Bond Valuation & Duration

8/11/2019 Bond Valuation & Duration

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INVESTMENTS

Instructor:

Dr. Kumail Rizvi, PhD, CFA, FRM


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KEYCONCEPTS& SKILLS

Understand bond values and why they fluctuate

How Bond Prices Vary With Interest Rates

Four measures of bond price sensitivity to

interest rate Maturity

Macaulay Duration (Effective Maturity)

Modified Duration

Convexity

Understand the term structure of interest ratesand the determinants of bond yields

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VALUINGABOND

Bond Value = PV of coupons + PV of par

Bond Value = PV of annuity + PV of lump sum

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N

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VALUINGABOND

Example

If today is October 1, 2007, what is the value of the following

bond? An IBM Bond pays $115 every September 30 for 5 years.

In September 2012 it pays an additional $1000 and retires the

bond. The bond is rated AAA (AAA YTM is 7.5%)

Cash Flows

Sept 08 09 10 11 12115 115 115 115 1115

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VALUINGABOND

Example continued

If today is October 1, 2007, what is the value of the following bond? An

IBM Bond pays $115 every September 30 for 5 years. In September

2012 it pays an additional $1000 and retires the bond. The bond is rated

AAA (AAA YTM is 7.5%)

84.161,1$

075.1

115,1

075.1

115

075.1

115

075.1

115

075.1

1155432

PV

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WHYBONDS PRICESFLUCTUATE?

The price of a bond is a function of the promised

payments and the market required rate of return.

Since the promised payments are fixed, bond prices

change in response to the changes in the market

determined required rate of return.

Bond price = f (required rate of return)

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HOWBONDPRICESFLUCTUATE?

Interest Rates, %

Required rate of return

YTM

BondPrice,%

80.00

85.00

90.00

95.00

100.00

105.00

110.00

115.00

0 1 2 3 4 5 6 7 8 9 10

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HOWBONDPRICESFLUCTUATE?

Bond prices or present values decrease as rates

increase. It means,

if we increase our yield above the coupon, the present

value (price) must decrease below par.

On the other hand, if we decrease our yield below the

coupon, the present value (price) must increase above

par.

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WHYTHERELATIONSHIPISINVERSE?

Think that

the yield-to-maturity is the interest rate required on

newly issued debt of the same risk and that debt

would be issued so that the coupon = yield.

Then, suppose that the coupon rate on your bond is8% and the yield is 9%.

Which bond you would be willing to pay more for?

You would pay more for new bond since it is priced to

sell at $1,000, the 8% bond must sell for less than$1,000.

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WHYTHERELATIONSHIPISINVERSE?

Another way to look at it is that

return = dividend yield + capital gains yield.

The dividend yield in this case is just the coupon

rate. The capital gains yield has to make up the

difference to reach the yield to maturity. Therefore, ifthe coupon rate is 8% and the YTM is 9%, the capital

gains yield must equal approximately 1%. The only

way to have a capital gains yield of 1% is if the bond

is selling for less than par value. (If price = par, there

is no capital gain.)

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HOWTOMEASUREBONDPRICE

SENSITIVITYTOYIELDCHANGES?

Four measures of bond price sensitivity to

interest rate1) Maturity

2) Macaulay Duration (Effective Maturity)

3) Modified Duration4) Convexity

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MATURITY: SENSITIVITYMEASURE(1)

Simple maturity is just the time left to maturity on a bond.

We generally think of 5-year bonds or 10-year bonds. It is

straightforward and requires no calculation.

The longer the time to maturity the more sensitive aparticular bond is to changes in the required rate of return.

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Consider two zero coupon bonds, each with a face

value of $1,000. Bond A matures in 10 years and has

a required rate of return of 10%. The price of Bond A

is $376.89, where

Bond B has a maturity of 5 years and also has a

required rate of return of 10%. Its price is $613.91 or

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If the required rate of return for each bond was to

increase by 100 basis points to 11%, the prices would

then be $342.73 for Bond A and $585.43 for Bond B.

This translates into a -9.1% change in price for Bond

A and -4.6% for Bond B.

Thus, for zero coupon bonds simple maturity can be

used to compare price sensitivity

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MACAULAYDURATION: SENSITIVITY

MEASURE(2)

The relationship between price and maturity is not as clear

when you consider non-zero coupon bonds.

For a coupon-paying bond, many of the cash flows occur

before the actual maturity of the bond and the relative

timing of these cash flows will affect the pricing of thebond.

In order to deal with this, Frederick Macaulay in 1938

suggested that investors use the effective maturity of a

bond as a measure of interest rate sensitivity. He called

this duration and defined it as avalue-weighted average ofthe timing of the cash flows.

