1. The Valuation of Bonds
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2. Bond values are discussed in one of two ways:
The dollar price
The yield to maturity
These two methods are equivalent since a price implies a yield, and vice-versa

3. Bond Yields
There are several ways that we can describe the rate of return on a bond:
Coupon rate
Current yield
Yield to maturity
Modified yield to maturity
Yield to call
Realized Yield

4. The Coupon Rate
The coupon rate of a bond is the stated rate of interest that the bond will pay
The coupon rate does not normally change during the life of the bond, instead the price of the bond changes as the coupon rate becomes more or less attractive relative to other interest rates
The coupon rate determines the dollar amount of the annual interest payment:

5. The Current Yield
The current yield is a measure of the current income from owning the bond
It is calculated as:

6. The Yield to Maturity
The yield to maturity is the average annual rate of return that a bondholder will earn under the following assumptions:
The bond is held to maturity
The interest payments are reinvested at the YTM
The yield to maturity is the same as the bond’s internal rate of return (IRR)

7. The Modified Yield to Maturity
The assumptions behind the calculation of the YTM are often not met in practice
This is particularly true of the reinvestment assumption
To more accurately calculate the yield, we can change the assumed reinvestment rate to the actual rate at which we expect to reinvest
The resulting yield measure is referred to as the modified YTM, and is the same as the MIRR for the bond

8. The Yield to Call
Most corporate bonds, and many older government bonds, have provisions which allow them to be called if interest rates should drop during the life of the bond
Normally, if a bond is called, the bondholder is paid a premium over the face value (known as the call premium)
The YTC is calculated exactly the same as YTM, except:
The call premium is added to the face value, and
The first call date is used instead of the maturity date

9. The Realized Yield
The realized yield is an ex-post measure of the bond’s returns
The realized yield is simply the average annual rate of return that was actually earned on the investment
If you know the future selling price, reinvestment rate, and the holding period, you can calculate an ex-ante realized yield which can be used in place of the YTM (this might be called the expected yield)

10. Calculating Bond Yield Measures
As an example of the calculation of the bond return measures, consider the following:
You are considering the purchase of a 2-year bond (semiannual interest payments) with a coupon rate of 8% and a current price of $ The bond is callable in one year at a premium of 3% over the face value. Assume that interest payments will be reinvested at 9% per year, and that the most recent interest payment occurred immediately before you purchase the bond. Calculate the various return measures.
Now, assume that the bond has matured (it was not called). You purchased the bond for $ and reinvested your interest payments at 9%. What was your realized yield?

11. Calculating Bond Yield Measures (cont.)1 2 3 4 40
1,000
Timeline if not called 1 2 40
1,030
Timeline if called

12. Calculating Bond Yield Measures (cont.)
The yields for the example bond are:
Current yield = 8.294%
YTM = 5% per period, or 10% per year
Modified YTM = 4.971% per period, or 9.943% per year
YTC = 7.42% per period, or 14.84% per year
Realized Yield:
if called = 7.363% per period, or % per year
if not called = 4.971% per period, or 9.943% per year

13. Bond Valuation in Practice
The preceding examples ignore a couple of important details that are important in the real world:
Those equations only work on a payment date. In reality, most bonds are purchased in between coupon payment dates. Therefore, the purchaser must pay the seller the accrued interest on the bond in addition to the quoted price.
Various types of bonds use different assumptions regarding the number of days in a month and year.

14. Valuing Bonds Between Coupon Dates (cont.)
Imagine that we are halfway between coupon dates. We know how to value the bond as of the previous (or next even) coupon date, but what about accrued interest?
Accrued interest is assumed to be earned equally throughout the period, so that if we bought the bond today, we’d have to pay the seller one-half of the period’s interest.
Bonds are generally quoted “flat,” that is, without the accrued interest. So, the total price you’ll pay is the quoted price plus the accrued interest (unless the bond is in default, in which case you do not pay accrued interest, but you will receive the interest if it is ever paid).

