Question

Consider a bond with 10,000 USD par value, 8% coupon rate paid semi annually and 10 years to maturity. Assuming a 10% required return, answer the following questions:

Find the PV of the bond

Find the PV of the bond given it’s a Zero-Coupon Bond.

What is the bond’s price elasticity if the required return changed to 12%?

Calculate the duration of the bond.

What is the modified duration at an 8% yield?

What is the percentage change in bond’s price for an increase in yield for 0.3 percentage point.

Answer 1

PV of the bond

we can use financial calculator for calculation of PV of the bond with below key strokes:

N = semi-annual maturity = 10*2 = 20; I/Y = semi-annual required return = 10%/2 = 5%; FV = par value = $10,000; PMT = semi-annual coupon = $10,000*8%/2 = $400 > CPT = compute > PV = value of the bond = $8,753.78

the PV of the bond is $8,753.78.

PV of the bond given it’s a Zero-Coupon Bond

PV of the bond = par value/(1+required return/2)maturity*2 = $10,000/(1+0.10/2)10*2 = $10,000/1.0520 = $10,000/2.653297705144420133945430765152‬ = $3,768.89

PV of the bond given it’s a Zero-Coupon Bond is $3,768.89.

the bond’s price elasticity if the required return changed to 12%

bond’s price elasticity = % change in price/% change in required return

for % change in price, we need to calculate PV of bond at required return of 12%.

N = semi-annual maturity = 10*2 = 20; I/Y = semi-annual required return = 12%/2 = 6%; FV = par value = $10,000; PMT = semi-annual coupon = $10,000*8%/2 = $400 > CPT = compute > PV = value of the bond = $7,706.02

% change in price = (PV of bond at required return of 12%/PV of bond at required return of 10%) - 1 = ($7,706.02/$8,753.78) - 1 = 0.8988 - 1 = -0.1012 or -10.12%

% change in required return = (12%/10%) - 1 = 1.2 - 1 = 0.2 or 20%

bond’s price elasticity = -10.12%/20% = -0.506‬

duration of the bond

Duration = [(1+YTM/2)/YTM] - [(1+YTM/2) + M(C-YTM)]/YTM + C[(1+YTM/2)2M - 1]

C = annual coupon rate i.e.8%; YTM = yield to maturity or required return i.e. 10%; M = bond maturity i.e. 10 years

Duration = [(1+0.10/2)/0.10] - [(1+0.10/2) + 10(0.08-0.10)]/0.10 + 0.08[(1+0.10/2)2*10 - 1]

Duration = [(1+0.05)/0.10] - [(1+0.05) + 10(-0.02)]/0.10 + 0.08[(1+0.05)20 - 1]

Duration = (1.05/0.10) - (1.05 - 0.2)/0.10 + 0.08[(1.05)20 - 1]

Duration = 10.5 - (0.85)/0.10 + 0.08(2.65 - 1)

Duration = 10.5 - (0.85)/0.10 + 0.08*1.65 = 10.5 - (0.85)/0.10 + 0.132 = 10.5 - 8.5 + 0.132 = 2.132‬ years