Question

Spacely’s Sprockets wants to raise funding to develop software for a new Space Transport System. They have chosen to pursue debt financing and will sell bonds. Assume the following regarding their funding:

• The company needs to raise $500,000 for the project
• They will pay a 5% coupon annually
• The bond’s yield-to-maturity is 4%
• The bond will last 5 years
• The risk-free rate of return is 3% annually

1. What is the price of the bond? (4 points)
2. Assume the yield-to-maturity changes from 4% to 7 %. What is the bond’s price change (in percentage) in reaction to this change in yield? (4 points)
3. What is the Duration of this bond (assume interest rates are once again at 4%)? (5 points)
4. What is the Modified Duration of this bond? (4 points)
5. What is the Duration approximated change in the value of the bond if the yield increases from 4% to 7%? (4 points)
6. What is the Convexity of this bond? (5 points)
7. What does Convexity Measure? (4 points)

Now, incorporate convexity to calculate the approximated change in the price of the bond if the yield increases from 4% to 7%

Answer 1

a)Calculation of Price of Bond:

Coupon rate(CR)=5%, Kd=7%(4%+3%), n=5 years

B₀=(500000*5%)*PVAF(7%,5 years)+(500000*PVF(7%,5 years))

=25000*4.1+500000*0.713= 459000

b)Calculation of Price of Bond:

Coupon rate(CR)=4%, Kd =10%(7%+3%), n=5 years

B₀=(500000*5%)*PVAF(10%,5 years)+(500000*PVF(10%,5 years))

=25000*3.79+500000*0.621= 405250

c) Duration of Bond:

Coupon rate(CR)=5%, Kd =7%(4%+3%), n=5 years

DOB=1/B₀*(1*Int/(1+kd)¹+2*Int/(1+kd)²+3*Int/(1+kd)³+4*Int/(1+kd)⁴+5*Int/(1+kd)⁵)

=1/459000*(1*25000/(1+0.07)¹+2*25000/(1+.07)²+3*25000/(1+.07)³+4*25000/(1+.07)⁴+5*(25000+500000)/(1+.07)⁵

=1/459000*(20761150)

=4.52 years

d) Modified Duration= DOB/(1+kd)

=4.52/(1.07)=4.22 years

e) Duration of Bond:

Kd=4%, Coupon rate(CR) =10%(7%+3%), n=5 years

DOB=1/B₀*(1*Int/(1+kd)¹+2*Int/(1+kd)²+3*Int/(1+kd)³+4*Int/(1+kd)⁴+5*Int/(1+kd)⁵)

=1/405250*(1*25000/(1+0.1)¹+2*25000/(1+0.1)²+3*25000/(1+0.1)³+4*25000/(1+0.1)⁴+5*(25000+500000)/(1+0.1)⁵

=1/405250*(1818775)

=4.488 years