CHAPTER 8
VALUATION OF KNOWN CASH FLOWS: BONDS
8.1 Using Present Value Formulas to Value Known (or Riskless) Cash
Flows
Example 1.
Suppose you buy a security that pays you $ 100 at the end of each year for the next three years. The appropriate discount rate is 6%. This is an ordinary annuity, and its present value is given by the following formula (see Chapter 4):
PV
OA
= PMT [1 – 1/(1+i) n ]/i
The present value would work out to $ 267.30. This would therefore be the price you would pay for this security. You would get the same price, even if you did not use the formula, but valued each cash flow by discounting it for the appropriate number of years, at the yield of 6%:
Time Cash Flow Present value of the Cash flow
1
2
3
$100
$100
$100
$100/(1.06) = $94.34
$100/(1.06) 2 = $89.00
$100/(1.06) 3 = $83.96
$267.30
Let us assume that a few minutes after you bought the security, the risk-free interest rate rises from 6% to 7%. Now if you want to sell the security, how much will you get for it?
You would get the present value of the security, which now will be different.
This is because the discount rate that is applicable has changed from 6% to 7%.
If you now recalculate the PV using 7% you will find that it equals $262.43.

Thus, a rise in the interest rate causes a fall in the price of this security.
Similarly, a fall in interest rate will cause a rise in the price of this security. In general, this principal applies to most financial securities, such as stocks and bonds. The exceptions are some complicated financial derivatives, where future cash flows are not fixed and change in the same direction as that of the interest rates.
8.2 The Basic Building Blocks: Pure Discount Bonds
Pure discount bonds or zero-coupon bonds are bonds that promise a single payment of cash at some date in the future called the maturity date.
The promised cash payment on a pure discount bond is called its face value or par value.
The yield (interest rate) on a pure discount bond is the annualized rate of return to investors who buy it and hold it until it matures.
Example 2.
Suppose you were given three pure discount bonds as follows:
Maturity Cash Flow Present value of the Cash flow
Bond 1 1
Bond 2
Bond 3
2
3
$100
$100
$100
$100/(1.05) = $95.24
$100/(1.06) 2 = $89.00
$100/(1.0638) 3 = $83.06
Price of Bond 1 = $95.24
Price of Bond 2 = $89.00
Price of Bond 3 = $83.06
Now, suppose you hold one unit of each of the above pure discount bonds. The total price or value of your portfolio equals $267.30. Is this portfolio of three different pure discount bonds any different from the security considered in example 1. Any similarities? Any differences? Spend a few minutes with your group to think about this issue. Share any insights.

Each bond in the above example is discounted at a different yield.
One-Year Yield = 5%
Two-Year Yield = 6%
Three-Year Yield = 6.38%
These yields are called zero coupon yields and together they define the term structure of zero coupon yields or the zero-coupon yield curve . The yield given in example 1 is the yield to maturity of a coupon bond. The yield to maturity of a coupon bond must be some geometric average of the corresponding zero coupon yields.
All fixed income securities with known cash flows can be considered portfolios of pure discount bonds of varying maturities.
When the yield curve is not flat i.e. the observed yields are not the same for all maturities, the correct procedure for valuing a security is to discount each cash flow at the rate corresponding to a pure discount bond of its maturity and adding the resulting individual present values
8.3 Coupon Bonds, Current Yield and Yield to Maturity
A coupon bond obligates the issuer to make periodic payments of interest – called coupon payments – to the bondholder for the life of the bond and pay the face value of the bond upon maturity. The periodic payments of interest are called coupons.
The coupon rate is the interest rate applied to the face value of the bond to compute the coupon payment.
Example 3.
Consider a six-year bond with annual payments, a 10% coupon rate, and a face value of $1,000.
Year
Coupon
Face Value
0 1 2 3 4 5 6
$100 $100 $100 $100 $100 $100 $100
$1,000

Definitions:
Coupon Rate = Annual Coupon/Face value = 100/1000 = 10%
Current Yield = Annual Coupon/Price
Yield to Maturity of a bond is defined as that discount rate that makes the present value of the bond’s stream of promised cash payments equal to its price.
Further, assume that the yield to maturity on this bond is 10%. Since the yield to maturity equals the coupon rate of 10%, the price of the bond must equal its face value of $1,000. In fact, there exist three general principles can be stated as follows:
Principle 1: Par Bonds
A coupon bond with a market price equal to its face value is called a par bond .
For a par bond, Yield to Maturity = Current Yield = Coupon Rate
Principle 2: Premium Bonds
A coupon bond with a market price greater than its face value is called a premium bond .
For a premium bond, Yield to Maturity < Current Yield < Coupon Rate
Principle 3: Discount Bonds
A coupon bond with a market price less than its face value is called a discount bond .
For a discount bond, Yield to Maturity > Current Yield > Coupon Rate
BOND PRICE FORMULA:
Bond Price = PMT [1 – 1/(1+i) n ]/i + FV/(1+i) n
PMT = Annual Coupon
FV = Face value i = Yield to maturity n = Bond maturity

