Chapter 4Bond Price Volatility

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.

Price-Yield Relationship - Maturity 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.

Price-Yield Relationship - Coupon Rate 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

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

exhibit greater price volatility when interest rates change. Why?

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

exhibit greater price volatility when interest rates change.

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.

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.

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

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 The change in the price of a bond if the required yield

changes by 1bp. 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 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 YTMold 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 we need:

1dPdy 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 (100bp) change in yield.

Duration is measured on a per period basis. For semi-annual cash flows, we adjust the duration to an annual figure by dividing by 2:

Duration in years2

Duration over a single period

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

11(1 )

(1 )

n

n

MyP Cy 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)

Duration duration is less than (coupon bond) or equal to (zero

coupon bond) the term to maturity all else equal,

the lower the coupon, the larger the duration the longer the maturity, the larger the duration the lower the yield, the larger the duration

the longer the duration, the greater the price volatility

Properties of DurationBond

Macaulay Duration

Modified Duration

9% 5-year 4.13 3.969% 25-year 10.33 9.886% 5-year 4.35 4.166% 25-year 11.10 10.620% 5-year 5.00 4.780% 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 (duration). The lower the coupon the greater the bond’s price volatility (duration).

Duration and Maturity: Duration increases with maturity.

Duration and Coupon: The lower the coupon the greater the

duration.

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 lower the yield the higher the duration.

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

Approximating Dollar Price Changes How do we measure dollar price changes for a given change in yield? We use dollar duration: approximate price change for 100 bp change in yield.

Recall: 1 = - modified durationdP

dy P

Solve for dP/dy:

_

(modified duration)dollar duration

dP Pdy

(dollar duration)dP dy

Solve for dP:

Example of Dollar Duration A 6% 25-year bond priced to yield 9% at 70.3570.

Dollar duration = 747.2009 What happens to bond price if yield increases by 100 bp?

(dollar duration)dP dy

(747.2009) 0.01dP

$7.47dP A 100 bp increase in yield reduces the bond’s price by

$7.47 dollars (per $100 in par value) 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 $22.4161 dollars (per $100 in par value)

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.

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 the yields affecting the four bonds change by 100 bps, the

value of the portfolio will change by about 5.4%.

Price-Yield Relationship

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 Dy.

Duration treats the price/yield relationship as a linear.

Error is small for small Dy.

The error occurs because of convexity.

P2, Actual

P2, Estimated

y3

P3, Actual

P3, Estimated

Error

The error is larger for yield decreases.

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.

How Do We Measure Convexity?1. Recall a Taylor Series Expansion from Calculus:

22

2

1 ( )2

dP d PdP dy dy errordy dy

2. Divide both sides by P to get percentage price change:2

22

1 1 1 ( )2

dP dP d P errordy dyP dy P dy P P

Note: First term on RHS of (1) is the dollar price change for a given

change in yield based on dollar duration. First term on RHS of (2) is the percentage price change for a given

change in yield based on modified duration. The second term on RHS of (1) and (2) includes the second

derivative of the price-yield relationship (this measures convexity)

Measuring Convexity The first derivative measures slope (duration). The second derivatives 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 percentage convexity measure of a bond:2

2

1convexity measure d Pdy P

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 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

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

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

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

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

2

51,476.26 12,869.0654

d 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(modified duration)( ) (convexity measure)( )2

dy dy

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?

Percentage price change due to duration and convexity

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

dy dy

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

21.24% 3.66% 17.58%

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.

Why Is It Still an Approximation? Recall the “error” in the our Taylor Series expansion?

22

2

1 ( )2

dP d PdP dy dy errordy dy

The “error” includes 3rd, 4th, and higher derivatives: The more derivatives we include in our equation, the more accurate

our measure becomes. Remember, duration is based on the first derivative and convexity is

based on the 2nd derivative.

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.

Approximation Methods We can approximate the duration and convexity for any

bond or more complex instrument using the following:

0

approximate duration =2( )( )

P PP y

D

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 in decimal form.

02

0

2approximate convexity =

( )( )P P P

P y

D

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

D71.1105 69.6164= 10.622(70.357)(0.001)

02

0

2approximate convexity =( )( )

P P PP y

D 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.