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Discounted Cash Flow Valuation
Bond Valuation, Bond Yields and Bond Prices
Vassil Mihov
Vassil Mihov
Bond Features
Bond - evidence of debt issued by a corporation or a governmental body. A bond represents a loan made by investors to the issuer. In return for his/her money, the investor receives a legal claim on future cash flows of the borrower. The issuer promises to:
Make regular coupon payments every period until the bond matures, and
Pay the face /par/ value of the bond when it matures.
Default - since the abovementioned promises are contractual obligations, an issuer who fails to keep them is subject to legal action on behalf of the lenders (bondholders).
Vassil Mihov
You buy a 7% Treasury bond maturing in 5 years. The the bond has a $70 annual coupon, $1000 face value, and its cash flows would look like this:
Time 0 1 2 3 4 5
Coupons $70 $70 $70 $70 $70
Face (Par) Value $ 1000
$______
How much is this bond worth? It depends on the level of current market interest rates. If the going rate on bonds like this one is 7%, then this bond is worth $1000. Why?
Bond Features
Vassil Mihov
The value of a bond is equal to the present value of its future cash flows!
The bonds have a $1,000 face value
The promised annual coupon is $70
The bonds mature in 5 years
The market’s required return on similar bonds is 7%
1. Calculate the present value of the face value
=$1,000 x (1/1.075) = $712.99
2. Calculate the present value of the coupon payments
= $70 x [1/0.07 – 1/ 0.07 (1.07) 5] = $287.01
3. The value of each bond = 712.99+287.01 = $1,000
Vassil Mihov
The Bond Pricing Equation
In general,
Bond Value = Present Value of the Coupons + Present Value of the Face Value =
where: CPN = the promised coupon payment
FV = the promised face value
N= number of periods until the bond matures
y = the market’s required return, YTM
Vassil Mihov
Bond Rates and Yields
Suppose a bond currently sells for $1,095.78. It pays an annual coupon of $70, and it matures in 5 years. It has a face value of $1,000. What are its coupon rate, current yield, and yield to maturity (YTM)?
The coupon rate (or just “coupon”) is the annual dollar coupon expressed as a percentage of the face value:
Coupon rate = $70 /$_____ = 7%
The current yield is the annual coupon divided by the current market price of the bond:
Current yield = $___ / 1,095.78 = 6.39%
Under what conditions will the coupon rate and current yield be the same? Stay tuned.
Vassil Mihov
Bond Rates and Yields
3. The yield to maturity (or “YTM”) is the rate that makes the price of the bond just equal to the present value of its future cash flows. It is the unknown r in:
$1,095.78 = $_______ x [1/r - 1/(r(1 + r)5) + $_______ /(1 + r)5
The yield to maturity is 4.8% in this case
Vassil Mihov
A Discount Bond
Assume you have the following information.
A company issued bonds have a $1,000 face value
The promised annual coupon is $70
The bonds mature in 5 years
The market’s required return on similar bonds is 10%
1. Calculate the present value of the face value
=$1,000 x (1/1.105) = $620.92 2. Calculate the present value of the coupon payments
= $70 x [1/0.10 – 1/(0.10(1.105))] = $265.36
3. The value of each bond = $620.92 + $265.35 = $886.28
Vassil Mihov
Example: A Premium Bond
Assume you have the following information.
A company issued bonds have a $1,000 face value
The promised annual coupon is $70
The bonds mature in 5 years
The market’s required return on similar bonds is 2%
1. Calculate the present value of the face value
=$1,000 x (1/1.025) = $905.73
2. Calculate the present value of the coupon payments
= $70 x [1/0.02 – 1/(0.02(1.025))] = $329.94
3. The value of each bond = $905.73 + $329.94 = $1,235.67
Why do the bonds in this and the preceding example have prices that are different from par?
Vassil Mihov
Bond Yields and Prices
$600
$700
$800
$900
$1,000
$1,100
$1,200
$1,300
$1,400
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16
YTM
Pri
ce
A 5-year, 7%-coupon,
1000-par bond
Vassil Mihov
Bond Pricing Theorems
1. Bond prices and market interest rates move in opposite directions.
2. When a bond’s coupon rate is (greater than / equal to / less than) the market’s required return, the bond’s market value will be (greater than / equal to / less than) its par value.
3. Given two bonds identical but for maturity, the price of the longer-term bond will change more than that of the shorter-term bond, for a given change in market interest rates.
4. Given two bonds identical but for coupon, the price of the lower-coupon bond will change more than that of the higher-coupon bond, for a given change in market interest rates.
Vassil Mihov
Vassil Mihov
Semiannual Interest and Bond Prices
In actuality, most bond make coupon payments twice a year. Suppose you have a 7 percent, $1000 par value bond with maturity of 5 years, making semi-annual payments. If the yield to maturity is 12 percent, what is the price of the bond?
These is nothing new here: you will make the adjustments that we made before when you had compounding more than once a year.
1.Increase the number of periods: t = 5 x 2 = 10.
2.Decrease the yield to maturity by dividing by two:
r = 12%/2 = 6% per six months.
3. Finally, you will divide the coupon payment by two: C = 70/2 = 35 per period.
P = PV(Par) + PV(coupon) =$816
What is the price of the same bond if compounding is done annually? P = 819.76
Vassil Mihov
Zero-coupon bonds
A company has issued a bond that pays no coupons. It has a $1,000 par value, and there are 5 years left to its maturity. The market yield on similar bonds is 10%. What is the price of the bond today?
P = PV (Par) = 1000 / 1.15 = $620.92
This type of bond does not pay any coupons. It is issued at a price below par, and when it matures, the issuer pays only the par value. A zero-coupon bond always trades below par!
Vassil Mihov
Factors Affecting Bond Yields
What factors affect observed bond yields?
Real rate of interest Expected future inflation Interest rate risk Default risk premium Taxability premium Liquidity premium