Problem 1
Here we are told that the Treasury bond matures in 10 years. The coupon rate is
7% and interest payments are annual. The yield to maturity is 8%.
(a) Price of the bond n = 10, PMT = 7, FV = 100, i = 8 %
We need to calculate the present value of the cash flows from the bond using:
Bond Price = PMT [1 – 1/(1+i) n ]/i + FV/(1+i) n
Bond Price = $93.2893
(b) Coupons paid twice a year.
If coupons are paid m times a year, then the face value remains the same, but n, i, and PMT are changed as follows:
Bond Price Formula Adjustment Rules: n = Bond maturity (in Years) multiplied by m i = Yield to maturity divided by m
PMT = Annual coupon divided by m
Hence, if coupons are paid twice a year in the above example, then n = 20, i =
0.04, and PMT = $3.5.
Using exactly the same formula used in (a) we will get a price of $ 93.20
(c ) What is the price if the yield to maturity is 7 %?
Here we do not need to do any calculations. Whenever the coupon rate and the
Yield to Maturity are the same, the bonds will sell at par value (face value).
This remains true even if coupons are paid semi-annually. In this case the price will be $100. Of course, you may wish to prove this by doing the calculations.

Problem 2
6 months ago:
The US Treasury yield curve was flat at a rate of 4% (annually compounded).
You bought a 30-year U.S Treasury bond.
Today:
Yield curve is flat at 5%
Note that the remaining maturity on the bond today is 29.5 years.
We need to find the return on investment under two different assumptions.
(a) The bond is a 4% coupon bond
(b) It is a zero-coupon bond
We use the same principle in finding the rate of return in either case. The first step is to find the price of the bond 6 months ago. This would be the price at which you purchased the bond. We calculate the present value (6 months ago) of the bond as we have done earlier.
For the 4% coupon bond, since the Treasury yield was 4% when you bought the bond, the price would be the face value ($100).
Price of 4 % coupon bond = $100
For the zero coupon bond we can use the above formula. However there are no payments during the life of a zero coupon bond except at maturity when you receive the face value. i = 4 %, n = 30, FV =100
Price of Zero Coupon = FV/(1+i) n = 100/ (1 .04) 30
= 30.8318

Now we need to calculate the prices today.
However today the Treasury yield is 5 % and this would be the discount rate used to find the price.
4% coupon bond: i = 0.05, n = 29.5, PMT = 4, FV = 100
We find that the PV (price of the bond) today is 84.74
For the zero-coupon bond:
Price = FV/ (1+i ) n
= 100/(1.05) 29.5
= 23.71
We can now calculate the rate of return in both cases:
Rate of Return (r) =
[Coupon or Accrued Interest + Change in Price] / Initial Price
4% Coupon bond: r = [2 + (84.74 - 100)]/ 100 = - 0.1326 or – 13.26%
Zero coupon : r = [0 + (23.71 - 30.83)] /30.83 = - 0.2309 or –23.09%
(c)
What happens if the compounding was semi-annual?
We use the same procedure to calculate the rate of return. The differences are in the number of periods, the yield (discount rate) applicable for each period and the coupon for each period in the case of a coupon bond.

The number of periods doubles since we have two periods (6 months each) in each year. The coupon and yield applicable for each period will be half of that applicable for a full year.
6 months ago:
4% annual coupon. n = 60 , i = 2 % , PMT =2 , FV=100
Since the yield to maturity and the coupon are the same we do not need to calculate the price. It will be the face value ($100 )
Zero coupon:
N= 60, i = 0.02, FV=100.
There is no coupon. The price will be = 100/(1.02) 60 = 30.48
Today:
4% annual coupon: n = 59, i = 2.5 % , PMT =2 , FV=100
Using the same formula as before, the price of the bond is = $ 84.66
Zero-coupon bond: n=59, i = 0.025 (or 2.5%), FV =100
Price = 100/(1.025) 59
= 23.30
Rate of Return (r) =
[Coupon or Accrued Interest + Change in Price] / Initial Price
4% Coupon bond: r = [2 + (84.66 -100)] / 100 = -0.1334 or –13.34%
Zero coupon : r = [0 + (23.30 - 30.48)] / 30.48 = - 0.23556

