Question

Treasury is: Spot 1-year Treasury is 4% Spot 2-year treasury is 4.5%

B Corporate Debt interest annually Spot 1 year 8.5%, Spot 2 years 9.5%

Forward 1 Year maturity

Let’s call the Treasury x

The Corporate debt is called y

1. What is the value of x the forward rate on one-year maturity Treasuries delivered in one year?
2. What is the value of y the forward rate on one-year maturity B corporate debt delivered in one year?
3. Exercising the term structure of default probabilities, the implied default probability for B corporate debt during the current year is?
4. Exercising the term structure of default probabilities, the implied default probability for B corporate debt during the second year is what?

Answer 1

a. Forward rate on 1 year maturity Treasuries delivered in one year (x)

Forward rate (₂f₁ or x ) = [(1 + Spot rate for year 2)^² / (1 + Spot rate for year 1)] - 1

= [( 1 + 0.045)^2 / (1 + 0.04) ] - 1 = 5.00%

i.e Expected 1 year spot rate on treasury after 1 year is 5.00%.

b. Forward rate on 1 year maturity B Corporate Debt delivered in one year (y)

Forward rate (₂f₁ or y) = [(1 + Spot rate for year 2)^² / (1 + Spot rate for year 1)] - 1

= [( 1 + 0.095)^2 / (1 + 0.085) ] - 1 = 10.51%

i.e Expected 1 year spot rate on B Corporate Debt after 1 year is 10.51%.

c. Implied default probability for B corporate debt during the current year.

r_f = 1 -year spot rate on Treasury ; c = 1-year spot rate on Corporate debt

p₁ = probability of default; r = recovery rate

It is intuitive that (1 + r_f) = (1 + c)*(1-p₁) + (1+c)*p₁*r

Assuming recovery rate (r) as Nil i.e. no recovery in case of default

(1 + r_f) = (1 + c)*(1 - p₁) = (1.04) = (1.085)*(1 - p₁)

1 - p₁ = 0.9585 or p₁ = 0.0415 or 4.15%

d. Implied default probability for B corporate debt during the second year

x = Forward rate on 1 year maturity Treasuries delivered in one year

y = Forward rate on 1 year maturity B Corporate Debt delivered in one year (y)

p₂ = probability of default;

Continuing our assumption of Nil recovery rate (r)

(1 + x) = (1 + y)*(1 - p₂) = (1.05) = (1.1051)*(1 - p₂)

1 - p₂ = 0.9502 or p₂ = 0.0498 or 4.98%

* Please note that p is the implied probability of default in year 2 and not the cumulative probability of default