Chapter 4

Chapter 4Bond Price VolatilityChapter Pages 58-85,89-91

Introduction Bond volatility is a result of interest rate

volatility: When interest rates go up bond prices go down and

vice versa. Goals of the chapter:

To understand a bond’s price volatility characteristics.

Quantify price volatility.

Review of Price-Yield Relationships Consider two 9% coupon semiannual pay bonds:

Bond A: 5 years to maturity. Bond B: 25 years to maturity.

Yield 5 Years 25 Years6 1,127.95 1,385.95 7 1,083.17 1,234.56 8 1,040.55 1,107.41 9 1,000.00 1,000.00

10 961.39 908.72 11 924.62 830.68 12 889.60 763.57

The long-term bond price is more sensitive to interest rate changes than the short-term bond price.

Review of Price-Yield Relationships Consider three 25 year semiannual pay bonds:

9%, 6%, and 0% coupon bonds Notice what happens as yields increase from 6% to 12%:

Yield 9% 6% 0% 9% 6% 0%6% 1,127.95 1,000.00 228.11 0% 0% 0%7% 1,083.17 882.72 179.05 -4% -12% -22%8% 1,040.55 785.18 140.71 -8% -21% -38%9% 1,000.00 703.57 110.71 -11% -30% -51%

10% 961.39 634.88 87.20 -15% -37% -62%11% 924.62 576.71 68.77 -18% -42% -70%12% 889.60 527.14 54.29 -21% -47% -76%

Bond Characteristics That Influence Price Volatility

Coupon Rate: For a given maturity and yield, bonds with lower coupon rates

exhibit greater price volatility when interest rates change. Why?

Maturity: For a given coupon rate and yield, bonds with longer maturity

exhibit greater price volatility when interest rates change. Why?

Note: The higher the yield on the bond, the lower its volatility.

Shape of the Price-Yield Curve If we were to graph price-yield changes for bonds we

would get something like this:

Yield

Price

What do you notice about this graph?

It isn’t linear…it is convex. It looks like there is more

“upside” than “downside” for a given change in yield.

Quick Review: Why Do Yields Change? The required return on any security equals:

r = rreal + Expected Inflation + RP

Yields can change for three reasons: Change in the real rate—compensation for deferring

consumption. Change in expected inflation—i.e., erosion of purchasing

power (important). Change in risk—e.g., credit risk, liquidity risk, etc.

Price Volatility Properties of Bonds Exhibit 4-3 from Fabozzi text, p. 61 (required yield is 9%):

Price Volatility Properties of Bonds

Properties of option-free bonds: All bond prices move opposite direction of yields, but the

percentage price change is different for each bond, depending on maturity and coupon

For very small changes in yield, the percentage price change for a given bond is roughly the same whether yields increase or decrease.

For large changes in yield, the percentage price increase is greater than a price decrease, for a given yield change.

Measures of Bond Price Volatility

Three measures are commonly used in practice:1. Price value of a basis point

(also called dollar value of an 01)2. Yield value of a price change3. Duration

Price Value of a Basis Point Change in the dollar price of the bond if the required

yield changes by 1 bp. Recall that small changes in yield produce a similar

price change regardless of whether yields increase or decrease.

Therefore, the Price Value of a Basis Point is the same for yield increases and decreases.

Price Value of a Basis Point - pg 63

We examine the price of six bonds assuming yields are 9%. We then assume 1 bp increase in yields (to 9.01%)

BondInitial Price

(at 9% yield)New Price(at 9.01%)

Price Value of a BP

5-year, 9% coupon 100.0000 99.9604 0.0396

25-year, 9% coupon 100.0000 99.9013 0.0987

5-year, 6% coupon 88.1309 88.0945 0.036425-year, 6% coupon 70.3570 70.2824 0.0746

5-year, 0% coupon 64.3928 64.362 0.0308

25-year, 0% coupon 11.0710 11.0445 0.0265

Yield Value of a Price Change

Procedure: Calculate YTM. Reduce the bond price by X dollars. Calculate the new YTM. The difference between the YTMnew and YTMinitial is the

yield value of an X dollar price change.

