1. Chapter 4
Bond Price Volatility

2. 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.

3. 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 Years 6
1,127.95
1,385.95 7
1,083.17
1,234.56 8
1,040.55
1,107.41 9
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.

4. 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%
1,127.95
1,000.00
228.11 7%
1,083.17
882.72
179.05
-4%
-12%
-22% 8%
1,040.55
785.18
140.71
-8%
-21%
-38%
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
-47%
-76%

5. 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.

6. Shape of the Price-Yield Curve
If we were to graph price-yield changes for bonds we would get something like this:
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
Yield

7. 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.

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

9. Measures of Bond Price Volatility
Three measures are commonly used in practice:
Price value of a basis point (also called dollar value of an 01)
Yield value of a price change
Duration

10. 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.

11. 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%)
Bond
Initial Price (at 9% yield)
New Price (at 9.01%)
Price Value of a BP
5-year, 9% coupon
0.0396
25-year, 9% coupon
0.0987
5-year, 6% coupon
0.0364
25-year, 6% coupon
0.0746
5-year, 0% coupon
64.362
0.0308
25-year, 0% coupon
0.0265

12. 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.

13. Duration
The concept of duration is based on the slope of the price-yield relationship:
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.
Price
Yield

14. Two Types of Duration Modified duration: Dollar 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.

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

16. Duration, con’t
The first derivative of bond price (P) with respect to yield (y) is:
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:
Macaulay Duration

17. Duration, con’t Therefore we get:
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:

18. Calculating Duration
Recall that the price of a bond can be expressed as:
Taking the first derivative of P with respect to y and multiplying by 1/P we get:

19. Example
Consider a 25-year 6% coupon bond selling at (par value is $100) and priced to yield 9%.
(in number of semiannual periods)
To get modified duration in years we divide by 2:

20. 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

21. Properties of Duration
Bond
Macaulay Duration
Modified Duration 9%
5-year
4.13
3.96
25-year
10.33
9.88 6%
4.35
4.16
11.10
10.62 0%
5.00
4.78
25.00
23.98
Duration and Maturity:
Duration increases with maturity.
Duration and Coupon:
The lower the coupon the greater the duration.
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).

22. Properties of Duration, con’t
What is the relationship between duration and yield?
The lower the yield the higher the duration.
Therefore, the lower the yield the higher the bond’s price volatility.
Yield (%)
Modified Duration 7
11.21 8
10.53 9
9.88 10
9.27 11
8.7 12
8.16 13
7.66 14
7.21

23. 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:
Solve for dP/dy:
Solve for dP:

24. Example of Dollar Duration
A 6% 25-year bond priced to yield 9% at
Dollar duration =
What happens to bond price if yield increases by 100 bp?
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.

25. Example, con’t Suppose yields increased by 300 bps:
A 300 bp increase in yield reduces the bond’s price by $ 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.

26. 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
Duration A
$10 million
0.10 4 B
$40 million
0.40 7 C
$30 million
0.30 6 D
$20 million
0.20 2
Portfolio duration is:
If all the yields affecting the four bonds change by 100 bps, the value of the portfolio will change by about 5.4%.

27. Price-Yield Relationship

28. 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:
Price
P3, Actual
Duration treats the price/yield relationship as a linear.
Error
Error is small for small Dy.
P3, Estimated
Error is large for large Dy.
The error is larger for yield decreases. P0
The error occurs because of convexity. P1
P2, Actual
P2, Estimated
Error y3 y0 y1 y2
Yield

29. 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.

30. How Do We Measure Convexity?
Recall a Taylor Series Expansion from Calculus:
Divide both sides by P to get percentage price change:
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)

31. 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:
The percentage convexity measure of a bond:

32. Calculating Convexity
How do we actually get a convexity number?
Start with the simple bond price equation:
Take the second derivative of P with respect to y:
Or using the PV of an annuity equation, we get:

33. Convexity Example
Consider a 25-year 6% coupon bond priced at (per $100 of par value) to yield 9%. Find convexity.
Note: Convexity is measured in time units of the coupons.
To get convexity in years, divide by m2 (typically m = 2)

34. 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:

35. Example A 25-year 6% bond is priced to yield 9%.
Modified duration = 10.62
Convexity measure =
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

36. 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
-21.24
Duration & Convexity
-17.58
Actual
-18.03
Note: Duration & convexity provides a better approximation than duration alone.
But duration & convexity together is still just an approximation.

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

38. 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.

39. The Value of Convexity
Suppose we have two bonds with the same duration and the same required yield:
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.
Price
Bond B
Bond A y0
Yield

40. 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.

41. Approximation Methods
We can approximate the duration and convexity for any bond or more complex instrument using the following:
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.

42. Example of Approximation
Consider a 25-year 6% coupon bond priced at to yield 9%.
Increase yield by 10 bps (from 9% to 9.1%): P+ =
Decrease yield by 10 bps (from 9% to 8.9%): P- =
How accurate are these approximations?
Actual duration = 10.62
Actual convexity =
These equations do a fine job approximating duration & convexity.