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The liabilities side of the balance sheet we saw last time, contained examples of several financial instruments - notes payable, long-term debt, shareholder's equity, etc. The long-term debt of the company usually consists of bonds. This lesson tells us how we should value these bonds. Lets start by looking at the following article in the Wall Street Journal of February 4th, 1998.
Treasurys were mixed on Wednesday, as investors awaited the Federal Reserve policy committee's rate announcement later in the day. The benchmark 30-year bond was down 6/32, or about $1.25 for each $1,000 of face value, at 103 13/32 in midday trading. Its yield, which moves in the opposite direction from its price, rose to 5.88% from 5.86% late Tuesday.
Shorter-term bonds posted slim gains, however, with the two-year note up 1/32 at 100 3/32, yielding 5.31%.
Bonds saw some profit-taking in Asian and European trading following Tuesday's modest gains. In New York, investors' attention was fixed on the Federal Open Market Committee's rate announcement, expected around 2:15 p.m. EST, as policy makers wrap up a two-day meeting.
Fed policy makers are expected to leave rates unchanged, as the financial crisis in Asia is seen putting a drag on otherwise robust U.S. economic growth. That in turn is seen helping to keep inflation under control amid a tight labor market. Bond investors fear higher rates, which decrease the value of their fixed-rate holdings.
There are several puzzling features in this statement. Why is the Federal Policy statement so important? What is the benchmark 30-year bond? All in all, bonds are pretty hot news. There have even been movies where a bond has played the pivotal role
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Lets start by defining what a bond is.
What is a Bond?
A bond is basically a certificate that gives 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:
We generally talk about four main characteristics when we talk about bonds:
C : The coupon or interest payment
F : The face value (principal) of the bond
c : Coupon rate of interest = C/F
TN : The maturity date of the bond
The structure of a bond is then:
For example, if a bond has five years to maturity, an $80 annual coupon, and a $1000 face value, its cash flows would look like this:
Time 0 1 2 3 4 5
Coupons $80 $80 $80 $80 $80
Face 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 10%, then this bond is worth $924.18. Why? What does this stream of cash flows look like? An annuity for five years of $80 plus a single lump sum at the end for $1000. We know the formulae for both the annuity and the lump sum. So we can combine them together to get the value today. That's all there is to bond valuation.
One example of a real bond is the bond displayed in Himeji castle in Japan. Himeji was used in scenes in Akira Kurosawa's film Ran. In this bond, the holder promises to repay the sum of 20,000 ryu within 5 years at an annual interest rate of 1%. Why is the interest rate so low?
But before we close the chapter, let's discuss some of the jargon we need to know to read the Wall Street Journal properly! This lesson will cover some of the jargon we need to know and then wind up by learning how to read the newspaper. All you have to remember is that a bond is simply a combination of a lump sum and an annuity. However, the one complicating feature is that interest rates may change in the period and so we have to adjust for that.
Bonds: The Jargon
Coupon Rate
The coupon rate is the rate of interest printed on the face of the bond. It is the rate of return that the bond promises to pay on the face value of the bond. For example, suppose a bond currently sells for $932.90. It pays an annual coupon of $70, and it matures in 10 years. It has a face value of $1000. What is its coupon rate?
The coupon rate (or just "coupon") is the annual dollar coupon expressed as a percentage of the face value:
Coupon rate = $70 /$_____ = ___%
Current Yield
The current yield is the annual coupon divided by the current market price of the bond. In this example, the current yield is given by:
Current yield = $___ /_____ = 7.5%
Yield to maturity
The yield to maturity of a bond or redemption yield, is the internal rate of return on the bond. It is the discount rate which sets the net present value of an investment in the bond equal to zero. In other words, 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 rate of interest Y in the formula:
$932.90 = $_______/Y × [1 - 1/(1 + Y)10] + $_______ /(1 + Y)10
The only way to find the YTM is trial and error:
a. Try 10%: $70 × [(1 - 1/(1.10)10]/.10
+ $1000/(1.10)10= $816
b. Try 9%: $70 × [1 - 1/(1.09)10]/.09
+ $1000/(1.09)10= $872
c. Try 8%: $70 × [1 - 1/(1.08)10]/.08
+ $1000/(1.08)10 = $933
The yield to maturity is 8%
Bond Valuation
Lets start with a simple example. Suppose you have the following information:
2. Calculate the present value of the coupon payments
3. The value of each bond is therefore given by:
Now lets change the interest rate required on similar bonds to 8%. What happens to the price of the bond? And if the interest rate becomes 12%?
Note that in the last two cases, the price of the bond becomes very
different from the face value. In general, the price of a bond is very
sensitive to its YTM.
In general, the value of a bond is given by the equation:
Bond Value = Present Value of the Coupons+ Present Value of the Face Value
The first part is an annuity and the second part a lump sum. So we apply the equations we already know and get the price of the bond as:
where: C = the promised coupon payment
F = the promised face value
t = number of periods until the bond matures
Y = the market's required return, YTM
The major risk that arises for bond holders is interest rate risk. As can be seen from the above equation, bond prices and interest rates always move in opposite directions. When interest rates rise, bond prices fall. How much interest rate risk exists in a bond depends on how sensitive it is to interest rate changes. Generally however,
As can be seen, the price of a one-year bond does not change very much if the interest rate changes. The price of a 30-year bond however, changes dramatically. Why is this?