Duration The concept of duration is based on the slope of the

price-yield relationship:

Yield

Price

What does slope of a curve tell us? How much the y-axis changes

for a small change in the x-axis. Slope = dP/dy Duration—tells us how much

bond price changes for a given change in yield.

Note: there are different types of duration.

Two Types of Duration Modified duration:

Tells us how much a bond’s price changes (in percent) for a given change in yield.

Dollar duration: Tells us how much a bond’s price changes (in dollars)

for a given change in yield. We will start with modified duration.

Deriving Duration The price of an option-free bond is:

2 3(1 ) (1 ) (1 ) (1 ) (1 )n n

C C C C MPy y y y y

P = bond’s price C = semiannual coupon payment M = maturity value (Note: we will assume M = $100) n = number of semiannual payments (#years 2). y = one-half the required yield

How do we get dP/dy?

Duration, con’t

2

1 21 (1 ) (1 ) (1 ) (1 )n n

dP C C nC nMdy y y y y y

This tells us the approximate dollar price change of the bond for a small change in yield.

To determine the percentage price change in a bond for a given change in yield (called modified duration) we need:

1dP dPPdy dy P

2

1 1 21 (1 ) (1 ) (1 ) (1 )n n

C C nC nMy P y y y y

The first derivative of bond price (P) with respect to yield (y) is:

Macaulay Duration

Duration, con’t Therefore we get:

Modified Duration1

Macaulay Durationy

Modified duration gives us a bond’s approximate percentage price change for a small change in yield.

The negative sign reflects the inverse relation between bond price and yield. Duration is measured over the time horizon of the m periodic CFs that occur

during a year (typically m = 2). To get an annual duration:

Duration in years Duration over a single periodm

Calculating Duration Recall that the price of a bond can be expressed as:

11(1 )

(1 )

n

n

MyPy y

Taking the first derivative of P with respect to y and multiplying by 1/P we get:

2 11 ( / )1

(1 ) (1 )Modified Duration

n nC n M C yy y y

P

Example Consider a 25-year 6% coupon bond selling at 70.357

(par value is $100) and priced to yield 9%.

2 1

1 ( / )1(1 ) (1 )

Modified Durationn n

C n M C yy y y

P

2 50 513 1 50(100 3 / 0.045)1

0.045 (1.045) (1.045)Modified Duration

70.357

Modified Duration 21.23508

To get modified duration in years we divide by 2:

Modified Duration 10.62

(in number of semiannual periods)

(what is Macaulay duration?)

Properties of DurationBond

Macaulay Duration

Modified Duration

9% 5-year 4.13 3.96

9% 25-year 10.33 9.88

6% 5-year 4.35 4.16

6% 25-year 11.10 10.62

0% 5-year 5.00 4.78

0% 25-year 25.00 23.98

Earlier we showed that holding all else constant: The longer the maturity the greater the bond’s price volatility. The lower the coupon the greater the bond’s price volatility.

So, the greater a bond’s duration, the greater its volatility: So duration is a measure of a bond’s volatility.

Duration and Maturity: Duration increases with maturity. Coupon bonds: duration < maturity. Zeros: Macaulay duration = maturity

Modified duration < maturity. Duration and Coupon:

The lower the coupon the greater the duration (exception is long-maturity deep-discount bonds)

Properties of Duration, con’t What is the relationship between duration and yield?

Yield(%)

Modified Duration

7 11.218 10.539 9.88

10 9.2711 8.712 8.1613 7.6614 7.21

The higher the yield the lower the duration.

Therefore, the higher the yield the lower the bond’s price volatility.

Duration In Action! Recall:

1modified duration dPdy P

Solve for dP/P (the % price change): Formula 4.11

modified durationdP dyP

We can use this to approximate the % price change in a bond for a given change in yield.