or –23.56%
Problem 5
Bond Coupon rate Maturity
1 6% 2
2
3
0%
7%
2
2
Price
1.0092
0.8983
1.0277
Yield to maturity(i)
5.5 %
5.51%
5.50
4 0% 1 $ 0.95 5.26%
We are asked to assume that coupon payments are annual, the face value of the bonds is $1 and that the law of one price holds.
Let us first consider bonds 1 and 4. The only unknowns here are the price for bond 1 and the YTM for bond 4.
For bond 1:
Price = 0.06/1.055 + 1.06/1.055
2
= 0.05687 + 0.95236
= 1.0092
For bond 4 which is a zero-coupon bond
Price = FV/(1+i) n
0.95
= 1/ (1+i)
1+ i = 1.05263 i = .05263 or 5.26%
We know from bond 4 that the yield on a zero coupon bond with 1 year maturity is 5.26 %
We can find the yield on a zero coupon bond with 2 year maturity as follows:
For the coupon bond with 2-year maturity we have 2 cash flows:
At the end of 1 year a cash flow of 0.06. The yield applicable to this cash flow will be 5.26 %. We know that the price of this bond is 1.0092. We also know that the 2 nd cash flow from this bond that arises after 2 years is 1.06. We can

now find the only unknown here, which is the yield (discount rate) applicable to this cash flow of 1.06.
1.0092 = 0.06/ 1.0526 + 1.06/ (1 +z2) 2
1.0092 = 0.057002 + 1.06/(1+z2) 2
(1 +z2) 2 = 1.113213 z2 = .055089 or 5.51 %
This is the yield on a 2-year zero-coupon bond
Now we can find the price of the 2-year zero-coupon bond. We have i =
5.51%, FV=1
PV = FV/(1+i) n
= 1/(1.0551) 2 = 0.8983
Bond 3:
Coupon 7%, maturity 2 years.
We know the yields for the 1 and 2 year zero-coupon bonds. We use this to calculate the present values of the 1 st and 2 nd year cash flows for this bond and add them up to get the price.
Price = 0.07/1.0526 + 1.07/1.0551
2
=0.06650 +0.96116
= 1.0277
8.4 Reading Bond Listings – Read at home
8.5 Why Yields For the Same Maturity May Differ
8.5.1. The Effect of Coupon Rate
When the yield curve is not flat, bonds of the same maturity with different coupon rates have different yields to maturity.
8.5.2 The Effect of Default Risk and Taxes

8.5.3 Other Effects on Bond Yields
Callability: This feature gives the issuer of the bond the right to redeem it before the final maturity date.
Convertibility: This feature gives the holder of a bond issued by a corporation a right to convert the bond to a prespecified number of shares of common stock.
8.6 The Behavior Of Bond Prices Over Time
8.6.1. The Effect of the Passage of Time
If the yield curve were flat and interest rates did not change, any default-free discount bonds price would rise with the passage of time, and any premium bond’s price would fall. This is because eventually, bonds mature, and their price must equal their face value at maturity.
In the absence of changes in interest rates and with a flat term structure, the price of a zero-coupon bond might be expected to rise over time at a rate equal to its yield to maturity.
8.6.2. Interest Rate Risk
Maturity Effect – Longer maturity bonds have higher interest rate risk
Coupon Effect – Lower coupon bonds have higher interest rate risk (see
Figure 8.4)
Here is a book on interest rate risk: http://www.amazon.com/exec/obidos/ASIN/0961944692/o/qid%3D946585883/s r%3D8-1/103/002-6519569-7037626
Problem 7
Essentially here we need to find the price of the Dafalto Corp bond had it been a risk free bond. The difference between the price of the risk free bond and the
Dafalto bond that has some default risk inherent in it, will be the price an investor will pay for a guarantee against default.

First calculate the price of the bond assuming it was free of default risk. n= 2, i = 6% (risk-free rate), PMT= 100, FV=1000
Price = 100/1.06 + 1100/1.06
2
Price = 94.3396 + 978.996
Price = 1073.335
Price investor would pay for a guarantee against default = 1073.3- 918 = $155.3
Problem 9
All else being equal if interest rates rise along the entire yield curve, you should expect that
(1) Bond prices will fall
(2) Bond prices will rise
(3) Prices on long term bonds will fall more than prices on short term bonds
(4) Prices on long term bonds will rise more than prices on short term bonds
Choices
(a) 2 and 4 are correct
(b) We cant be certain that prices will change
(c) Only 1 is correct
(d) Only 2 is correct
(e) 1 and 3 are correct
The correct answer is (e). Bond prices are inversely proportional to yields.
Hence when yields rise, bond prices will fall. Also long-term bonds are more sensitive to yield changes than short-term bonds.