15. Valuing Bonds Between Coupon Dates (cont.)
The procedure for determining the quoted price of the bonds is:
Value the bond as of the last payment date.
Take that value forward to the current point in time. This is the total price that you will actually pay.
To get the quoted price, subtract the accrued interest.
We can also start by valuing the bond as of the next coupon date, and then discount that value for the fraction of the period remaining.

16. Valuing Bonds Between Coupon Dates (cont.)
Let’s return to our original example (3 years, semiannual payments of $50, and a required return of 7% per year).
As of period 0 (today), the bond is worth $1, As of next period (with only 5 remaining payments) the bond will be worth $1, Note that:
So, if we take the period zero value forward one period, you will get the value of the bond at the next period including the interest earned over the period. P1 P0
Interest earned

17. Valuing Bonds Between Coupon Dates (cont.)
Now, suppose that only half of the period has gone by. If we use the same logic, the total price of the bond (including accrued interest) is:
Now, to get the quoted price we merely subtract the accrued interest:
If you bought the bond, you’d get quoted $1, but you’d also have to pay $25 in accrued interest for a total of $1,

18. Day Count Conventions
Historically, there are several different assumptions that have been made regarding the number of days in a month and year. Not all fixed-income markets use the same convention:
30/360 – 30 days in a month, 360 days in a year. This is used in the corporate, agency, and municipal markets.
Actual/Actual – Uses the actual number of days in a month and year. This convention is used in the U.S. Treasury markets.
Two other possible day count conventions are:
Actual/360
Actual/365
Obviously, when valuing bonds between coupon dates the day count convention will affect the amount of accrued interest.

19. The Term Structure of Interest Rates
Interest rates for bonds vary by term to maturity, among other factors
The yield curve provides describes the yield differential among treasury issues of differing maturities
Thus, the yield curve can be useful in determining the required rates of return for loans of varying maturity

20. Types of Yield Curves

21. Today’s Actual Yield Curve
Data Source:

22. Explanations of the Term Structure
There are three popular explanations of the term structure of interest rates (i.e., why the yield curve is shaped the way it is):
The expectations hypothesis
The liquidity preference hypothesis
The market segmentation hypothesis (preferred habitats)
Note that there is probably some truth in each of these hypotheses, but the expectations hypothesis is probably the most accepted

23. The Expectations Hypothesis
The expectations hypothesis says that long-term interest rates are geometric means of the shorter-term interest rates
For example, a ten-year rate can be considered to be the average of two consecutive five-year rates (the current five-year rate, and the five-year rate five years hence)
Therefore, the current ten-year rate must be:

24. The Expectations Hypothesis (cont.)
For example, if the current five-year rate is 8% and the expected five-year rate five years from now is 10%, then the current ten-year rate must be:
In an efficient market, if the ten-year rate is anything other than 8.995%, then arbitrage will bring it back into line
If the ten-year rate was 9.5%, then people would buy ten-year bonds and sell five-year bonds until the rates came back into line

25. The Expectations Hypothesis (cont.)
The ten-year rate can also be thought of a series of five two-year rates, ten one-year rates, etc.
Note that since the ten-year rate is observable, we normally would solve for an expected future rate
In the previous example, we would usually solve for the expected five-year rate five years from now:

26. The Liquidity Preference Hypothesis
The liquidity preference hypothesis contends that investors require a premium for the increased volatility of long-term investments
Thus, it suggests that, all other things being equal, long-term rates should be higher than short-term rates
Note that long-term rates may contain a premium, even if they are lower than short-term rates
There is good evidence that such premiums exist

27. The Market Segmentation Hypothesis
This theory is also known as the preferred habitat hypothesis because it contends that interest rates are determined by supply and demand and that different investors have preferred maturities from which they do no stray
There is not much support for this hypothesis