Example: Consider the 25-year 6% bond priced at 70.3570 to yield 9%. Modified duration = 10.62.

By how much will the bond price change (in percentage terms) if yields increase from 9% to 9.10%?

Solution Using our formula: modified durationdP dy

P

Here, y is changing from 0.09 to 0.091 so dy = +0.001: 10.62 0.001dP

P 0.01062

Thus, a 10 bp increase in yield will result (approximately) in a 1.06% decline in bond price.

Note this effect is symmetric: A 10 bp decline in yield (from 9% to 8.90%) result in a 1.06% price

increase.

One More Example

Assume the yield increases by 300 bps.

10.62 0.03dPP

(or –31.86%)0.3186

Likewise, a 300 bps decline in yield will change the bond’s price by +31.86%

Are these approximations accurate?

Accuracy of Duration (Exhibit 4-3)

Change (bp)

Duration %∆

Actual %∆

Abs Dif

+10 -1.06 -1.05 0.01-10 +1.06 +1.07 0.01

+300 -31.86 -25.08 6.78-300 +31.86 42.13 10.27

Problems with duration: It assumes symmetric changes in bond price (not true in reality). The greater the yield change the larger the approximation error. Duration works well for small yield changes but is problematic for

large yield changes.

Approximating Dollar Changes How do we measure dollar price changes for a given

change in yield? Recall:

1Modified Duration dPdy P

Solve for dP/dy:

(modified duration)dP Pdy

(This is called Dollar Duration)

(dollar duration)dP dy

Solve for dP:

Consider Previous Example A 6% 25-year bond priced to yield 9% at 70.3570.

Dollar duration = 747.2009 (= 10.62 x 70.3570) What happens to bond price if yield increases by 1 bp?

(dollar duration)dP dy

(747.2009) 0.0001dP

$0.0747dP A 1 bp increase in yield reduces the bond’s price by

$0.0747 dollars (per $100 of face value) If an investor had $1,000,000 in face value of the bond, a 1 bp increase

in yield would reduce the value of the holdings by $747. This is a symmetric measurement.

Example, con’t Suppose yields increased by 300 bps:

(dollar duration)dP dy(747.2009) 0.03dP $22.4161dP

A 300 bp increase in yield reduces the bond’s price by $22.42 dollars (per $100 in par value)

A $1,000,000 face value in bond holding would decline in value by $224,161 if the yield were to increase by 300 bps.

Again, this is symmetric.

How accurate is this approximation? As with modified duration, the approximation is good for small yield

changes, but not good for large yield changes.

Accuracy of Duration Why is duration more accurate for small changes in yield than for

large changes? Because duration is a linear approximation of a curvilinear (or convex) relation:

Yield

Price

y0

P0P1

y1 y2

Error

Error is large for large y.

Duration treats the price/yield relationship as a linear.

Error is small for small y.

The error occurs because of convexity.

P2, Actual

P2, Estimated

y3

P3, Actual

P3, Estimated

Error

The error is larger for yield decreases.

Portfolio Duration The duration of a portfolio of bonds is the weighted average of

the durations of the bonds in the portfolio. Example:

Bond Market Value Weight DurationA $10 million 0.10 4B $40 million 0.40 7C $30 million 0.30 6D $20 million 0.20 2

Portfolio duration is: 0.10 4 0.40 7 0.30 6 0.20 2 5.40 If all yields affecting all bonds change by 100 bps, the value of

the portfolio will change by about 5.4%.

Convexity Duration is a good approximation of the price yield-

relationship for small changes in y. For large changes in y duration is a poor approximation.

Why? Because the tangent line to the curve can’t capture the appropriate price change when ∆y is large.

Also keep in mind that there is a different duration for every different yield for a bond.

This means each time we get a new yield, we need to calculate a new duration.

Measuring Convexity The first derivative measures slope (duration). The second derivative measures the change in slope (convexity). As with duration, there are two convexity measures:

Dollar convexity measure – Dollar price change of a bond due to convexity. Convexity measure – Percentage price change of a bond due to convexity.