Bond Pricing Theorems
The following statements about bond pricing are always true.
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.
General Formula for Bond Valuation
In general, the formula for the price of a bond can be written as
E(irj) is the rate of return that investors require to hold securities of similar risk from time i to time j. This is also called the short term rate. In the special case of constant interest rates, where E(irj) = E(r) for all i, j, we have
Bond Quotations
Consider the following page from the Wall Street Journal.
What does all this stuff mean? First, bonds are quoted as a percentage of face value. Also, bonds are quoted without accrued interest. This means that the invoice price of a bond is the quoted price plus any accrued interest.
Consider the ATT 6s00 bond. Here, the bond has been issued by AT&T with a coupon rate of 6% of the face value. If we assume that the face value is $1000, the annual coupon on the bond is $6. The bond matures in 2000 and its current yield is 6.1%. 5 bonds were traded yesterday and the closing price was 98¼% of the face value. The last column tells us that the closing price was 5/8 % less than the closing price the day before.
Similarly, we also have US Government bonds reported in the Wall Street Journal. Government bonds are of three types: T-bills, T-notes and T-bonds.
Bills are short term debt less than a year old. They are always discount securities or zero coupon bonds. The holder receives the face value of the bill at expiry and the face value is greater than the purchase price. These are the simplest bonds to value since there are no intermediate coupon payments; there is only a single lump sum received at the end.
Notes and bonds are both coupon bearing instruments. Notes mature in more than two but less than 10 years while bonds take more than 10 years to mature. They are valued in the usual way. Current quotes can always be obtained from the Interactive Wall Street Journal Edition. Consider the following page from the same section of the WSJ.
What does this stuff mean? Lets start by looking at the bill marked with the arrow. Here the bond's coupon rate is 8% while the bond matures in November 2021. Treasury bonds make semi-annual payments and have a face value of $1,000, so this bond will pay $40 every six months until it matures. The next two prices are the prices at which the dealer is willing to buy the bond from you (the bid price quoted in 32nds of 1% percent of face value) and the price at which the dealer is willing to sell the bond to you (the ask price, also quoted in 32nds of 1 percent of face value). The bond has declined by 14 ticks of 1/32 of 1% of face value since yesterday. The last number is the YTM which is less than the coupon rate. The bond trades at a premium. The last bond listed in this example, the 6¾ bond is called the benchmark bond. This is the bond quoted in the papers when they say that long-term interest rates rose or fell.
Dealing with Inflation
The key issues we're dealing with here are:
Suppose we have $1,000, and Diet Coke costs $2.00 per six pack. We can buy 500 six packs. Now suppose the rate of inflation is 5%, so that the price rises to $2.10 in one year. We invest the $1,000 and it grows to $1,100 in one year. What's our return in dollars? In six packs?
In Dollars:
Our return is
($1100 - $1000)/$1000 = $100/$1000 = ________.
The percentage increase in the amount of green
stuff is 10%; our return is 10%.
In Six packs:
We can buy $1100/$2.10 = ________ six packs, so our return is
(523.81 - 500)/500 = 23.81/500 = 4.76%
The percentage increase in the amount of brown
stuff is 4.76%; our return is 4.76%.
Real versus nominal returns
Remember,
Your nominal return is the percentage change in
the amount of money you have.
Your real return is the percentage change in the
amount of stuff you can actually buy.
The relationship between real and nominal returns is described by the
Fisher Effect. Let:
R = the nominal return
r = the real return
h = the inflation rate
According to the Fisher Effect:
From the example, the real return is 4.76%; the nominal return is 10%,
and the inflation rate is 5%:
(1 + R) = 1.10
(1 + r) x (1 + h) = 1.0476 x 1.05 = 1.10
The yield curve tells us how bond yields will change over the future. Yield curves are reported by the WSJ every day. Look at the following example.
What does it tell us? This yield curve is based on coupon bearing treasury bonds. It tells us that the rate of inflation is expected to rise gradually. At the same time, the interest rate risk premium is expected to increase at a decreasing rate, so the overall effect is a sharp rise in the graph after 3 years. Basically, what factors affect observed bond yields?
Inferring Interest Rates from
Bond Prices: Spot Rates
The spot rate in period t is the interest rate per period that the market is using to discount cash flows t periods from now.
If you know the spot rates you can value government bonds, without estimating
the short term rates E[trt+1]. How do we do this?
Define: Ri as the yield on a zero coupon bond with i years
to maturity. (If no zero coupon bonds are available infer the spot rates
from the price of adjacent bonds). Consider the following example. You
have the following bonds available:
| Bond | Maturity (years) | Market Price | c | Face Value |
| 1 | 1 | 997.50 | 5% | 1000 |
| 2 | 2 | 1,026.40 | 8% | 1000 |
| 3 | 3 | 947.20 | 6% | 1000 |
Then the cash flows are:
| End of year | 1 | 2 | 3 |
| Bond 1 | 1050 | ||
| Bond 2 | 80 | 1080 | |
| Bond 3 | 60 | 60 | 1060 |
| Bond 4 | 30 | 30 | 1030 |
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