28. Bond prices change as any of the variables change:
Bond Price Volatility
Bond prices change as any of the variables change:
Prices vary inversely with yields
The longer the term to maturity, the larger the change in price for a given change in yield
The lower the coupon, the larger the percentage change in price for a given change in yield
Price changes are greater (in absolute value) when rates fall than when rates rise

29. Measuring Term to Maturity
It is difficult to compare bonds with different maturities and different coupons, since bond price changes are related in opposite ways to these variables
Macaulay developed a way to measure the average term to maturity that also takes the coupon rate into account
This measure is known as duration, and is a better indicator of volatility than term to maturity alone

30. Duration
Duration is calculated as:
So, Macaulay’s duration is a weighted average of the time to receive the present value of the cash flows
The weights are the present values of the bond’s cash flows as a proportion of the bond price

31. Recall our earlier example bond with a YTM of 5% per six-months:
Calculating Duration
Recall our earlier example bond with a YTM of 5% per six-months:
Note that this is 3.77 six-month periods, which is about 1.89 years 1 2 3 4 40
1,000

32. Notes About Duration
Duration is less than term to maturity, except for zero coupon bonds where duration and maturity are equal
Higher coupons lead to lower durations
Longer terms to maturity usually lead to longer durations
Higher yields lead to lower durations
As a practical matter, duration is generally no longer than about 20 years even for perpetuities

33. Modified Duration
A measure of the volatility of bond prices is the modified duration (higher DMod = higher volatility)
Modified duration is equal to Macaulay’s duration divided by 1 + per period YTM
Note that this is the first partial derivative of the bond valuation equation wrt the yield

34. Why is Duration Better than Term?
Earlier, it was noted that duration is a better measure than term to maturity. To see why, look at the following example:
Suppose that you are comparing two five-year bonds, and are expecting a drop in yields of 1% almost immediately. Bond 1 has a 6% coupon and bond 2 has a 14% coupon. Which would provide you with the highest potential gain if your outlook for rates actually occurs? Assume that both bonds are currently yielding 8%.

35. Why is Duration Better than Term? (cont.)
Both bonds have equal maturity, so a superficial investigation would suggest that they will both have the same gain. However, as we’ll see bond 2 would actually gain more.

36. Why is Duration Better than Term? (cont.)
Note that the modified duration of bond 1 is longer than that of bond two, so you would expect bond 1 to gain more if rates actually drop.
Pbond 1, 8%= ; Pbond 1, 7%= ; gain = 38.85
Pbond 2, 8%= ; Pbond 2, 7%= ; gain = 45.30
Bond 1 has actually changed by less than bond 2. What happened? Well, if we figure the percentage change, we find that bond 1 actually gained by more than bond 2.
%Dbond 1 = 4.22%; %Dbond 2 = 3.91% so your gain is actually 31 basis points higher with bond 1.

37. Why is Duration Better than Term? (cont.)
Bond price volatility is proportionally related to the modified duration, as shown previously. Another way to look at this is by looking at how many of each bond you can purchase.
For example, if we assume that you have $100,000 to invest, you could buy about units of bond 1 and only units of bond 2.
Therefore, your dollar gain on bond 1 is $4, vs. $3, on bond 2. The net advantage to buying bond 1 is $ Obviously, bond 1 is the way to go.

38. Convexity
Convexity is a measure of the curvature of the price/yield relationship
Note that this is the second partial derivative of the bond valuation equation wrt the yield

39. Calculating Convexity
Convexity can be calculated with the following formula:
For the example bond, the convexity (per period) is:

40. Calculating Convexity (cont.)
To make the convexity of a semi-annual bond comparable to that of an annual bond, we can divide the convexity by 4
In general, to convert convexity to an annual figure, divide by m2, where m is the number of payments per year

41. Calculating Bond Price Changes
We can approximate the change in a bond’s price for a given change in yield by using duration and convexity:
If yields rise by 1% per period, then the price of the example bond will fall by 33.84, but the approximation is:

42. Solved Examples (on a payment date)