The dollar convexity measure of a bond is:2

2dollar convexity measure d Pdy

The convexity measure of a bond:2

2

1convexity measure d Pdy P

Measuring Convexity

Now we can measure the dollar price change of a bond due to convexity:

2(dollar convexity measure)( )dP dy

The percentage price change of a bond due to convexity:21 (dollar convexity measure)( )

2dP dyP

Calculating Convexity How do we actually get a convexity number? Start with the simple bond price equation:

2 3(1 ) (1 ) (1 ) (1 ) (1 )n n

C C C C MPy y y y y

Take the second derivative of P with respect to y:2

2 21

( 1) ( 1)(1 ) (1 )

n

t nt

d P t t C n n Mdy y y

Or using the PV of an annuity equation, we get:2

3 2 1 2

2 1 2 ( 1)( / )1(1 ) (1 ) (1 )n n n

d P C Cn n n M C ydy y y y y y

Convexity Example Consider a 25-year 6% coupon bond priced at 70.357

(per $100 of par value) to yield 9%. Find convexity.2

3 2 1 2

2 1 2 ( 1)( / )1(1 ) (1 ) (1 )n n n

d P C Cn n n M C ydy y y y y y

2

3 50 2 51 52

2(3) 1 2(3)(50) 50(51)(100 3 / 0.045)10.045 (1.045) 0.045 (1.045) (1.045)

d Pdy

2

51,476.26d Pdy

Note: Convexity is measured in time units of the coupons.

To get convexity in years, divide by m2 (typically m = 2)2 51,476.26 12,869.065

4d Pdy

Price Changes Using Both Duration and Convexity

% price change due to duration: = -(modified duration)(dy)

% price change due to convexity: = ½(convexity measure)(dy)2

Therefore, the percentage price change due to both duration and convexity is:

21 1(modified duration)( ) (convexity measure)( )2

dP dy dydy P

Example A 25-year 6% bond is priced to yield 9%.

Modified duration = 10.62 Convexity measure = 182.92

Suppose the required yield increases by 200 bps (from 9% to 11%). What happens to the price of the bond?

21 1(modified duration)( ) (convexity measure)( )2

dP dy dydy P

21 1(10.62)(0.02) (182.92)(0.02)2

dPdy P

1 21.24 3.66 17.58dPdy P

Important Question: How Accurate is Our Measure?

If yields increase by 200 bps, how much will the bond’s price actually change?

Measure of Percentage Price Change

Percentage Price Change

Duration -21.24Duration & Convexity -17.58Actual -18.03

Note: Duration & convexity provides a better approximation than duration alone.

But duration & convexity together is still just an approximation.

Some Notes On Convexity

Convexity refers to the curvature of the price-yield relationship.

The convexity measure is the quantification of this curvature

Duration is easy to interpret: it is the approximate % change in bond price due to a change in yield.

But how do we interpret convexity? It’s not straightforward like duration, since convexity is based on the

square of yield changes.

The Value of Convexity Suppose we have two bonds with the same duration and the same

required yield:

Yield

Price

Bond ABond B

Notice bond B is more curved (i.e., convex) than bond A. If yields rise, bond B will fall less than bond A. If yields fall, bond B will rise more than bond A. That is, if yields change from y0, bond B will always be

worth more than bond A! Convexity has value! Investors will pay for convexity (accept a lower yield) if

large interest rate changes are expected.

y0

Properties of Convexity All option-free bonds have the following properties

with regard to convexity. Property 1:

As bond yield increases, bond convexity decreases (and vice versa). This is called positive convexity.

Property 2: For a given yield and maturity, the lower the coupon the greater a

bond’s convexity.

Property 3: For a given yield and modified duration, the lower the coupon the

smaller the convexity (I disagree with this property – possible error)

Additional Concerns with Duration We know duration ignores convexity and may not be appropriate

when measuring price volatility. However, there are other concerns to address.

