Question

1. Portfolio A consists of $5,000 investment in a 1-year zero-coupon bond and $20,000 investment in a 20-year zero-coupon bond. Portfolio B consists of a 10-year zero-coupon bond with a face value of $17,000. The current yield on all bonds is 4% per annum (continuously compounded).
   1. Compute the actual percentage changes in the values of the two portfolios for a 20-basis point increase in yields.
   2. Compute the actual percentage changes in the values of the two portfolios for a 200-basis point increase in yields.
   3. Compute the duration for each portfolio. Use these durations to forecast the change in the value of each portfolio for a 20-basis point and a 200-basis point increase in yields.
   4. Compute the convexities for each portfolio. Use duration and convexity to forecast the change in the value of each portfolio for a 20-basis point and a 200-basis point increase in yields.
   5. Find the percentage forecast errors from c and d. Discuss your results.

Answer 1

PORTFOLIO A		PORTFOLIO B
I YEAR BOND	$5000	10 YEAE BOND	$17000
20 YEAR BOND	$20000	-	-
TOTAL	$25000		$17000

DURATION OF BOND=1+YTM/YTM -[(1+YTM +T(C-YTM))/C(1+YTM)^T-1+YTM)]

WERE,

YTM= YEILD TO MATURITY

C= COUPON RATE

T= NO OF YEARS

PORTFOLIO A

COMPUTAION OF DURATION OF 1 YEAR BOND

DURATION= 1.04/.04 -[1.04+1(0-.04)/0(1.04)¹-1+.04

= 26-1/.04   

= 1 YEAR

COMPUTAION OF DURATION OF 20 YEAR BOND

DURATION= 1.04/.04 -[(1.04+20(0-.04))/(0(1.04)²⁰-1+.04)]

= 26-.24/.04   

= 20 YEARE

PORTFOLIO B

COMPUTAION OF DURATION OF 10 YEAR BOND

DURATION= 1.04/.04 -[(1.04+10(0-.04))/(0(1.04)¹⁰-1+.04)]

= 26-.64/.04   

= 10 YEAR

DURATION OF PORTFOLIO =WEIGHT*DURATION

COMPUTATION DURATION OF PORTFOLIO A = $5000/$25000*1+$20000/25000*20

= 16.2 YEARS

COMPUTATION DURATION OF PORTFOLIO B  = 10 YEARS

VOLATILITY= DURATION OF PORTFOLIO/1+YTM

VOLATILILTY OF PORTFOLIO A = 16.2/1.04= 15.58%

VOLATILILTY OF PORTFOLIO B = 10/1.04 = 9.62%

ANSWER

A) % OF CHANGE IF CHANGES IS 20 BASIS POINT UPWARD

FOR PORTFOLO A= 15.58%*.2=3.12%

FOR PORTFOLIO B= 9.62%*.2=1.92%

B) % OF CHANGE IF CHANGES IS 200 BASIS POINT UPWARD

FOR PORTFOLO A= 15.58%*2=31.16%

FOR PORTFOLIO B= 9.62*2=19.24%

C) DURATION OF PORTFOLIO

FOR PORTFOLIO A= 16.2 YEARS ( AS COMPUTED ABOVE)

FOR PRORTFOLIO B= 10 YEARS (AS COMPUTED ABOVE)