©2015, College for Financial Planning, all rights reserved.

Session 14Duration – Concept and Calculation, Convexity

CERTIFIED FINANCIAL PLANNER CERTIFICATION PROFESSIONAL EDUCATION PROGRAMInvestment Planning

Session Details

Module 7

Chapter(s)

2, 3

LOs 7-3 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.

14-2

Duration

• Duration facilitates comparisons of price volatility of bonds witho different

coupons, and o different terms

to maturity

14-3

Duration Concept (1)

• Any change up or down in interest rates will cause price risk and reinvestment risk to pull in opposite directions.o 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).

o 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).

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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

14-5

Duration Example

Fund SEC yieldDuratio

n

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

Vanguard.com 12/7/2012

14-6

Duration

FulcrumPoint

14-7

Duration Matrix

Coupon

Current Market Interest Rates Maturity

Increases Duration Decrease Decrease Increase

Decreases Duration

Increase Increase Decrease

14-8

Duration Scenario

Scenario Bond Alpha Bond Omega

A 5% coupon with 10-year maturity

5% coupon with 15-year maturity

B 6% coupon with 8-year maturity

7% coupon with 8-year maturity

C 7% coupon with 15-year maturity

0% coupon with 15-year maturity

14-9

Duration Formula

y1]y)c[(1

y)t(cy)(1

y

y1Duration

t

14-10

Duration

Annual CompoundingAnnual 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: 1.06 to the 20th power keystrokes are: 1.06, SHIFT, yx (x key), 20, =

14-11

Duration

.061].06).08[(1

.06)20(.08.06)(1

.06

.061D

20

Annual Compounding

14-12

Duration

Semiannual CompoundingSemiannual 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: 1.03 to the 40th power keystrokes are: 1.03, SHIFT, yx (x key), 40, =

14-13

Duration

.031].03).04[(1

.03)40(.04.03)(1

.03

.031D

40

Semiannual Compounding

14-14

Duration Calculation

Price if yields decline Price if yields riseDuration

2(current price)(.01)

Alternative

14-15

Duration Calculation

AlternativeSemiannual compounding for a bond with a 20-year maturity and coupon rate of 8%. Current market rates are 6%.

1,377 1,107

2(1,231)(.01)

10.97

24.62270

Current Bond Price +1% in rates -1% in rates

40 pmt

1,000 FV

40 n

6 i 7 i 5 i

P V= 1231.15 PV = 1106.78 PV = 1376.54

14-16

Change in Bond Price

y1

ΔyDΔP

14-17

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?o 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.

o Once the percentage change has been determined, then multiply the percentage amount times the current bond price to obtain the approximate movement in price.

14-18

Change in Bond Price

3.24%.03241.08

.013.5ΔP

3.37%0337.1.04

.013.5ΔP

Annual & Semiannual Compounding

14-19

Convexity

Duration is not linear

Interest Rate

Bond

Price

Zero Convexity

Negative Convexity

Positive Convexity

14-20

Positive Convexity Example

Duration

% change in interest rates

Approximate price change w/o convexity

Approx. price change with + convexity

3 –1% +3% +3%

3 –2% +6% +6.5%

3 –3% +9% +10%

Straight Bond

14-21

Immunization

• Match duration (not maturity) to time horizon

• Bond ladders• Barbell portfolio• Bullet portfolio

14-22

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? o Option a: increase durations, 6% coupono Option b: decrease durations, 6% coupono Option c: increase durations, 7% coupono Option d: decrease durations, 7% coupon

a. Option ab. Option b c. Option cd. Option d

14-23

Question 2

Which of the following will fluctuate the least with a given change in interest rates?a. zero coupon, 9-year maturityb. zero coupon, 12-year maturityc. 7% coupon, 9-year maturityd. 7% coupon, 12-year maturity

14-24

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?a. 12.84b. 13.16c. 14.96d. 25.00

14-25

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?a. – 4.39%b. – 4.26%c. + 4.26%d. + 4.39%

14-26

Question 5

Which of the following statements about convexity is false?a. Positive convexity is a favorable

characteristic for bondholders.b. With convexity, there is a linear relationship

between duration and interest rates, and subsequent changes in bond prices.

c. Negative convexity would mean that as interest rates rise, the bond price would fall more than duration alone would indicate.

d. Convexity helps to measure the impact of interest rate changes greater than 1%.

14-27

Question 6

All of the following are strategies to immunize a portfolio excepta. ladder.b. barbell.c. intermarket.d. bullet.

14-28

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. a laddered portfolio with bond maturities of

12, 13, 14, and 15 yearsb. a barbell portfolio with maturities of 1 and 2

years and 12 and 14 yearsc. a laddered portfolio of zero coupon bonds of

12, 13, 14, and 15 yearsd. a barbell portfolio with durations of 1 and 2

years and 12 and 14 years

14-29

©2015, College for Financial Planning, all rights reserved.

Session 14End of Slides

CERTIFIED FINANCIAL PLANNER CERTIFICATION PROFESSIONAL EDUCATION PROGRAMInvestment Planning