Notice that duration is based on the simple bond pricing formula:

2 3(1 ) (1 ) (1 ) (1 ) (1 )n n

C C C C MPy y y y y

This formula assumes that yields for all maturities are the same (i.e., flat yield curve) and that all yield curve shifts are parallel. This is not true in general! Recall we can view a bond as a package of zeros, each with it’s own yield. We also know that the yield curve usually does not shift in a parallel fashion.

Our discussion of duration applies only to option-free bonds.

Duration as an Alternative Measure of “Maturity”

It is popular to interpret duration as the “weighted average” life of a bond.

This is true only with very simple bonds and is not true in general…be careful.

For example, there are 20 year bonds with durations greater than 20 years!

Obviously the interpretation as weighted average life does not hold.

Approximation Methods We can the approximate duration and convexity for any

bond or more complex instrument using the following:

0

approximate duration =2( )( )

P PP y

Where: P– = price of bond after decreasing yield by a small number of bps. P+ = price of bond after increasing yield by same small number of bps. P0 = initial price of bond. ∆y = change in yield.

02

0

2approximate convexity =

( )( )P P P

P y

Example of Approximation Consider a 25-year 6% coupon bond priced at 70.357 to yield 9%. Increase yield by 10 bps (from 9% to 9.1%): P+ = 69.6164 Decrease yield by 10 bps (from 9% to 8.9%): P- = 71.1105.

0

approximate duration =2( )( )

P PP y

71.1105 69.6164= 10.622(70.357)(0.001)

02

0

2approximate convexity =

( )( )P P P

P y

2

69.6164+71.1105 2(70.357)= 183.3(70.357)(0.001)

How accurate are these approximations? Actual duration = 10.62 Actual convexity = 182.92

These equations do a fine job approximating duration & convexity.

Additional Series Of Slides: Additional Slides on Duration & Convexity

Used by Dr. Shaffer in MBA derivatives class Additional explanations & clarifications B is used for P for notation

What is Duration? Measures how long, on average, it takes to receive the

cash flows from a bond: Is a weighted average of the “maturities” of a bond’s cash flows.

Recall, a bond is a package of zero-coupon bonds: Duration is the weighted average maturity of all of those zero-

coupon bonds.

n

iiitwD

1

Maturity of the ith cash flow (ith zero-coupon bond)

Weight given to the ith cash flow (ith zero coupon bond)

Duration What is the weight, wi, given to each cash flow?

The percentage contribution that each cash flow makes to the value of the bond.

The greater the impact a cash flow has on a bond’s value, the greater the weight assigned to that cash flow:

BeCFw

iyti

i

Example:

Consider two 5-year bonds, identical in every respect except the order in which the cash flows are received.

(for familiarity, we use discrete compounding).

Duration

The bond cash flows look like:

543211 )1(1001

)1(1

)1(1

)1(1

)1(1

yyyyyB

But, which bond has less interest rate risk? Why?

Both have the same maturity

543212 )1(1

)1(1

)1(1

)1(1

)1(1001

yyyyyB

Duration Example

Consider the following bond: Maturity = 5-years Coupon = 10% (coupons paid annually). Face Value = $1,000. Yield = 12% (and term structure is flat).

What is the bond’s price? What is the bond’s duration?

Duration Example The bond price is:

5544332211 twtwtwtwtwD

54321 54321 wwwwwD

Now we only need the wi

Duration is:

512.0412.0312.0212.0112.0 1100100100100100 eeeeeB

69.90269.60388.6177.6966.7869.88 B

Duration Example

%83.969.902

69.881 w

12.4)5(6688.0)4(0686.0)3(0773.0)2(0871.0)1(0983.0 D

Recall, the bond’s discounted cash flows:

Therefore:

Therefore:

69.90269.60388.6177.6966.7869.88 B

%88.6669.90269.603

5 w%86.669.90288.61

4 w

%71.869.90266.78

2 w

%73.769.90277.69

3 w

BeCFw

iyti

i

Duration Comments

We just found the bond’s Macaulay duration. Note that duration, D, is based on the first derivative

of bond price, B, with respect to y:

ByBD 1

Recall that a derivative the slope of a line tangent to a curve.

