1. Fixed Income Analysis Week 4 Measuring Price Risk
Com 4FJ3
Fixed Income Analysis
Week 4
Measuring Price Risk

2. Basics of Price Risk As YTM changes, bond prices change
Bond prices move in the opposite direction to the change in yield
Not all bonds react the same amount to a given change in yield
For large changes in yield, an increase has a higher change than a decrease

3. Time to Maturity Effect

4. Time to Maturity
All else held constant the longer the time to maturity the larger the price volatility of a bond with respect to changing yields
Intuition; if I am paying a premium to lock in an above average current yield, I am willing to pay more to lock it in for a longer period of time

5. Coupon Rate Consider two bonds, A and B;
$1,000 face value maturity = 10 years
YTM = 9%
Coupon rate; A = 5% B = 10%
Initial Prices = PVcoupons + PVface
A = = $740
B = = $1,065
What % change in price if YTM ↓ 8%?

6. Coupon Rate Price A, 8% = 340 + 456 = $796
Change = ( ) / 740 = 7.6%
Price B, 8% = = $1136
Change = (1, ,065) / 1,065 = 6.7%
In general, the larger the coupon payment, the less the change in price with a change in yield.

7. Effect of YTM

8. Level of YTM
As the level of interest rates rise, the sensitivity of bond prices to changes in the yield falls
Intuition; a change from 2% to 2.1% is much more significant than a change from 16% to 16.1% as a fraction of total return

9. Price Value of a Basis Point
One measure of price change is the dollar change in the price of a bond for a 1 basis point increase in the required yield
Also known as dollar value of an 01
Stated based on the pricing convention of quotes per $100 of face value
p63; 5 year 9% coupon par bond, 3.96¢

10. Yield Value of a Price Change
Pricing conventions used to quote prices in 32nds or 8ths of a point (fraction of a dollar per $100 of face value)
This measure converts the minimum price change into the effective change to YTM
5 year 9% par bond; -⅛ = $99.875
New YTM = 9.032%
Yield value of an 8th = 3.2 basis points

11. Macaulay’s Duration First published in 1938
A bond can be considered to be a package of zero coupon bonds
By taking a weighted average of the maturity of those zero coupon bonds, you can approximate the price sensitivity of the portfolio that the bond represents

12. Macaulay’s Duration
The average time that you wait for each payment, weighted by the percentage of the price that each payment represents.
Captures the effect of maturity, coupon rate and yield on interest rate risk.
The higher the duration the greater the level of interest rate risk in an investment.

13. Duration Calculation Two bonds; Find the duration. YTM = 8%
Maturity = 3 years
Coupon rate
A = 6%
B = 10%
Face Value = $1,000
Find the duration.

14. Price Elasticity
Using calculus on the price equation

15. Modified Duration
From the last line of the previous equation, the right hand side is
-1/(1+y) x Macaulay’s duration
The negative of this is called Modified Duration
Modified duration = Macaulay’s duration/(1+y)
Often used to approximate percentage price changes
Duration in years = D in six month periods/2
use 6 month rate for (1+y) in modified duration

16. Alternate Method
From the annuity formula for the price of a bond we can get a formula for modified duration instead of calculating weighted average (per $100 of face value)

17. Properties of Duration
Increases with time to maturity
Increases as coupon rate decreases to a maximum of time to maturity for a zero coupon bond
Decreases as YTM increases due to face value having less weight in portfolio
Modified Duration is similar, but lower max

18. Approximate Price Change
The change in price for a given change in yield can be calculated using modified duration (a.k.a. volatility)
The approximate percentage change in price = - modified duration x change in yield
Given MD = 7.66, calculate change in price for a 50 basis point increase in yield
DP% = x 0.5% = -3.83%

19. Dollar Price Change
The approximate dollar price change is simply the approximate percent price change times the price
Given the bond on the previous slide, if the initial price was $102.5 the decrease in value is 3.83%
In dollar terms, $3.926

20. How Close is This?
For small yield changes, the approximation is reasonable, p. 70 example, for a 1 basis point increase on a 25 year 6% coupon bond with an initial yield of 9%, the forecast change is -$ actual is -$0.0746
For large changes it is not as good
Reason: duration is a linear approximation of the price/yield relationship

21. Portfolio Duration
Since duration is simply a weighted average of the time to the coupon payments and face value, portfolio duration is simply the weighted average of the durations of the individual bonds
Portfolio managers look at the contribution to portfolio duration to assess their interest rate risk of a single bond issue

22. Convexity
Tangent line for estimated price

23. Convexity
Due to the shape of the yield curve, the predicted price will always be lower than the actual price
How close the approximation is depends on how convex the price/yield relationship is for a given bond

24. Measuring Convexity
Convexity is based on the rate of change of slope in the price/yield relationship
That means that we need the second derivative of the price of a bond
This is the dollar convexity

25. Convexity Measure
The convexity measure is the second derivative of the price divided by the price

26. Convexity Example

27. Price Change Example Given 25 year, 6% bond yielding 9%
Required yield increases to 11%
Mod. Duration = 10.62
change due to duration = x 2%=-21.24%
Convexity in years = 178
change due to convexity = 1/2 x 178 x 0.022=3.66%
Forecast change = = %
Actual change = %

28. Alternate Calculation
We could also take the second derivative of the annuity based price formula
Divide by price for convexity measure
Divide by m2 to convert to years

29. Note on Convexity
Different writers compute the convexity measure differently
One method moves the ½ into the measure

30. Value of Convexity
Price
Bond A
Bond B
Yield

31. Value of Convexity
Two bonds offering the same duration and yield, but with different convexity
Bond A will outperform bond B if the required yield changes
Bond A should have a higher price
Increase in value of A over B should be related to the volatility of interest rates

32. Positive Convexity As required yields increase convexity will decrease
As yields increase the slope of the tangent line will become flatter
Implication
as yield increases, prices fall and duration falls
as yield decreases, prices rise and duration rises

33. Properties of Convexity
For a given yield and maturity, the lower the coupon rate, the higher the convexity
For a given yield and modified duration, the higher the coupon rate, the higher the convexity
Although coupon rate has an impact on the convexity it has a bigger impact on duration

34. Effective Duration
If a bond has embedded options, that will change the bond’s price sensitivity to changes in required yields
The value of a call option on the bond decreases as yields increase, and increases as yields decrease
Effective duration can be calculated to account for the fact that expected cash flows may change in yields change

35. Duration vs. Time
With plain vanilla bonds, duration can be seen as a measure of time
With more complex instruments, this link is broken
Modified duration is a measure of the bond’s price volatility with respect to changes in the required yield

36. Duration of Floaters
A floating rate bond usually trades near par since the coupon rate adjusts to changes in interest rates
Therefore a floater’s duration is near zero
An inverse floater has a high duration (possibly greater than its maturity) since, when interest rates go up its coupon payments go down, exaggerating the impact of a change in yields
A double floater could have a negative duration

37. Approximating Duration
Instead of using duration to approximate price changes, we can use price changes to approximate duration
Potentially useful for complex instruments as a measure of price volatility
P-= price if yield down
P+= price if yield up
P0= original price

38. Approximating Convexity
We can also approximate convexity using a similar method
P-= price if yield down
P+= price if yield up
P0= original price

39. Changing Yield Curve
What happens if the shape of the yield curve changes?
It is possible that prices on 30 year bonds could change while short term rates are stable
Duration calculations can change to; key rate durations, duration vectors, partial durations, etc.
Key rate durations are illustrated in the text; they are calculated using the approximation formula