Question

Consider a 3-year 8% semiannual coupon bond. The YTM of this bond is 6%. Compute the following

a) Macaulay Duration (use  Mac Duration =

b) Modified Duration

c) Effective duration (assume a ±50 BP change of Yield)

d) Convexity Factor (use

e) Effective Convexity Factor (assume a ±50 BP change of Yield)

PLEASE ANSWER ALL PARTS

Answer 1

Without loss of generality, assume that the face value of the bond is 100

Semiannual Coupon =100*8%/2 = 4

No of semiannual periods = 3*2 = 6

Semiannual YTM = 6%/2 =3%

Price of bond =4/0.03*(1-1/1.03^6)+100/1.03^6 = 105.42

a)

Macaulay Duration = (4/1.03*0.5+4/1.03^2*1+4/1.03^3*1.5+4/1.03^4*2+4/1.03^5*2.5+104/1.03^6*3)/105.42

=2.7342 years

b) Modified Duration= Macaulay Duration/ (1+Semiannual YTM) = 2.7342/1.03 = 2.6546 years

c) If YTM increases by 50 BP

Semiannual YTM = 6.5%/2 =0.0325

Price of the bond (P-) = 4/0.0325*(1-1/1.0325^6)+100/1.0325^6 = 104.03

If YTM decreases by 50 BP

Semiannual YTM = 5.5%/2 =0.0275

Price of the bond (P+) = 4/0.0275*(1-1/1.0275^6)+100/1.0275^6 = 106.83

Effective Duration = ( (P+) - (P-) ) / (2*P0*Change in Yield)

=(106.83-104.03)/(2*105.42*0.005)

= 2.6547 years

d) Convexity factor

= (4/1.03*(0.5^2+0.5)+4/1.03^2*(1^2+1)+4/1.03^3*(1.5^2+1.5)+4/1.03^4*(2^2+2)+4/1.03^5*(2.5^2+2.5) +104/1.03^6*(3^2+3))/(105.42*1.03^2)

=10.02213

e) Effective Convexity = ((P+)+ (P-) -2*P0) / (2*P0*change in yield^2)

= (106.83+104.03-2*105.42)/(2*105.42*0.005^2)

= 4.367