • one for each coupon• This is called ‘stripping’ the bond
• The advantage of a ZCB is that there is no reinvestment risk
• For a ZCB, the yield, y, is the zero coupon rate denoted as z
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Bond Equation Applications
• Find the yield-to-maturity, y, from a known market price, P• Solve for y – nominal or effective
• Solve for the roots of a nonlinear equation• In this course use Excel Goal Seek
• Example: Compute both the effective and nominal yield for a bond with $1000 face value, current market price of $800, coupon rate of 7% paid semiannually, and 4.5 years to maturity.
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Bond Equation Applications
$1,000 F7.00% c nominal
13.434% y effective t CF DF DCF
0 $0 $0.000.5 $35 0.939 $32.86
1 $35 0.882 $30.851.5 $35 0.828 $28.97
2 $35 0.777 $27.202.5 $35 0.730 $25.54
3 $35 0.685 $23.983.5 $35 0.643 $22.51
4 $35 0.604 $21.144.5 $1,035 0.567 $586.94
Sum $1,315 P $800.00
13.011% y nominalt i CF DF DCF
0 0 $0 $0.000.5 1 $35 0.939 $32.86
1 2 $35 0.882 $30.851.5 3 $35 0.828 $28.97
2 4 $35 0.777 $27.202.5 5 $35 0.730 $25.54
3 6 $35 0.685 $23.983.5 7 $35 0.643 $22.51
4 8 $35 0.604 $21.144.5 9 $1,035 0.567 $586.94
Sum $1,315 P $800.00
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Bond Equation Applications
• Convert the nominal yield to the effective yield
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Bond Equation Applications• For a bond with a 12% effective annual yield and price $840.34
at time 0 here’s a plot of price as time progresses from 0 to 4.5 years assuming a constant yield of 12%
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Bond Price Quotes
• Dirty and clean prices • Dirty price
• Price from DCF formula • Transaction price• When the seller sells at this price she gets the
prorated share (accumulated interest) of the next coupon
• Clean price• Price quoted by bond dealer• Excludes accumulated interest on next coupon
payment• Clean price = Dirty price – accumulated
interest• Accumulated as simple interest using applicable
day count convention 29
Bond Price Quotes
• Bid and ask prices • The clean price is quoted for bid and ask prices
• The dealer will buy a bond at the bid price • The dealer will sell a bond at the ask (offer) price
• Bond prices are quoted relative to 100 regardless of actual par value
• Price is quoted as a percent of par • Example
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Yield Curve
• A corporate bond, say a AA rated bond, might have the following yield curve
y
0 T
AA Corporate Bond Yield Curve
U.S. Treasury Debt Yield Curve
risk premium and credit spread
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Types of Risk
• Reinvestment • Liquidity risk
• Results from a bond having few buyers and sellers
• The bond is “illiquid”• Older U.S. Treasuries (referred to as “off
the run”) can trade with a liquidity premium compared with newer issued U.S. Treasuries (referred to as “on the run”)
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Plot price v. yields to maturity
$700
$800
$900
$1,000
$1,100
$1,200
$1,300
0% 2% 4% 6% 8% 10% 12% 14% 16%
Yield
Pric
eF=$1000c=7% semiannualT=4.5 yrs
Bond “price – yield” or P-y curve
Illustrates how price changes as yield-to-maturity changes for a particular bond ( c, m, M, and F are constant)
Each point represents a DCF calculation
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Determine the fair price of a bond
• In this case c, m, T, and the relevant zero coupon yield curve are known
• Compute the fair value, P
zt
0 T for zero coupon bonds
ti for bond cash flows
CFt Cash flow diagram
Zero coupon bond yield curve
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Determine the fair price of a bond
Mortgage Example
• You wish to borrow $300,000 at 6.5% fixed for 30 years.
• The following excel table shows the calculations for the first 12 months and the last 5 months.
• The monthly payment of $1896 is determined using goal seek to force the sum of the last column to $300,000.
• Note that you will pay out $682,633 in principal and interest
• $300,000 in principal • $382,633 in interest
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Home Mortgage Calculation • Given the nominal interest rate, m=12, P, and N,
what is the monthly payment, C?
• C : monthly payment• Includes principal repayment and interest –
there is no return of principal “F”• N : number of years• m : number of compounding periods per year (12 for home loans)• r : nominal fixed interest rate for the loan• P : loan principal (the mortgage amount) • Solve for C using Excel Goal Seek
• Find the value of C that equates the left and right hand sides
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Mortgage Example
t i CF DF DCF0.000 0 -$ -$ 0.083 1 1,896$ 0.995 1,886$ 0.167 2 1,896$ 0.989 1,876$ 0.250 3 1,896$ 0.984 1,866$ 0.333 4 1,896$ 0.979 1,856$ 0.417 5 1,896$ 0.973 1,846$ 0.500 6 1,896$ 0.968 1,836$ 0.583 7 1,896$ 0.963 1,826$ 0.667 8 1,896$ 0.958 1,816$ 0.750 9 1,896$ 0.953 1,806$ 0.833 10 1,896$ 0.947 1,796$ 0.917 11 1,896$ 0.942 1,787$ 1.000 12 1,896$ 0.937 1,777$
29.667 356 1,896$ 0.146 277$ 29.750 357 1,896$ 0.145 276$ 29.833 358 1,896$ 0.145 274$ 29.917 359 1,896$ 0.144 273$ 30.000 360 1,896$ 0.143 271$
Sum 682,633$ P 300,000$
$300,000 P6.500% y nominal
12 m6.697% y annual effective0.542% y monthly effective
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Yield Curve
HW 8 A & B
Corporate Finance, RWJ
HW 8 C & D
Corporate Finance, RWJ
HW 8 E
Corporate Finance, RWJ