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The Valuation of Bonds

Valuation bond

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Page 1: Valuation bond

The Valuation of Bonds

Page 2: Valuation bond

Bond Values

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

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Bond Yields

The rate of return on a bond: Coupon rate Current yield Yield to maturity Modified yield to maturity Yield to call Realized Yield

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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:

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The Current Yield

The current yield is a measure of the current income from owning the bond

It is calculated as:

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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)

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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

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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

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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)

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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.

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Valuing Bonds Between Coupon Dates

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).

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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.

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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.

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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

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Types of Yield Curves

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Today’s Actual Yield Curve

Maturity YLDPRIME 4.75%DISC 1.25%FUNDS 1.75%90 DAY 1.71%180 DAY 1.88%YEAR 2.19%2 YR 3.23%3 YR 3.74%4 YR 4.18%5 YR 4.43%7 YR 4.91%10 YR 5.10%15YR 5.64%20 YR 5.76%30 YR 5.61%

U.S. Treasury Yield Curve24 April 2002

1.00%2.00%3.00%4.00%5.00%6.00%

90 D

AY

180 D

AY

YEAR2 Y

R3 Y

R4 Y

R5 Y

R7 Y

R10

YR

15YR

20 Y

R

30 Y

R

Term to Maturity

Yie

ld

Data Source: http://www.ratecurve.com/yc2.html

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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

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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:

10 555

5510 111 RRR t

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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

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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

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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

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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

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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

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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

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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

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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

Yield

D = Slope of Tangent LineMod

Convexity