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MEASURE(2)

Consider a six- year bond with face value of $1,000, and a

6.1% coupon rate (semi-annual payments). If the current

yield to maturity is 10%, the value of the bond is found by

discounting each of the semi-annual payments

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MEASURE(2)

Macaulay Duration takes the present value of each

payment and divides it by the total bond price, P. By

doing this, one has a percentage, wt, of the total bond

value that is received in each period, t.

The duration or effective maturity for the bond could then

be estimated by multiplying the weight, wt, times the time,tand then summing all of the weighted values, or

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MEASURE(2)2

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MEASURE(2)

The semi- annual duration for this bond is 10.014 six-

month periods. We usually use annual duration and we

annualize the semi-annual duration simply by dividing by 2

(the number of six month periods in a year). In this case,

the annualized duration would be 5.007 years.

Note that the Macaulay Durationfor a 5- year zero coupon

bondis the same as the simple maturity, 5.0 years.

Hence, we can expect that the original 6-year, 6.1% couponbond when interest rates change to behave in a manner

similar to a 5-year zero coupon bond, since their effective

maturity (Macaulay Duration) is essentially the same.

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MODIFIEDDURATION: SENSITIVITY

MEASURE(3)

If we want a more direct measure of the relationship

between changes in interest rates and changes in

bond prices, we can use Modified Duration. Modified

Duration, D, is defined as the following

wherePis the bond price, P is the change in bond price

and y is the change in the required rate of return (yield tomaturity).

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MEASURE(3)

We know price of a bond is:

Taking the derivative ofPwith respect toy,

Inserting this into the formula for Modified Durationyields

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MEASURE(3)

Rearranging the previous formula slightly

Comparing this to the definition of Macaulay

Duration and using that definition we can write

Modified Duration as

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MEASURE(3)

While it is easy calculate Modified Duration once you

have Macaulay Duration the interpretations of the

two are quite different.

Macaulay Duration is an average or effective

maturity.Modified Duration really measures how small

changes in the yield to maturity affect the price of the

bond.

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MEASURE(3)2

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MEASURE(3)

Modified Duration assumes that the price changes are

linear with respect to changes in the yield to

maturity.

From last table, the true relationship between thebond's price and the yield to maturity is not linear.

The Column with the differences is always positive

and increases as we move away from a yield tomaturity of 10%. The actual relationship between the

bond price and the yield to maturity is:

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MEASURE(3)29/03/2013

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CONVEXITY: SENSITIVITYMEASURE(4)

Modified Duration relationship does not fully capture

the true relationship between bond prices and yield to

maturity. In order to more fully capture this,

practitioners use Convexity. The definition of

Convexityis

The actual definition of Convexity that we can use is

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We can annualize the semi-annual convexity of

110.88 by dividing it by 22 or 4. Here it would be

27.72.

Convexity is useful to practitioners in a number of

ways. First it can be used in conjunction with duration to

get a more accurate estimate of the percentage price

change resulting from a change in the yield.

The formula is:

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Adjustment

factor


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Adding the convexity adjustment corrects for the fact

that Modified Duration understates the true bond

price.

For example, in our example, at a yield of 12% the

percentage price change using only Modified Durationwas -9.54%, while the actual was -9.01%. If we use

the Convexity value we just calculated, the predicted

percentage price change would be

This is -8.99%, which is much closer to the actual

percentage price change of -9.01%.

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Convexity provides insight into how a bond will react

to yield changes.

Earlier we saw that the price reaction to changes in

yield is not symmetric.

For a given change in yield, bond prices drop lessfor a given increase in yield and increase more

for the same decreases in yield. The downside is

less and the upside is more. This is clearly a desirable

property.

The higher the Convexity of a bond the more this is

true. Thus, bonds with high convexity are more

desirable.

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TERMSTRUCTUREOFINTERESTRATES

Term structure is the relationship between time tomaturity and yields, all else equal

It is important to recognize that we pull out the effect ofdefault risk, different coupons, etc.

Yield curve graphical representation of the termstructure

Normalupward-sloping, long-term yields are higher than

short-term yields Inverted downward-sloping, long-term yields are lower

than short-term yields

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FIGUREUPWARD-SLOPINGYIELDCURVE29/03/2013

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FIGUREDOWNWARD-SLOPINGYIELDCURVE29/03/2013

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http://www.bloomberg.com/markets/rates/index.html


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ASSIGNMENT

What is Effective Duration? Give an example to show

its calculation

What is the difference and similarity between

Modified Duration and Effective Duration?

Where it is appropriate to use Modified duration andwhere Effective duration is more suitable?

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Bond Valuation & Duration

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