Duration Relationship between bond prices and yields:

Bond price

Yield

dydB1

dydB2

The shape of this curve is convex.

Duration is based on the slope of this curve.

There is a different duration for every bond price (point on the curve).

Duration Recall duration: By

BD 1

ByBD 1

yDBB

oryBDB

We can express this as:

Now we can see how a bond’s price changes when yields change:

Dollar change in bond’s price Percentage change in bond’s price

Duration Example

Consider the example earlier. The bond was priced at $902.69 and its duration was 4.12. The term structure was flat at 12%.

How is the bond’s price related to changes in interest rates?

Duration Example Recall:

yBDB yB 12.469.902

Suppose interest rates increase by 100bp (i.e., 1% from 12% to 13%).

Then the bond’s price will drop by:19.37$01.008.719,3 B

y 08.719,3

The new bond price will be: $865.50 (= $902.69 - $37.19)

Duration Relationships Duration increases as:

(1) Maturity increases: There are more cash flows in the “out years” thus a higher

duration. (2) Coupons decrease:

Means distant cash flows to contribute more to the bond’s value. (3) YTM decreases:

The distant cash flows get discounted less and contribute more to the bond’s value.

Portfolio Duration Is the weighted average of the durations of the

bonds in the portfolio. For a portfolio of N bonds, the duration is:

N

iiip DwD

1

wi = Bond’s value in proportion to the value of the portfolio

Limitations of Duration

The application of duration is limited. Why? It makes two critical assumptions:

(1) The yield curve makes parallel shifts. (2) The shifts in the yield curve are small.

Changes in Yield are Small

When changes in yield are small, bond price changes can be approximated by duration.

However, if moderate or large shifts are experienced, duration is not accurate:

A second factor, convexity, must be considered. Recall, duration is based on first derivative of

price with respect to yield: Convexity is the based on second derivative.

Convexity: Small yBond Price

Yield

0dydBADuration (slope)

yy y0

In both cases:Duration approximates bond’s price change.

BD BNow suppose yields increase or decrease.

BD B

Convexity: Large y

Bond Price

Yield

Duration (slope)

Error is large for large y.

y0

B

BD

yy

BD

Duration treats price/yield relationship as a linear.

Error is small for small y.

B

(B – BD) is the error due to convexity.

More on ConvexityBond Price

Yield

Bond ABond B

Bonds A and B have same duration. Bond A is more convex than Bond B:

When y increases Bond A will decline less than Bond B.

When y decreases Bond A will increase more than Bond B.

Thus: Bond A performs better than Bond B. Convexity is valuable. High convexity bonds cost more than low

convexity bonds.

y0

Duration & Convexity If we include convexity, we can more accurately relate

the change in a bond’s price to changes in yield:

2

21 yCyD

BB

or

2

21 yCByDBB

Duration Convexity

Percentage change in bond price

Duration Convexity

Dollar change in bond price

Convexity Example

A 5-year 10% coupon bond is priced at $1,081.10 and has a duration of 3.94 and convexity of 19.37.

By what percentage will this bond’s price change if interest rates increase or decrease by 1% (i.e. 100 bp)?

Convexity Example

If interest rates rise by 1%:

2

21 rCrD

BB

2

21 rCrD

BB

If interest rates fall by 1%:

2)01.0(37.1921)01.0(94.3 %84.3

2)01.0(37.1921)01.0(94.3 %04.4

Convexity Properties

Convexity is: Greatest when bond cash flows are evenly distributed over life of bond. Lowest when bond cash flows are concentrated around a point in time.

The convexity of a portfolio of bonds is a weighted average of the convexities of the individual bonds in the portfolio:

N

iiip CwC

1