1. Session 14 Duration – Concept and Calculation, Convexity
CERTIFIED FINANCIAL PLANNER CERTIFICATION PROFESSIONAL EDUCATION PROGRAM Investment Planning
Session 14 Duration – Concept and Calculation, Convexity

2. Session Details Module 7 Chapter(s) 2, 3 LOs 7-3 7-4 7-5
Understand the concept of duration, and calculate change in price using duration.
7-4
Analyze the relationships among bond ratings, yields, maturities, and durations to determine comparative price volatility.
7-5
Assess how changes in variables affect bond risk and price volatility.
Let’s talk about the steps in the financial planning process…

3. Duration
Duration facilitates comparisons of price volatility of bonds with
different coupons, and
different terms to maturity

4. Duration Concept (1)
Any change up or down in interest rates will cause price risk and reinvestment risk to pull in opposite directions.
Increase in rates: bond price falls (not good), but reinvestment risk goes down since interest can be reinvested at a higher rate of return (good).
Decrease in rates: bond price rises (good), but reinvestment risk goes up since interest is being reinvested at a lower rate of return (not good).

5. Duration Concept (2)
The point in time when these two forces— interest rate risk (price risk) and reinvestment risk—offset each other is a bond’s duration
For testing purposes: understand how duration is a measure of risk, and what causes duration to increase or decrease

6. Duration Example Vanguard.com 12/7/2012 Vanguard Short-Term Bond Index
Fund
SEC yield
Duration
Vanguard Short-Term Bond Index
0.51%
2.7
Vanguard Intermediate-Term Bond Index
1.69%
6.4
Vanguard Long-Term Bond Index
3.38%
14.8

7. Duration
Fulcrum
Point

8. Current Market Interest Rates
Duration Matrix
Coupon
Current Market Interest Rates
Maturity
Increases Duration
Decrease
Increase
Decreases Duration

9. Duration Scenario A 5% coupon with 10-year maturity 15-year maturity B
Bond Alpha
Bond Omega A
5% coupon with
10-year maturity
15-year maturity B
6% coupon with
8-year maturity
7% coupon with C
0% coupon with

10. Duration Formula

11. Duration Annual Compounding
Annual compounding for a bond with a 20-year maturity and coupon rate of 8%. Current market rates are 6%.
y = .06
c = .08
t = 20
Note: to the 20th power keystrokes are:
1.06, SHIFT, yx (x key), 20, =

12. Duration Annual Compounding
Leave room to calculate beneath the equation.

13. Duration Semiannual Compounding
Semiannual compounding for a bond with a 20- year maturity and coupon rate of 8%. Current market rates are 6%.
y = .06/2 = .03
c = .08/2 = .04
t = 20 x 2 = 40
Note: to the 40th power keystrokes are: , SHIFT, yx (x key), 40, =

14. Duration Semiannual Compounding
Leave room to calculate beneath the equation.

15. Duration Calculation
Alternative

16. Duration Calculation Alternative
Semiannual compounding for a bond with a 20-year maturity and coupon rate of 8%. Current market rates are 6%.
Current Bond Price
+1% in rates
-1% in rates
40 pmt
1,000 FV
40 n
6 i
7 i
5 i
P V=
PV =
PV =

17. Change in Bond Price

18. Change in Bond Price
Example: $1,000 bond with a 10% coupon rate. Current yields (YTM) are at 8%, and the bond’s duration is 3.5.
What approximate price change would this bond have given a 1% decline (annual compounding), and 1% increase (semiannual compounding), in interest rates?
Note that you are entering the current yield (YTM) in this formula; you do not need the coupon rate for this calculation, that has already been taken into account when you calculated duration.
Once the percentage change has been determined, then multiply the percentage amount times the current bond price to obtain the approximate movement in price.

19. Change in Bond Price
Annual & Semiannual Compounding

20. Convexity Duration is not linear Bond Price Positive Convexity
Zero Convexity
Negative Convexity
Interest Rate

21. Positive Convexity Example
Straight Bond
Duration
% change in interest rates
Approximate price change w/o convexity
Approx. price change with + convexity 3
–1%
+3%
–2%
+6%
+6.5%
–3%
+9%
+10%

22. Immunization Match duration (not maturity) to time horizon
Bond ladders
Barbell portfolio
Bullet portfolio

23. Question 1
You believe interest rates are going to be falling for the foreseeable future. All of the following bonds have a 12- year maturity.
Which of the following combination of choices is best?
Option a: increase durations, 6% coupon
Option b: decrease durations, 6% coupon
Option c: increase durations, 7% coupon
Option d: decrease durations, 7% coupon
Option a
Option b
Option c
Option d
a. If you believe interest rates are going to fall, then you would want to increase (lengthen) durations. Also, the lower the coupon, the higher (longer) the duration.

24. Question 2
Which of the following will fluctuate the least with a given change in interest rates?
zero coupon, 9-year maturity
zero coupon, 12-year maturity
7% coupon, 9-year maturity
7% coupon, 12-year maturity
c. The higher the coupon, the lower (shorter) the duration; and the shorter the maturity, the lower (shorter) the duration.

25. Question 3
Dexter purchased a U.S. Treasury bond that matures in 25 years and has a 6% coupon. Assume that the coupon is paid annually. The current market interest rate for bonds with 25 years until maturity is 7%.
Calculate the duration for this bond. Which one of the following is the correct duration?
12.84
13.16
14.96
25.00
a is the correct answer. Note: To take the 1.07 to the 25th power, the keystrokes are 1.07, SHIFT, yx (the ‘x’ key), 25, =. This will then give you , then subtract the ‘1’ and you will have x .06 = Add this to .07 and you arrive at

26. Question 4
Bond DEF has a duration of 6, and a coupon rate of 7%. Current market interest rates are at 5%, and interest rates are expected to rise by .75%.
What would be the approximate price change of the bond using annual compounding?
– 4.39%
– 4.26%
+ 4.26%
+ 4.39%
b is the correct answer. The approximate change is down 4.26%. You would then multiply –.0426 by the bond price to come up with the approximate price change of the bond (not asked for here). The choice of –4.39% is the answer if you use semiannual compounding.

27. Question 5 Which of the following statements about convexity is false?
Positive convexity is a favorable characteristic for bondholders.
With convexity, there is a linear relationship between duration and interest rates, and subsequent changes in bond prices.
Negative convexity would mean that as interest rates rise, the bond price would fall more than duration alone would indicate.
Convexity helps to measure the impact of interest rate changes greater than 1%.
b. Option b. is incorrect; convexity is a curvilinear (not linear) relationship that explains changes in bond prices when the interest rate change is greater than 1%.

28. Question 6
All of the following are strategies to immunize a portfolio except
ladder.
barbell.
intermarket.
bullet.
c. Intermarket is a type of bond swap, not an immunization strategy.

29. Question 7
Jessica is saving for her son’s college education, which will be starting 12 years from now.
Which of the following choices would immunize her portfolio against interest rate and reinvestment rate risk?
a laddered portfolio with bond maturities of 12, 13, 14, and 15 years
a barbell portfolio with maturities of 1 and 2 years and 12 and 14 years
a laddered portfolio of zero coupon bonds of 12, 13, 14, and 15 years
a barbell portfolio with durations of 1 and 2 years and 12 and 14 years
c. To immunize a portfolio you match the duration, not maturity, to the time frame of the goal.

30. CERTIFIED FINANCIAL PLANNER CERTIFICATION PROFESSIONAL EDUCATION PROGRAM Investment Planning
Session 14 End of Slides