Solved by a verified expert :16. The risk-free rate of interest is constant and is 10%. The credit
spread for an issuer is also constant and is 3%. If the recovery rate is 40%,
then given continuous compounding,
(a)
What is the probability of default over a two-year period?
(b) What is the price of a two-year $1 zero-coupon bond issued by this
rm? Assume all cash ows occur at maturity, whether or not the bond defaults in
the interim period.
17.
The one-year risk-less interest rate and spread are 5% and 1%,
respectively. At the end of the year, the next year’s risk-less rates are
either 7% or 4% with equal risk-neutral probability. If the risk-less rate is
7%, then the spread over the next year will be 0.5%, and if the risk-less rate
is 4% then the spread will be 2%.
(a)
Depict the rates and spreads on a binomial tree.
(b) If the recovery rate is 40% (RMV), what is the price of a two-year
bond with an annual coupon of 6%? Assume that compounding and discounting are
continuous.
(c)
What is the probability of default over the rst period?
(d) What is the probability of default over the second period from
each of the nodes on the tree at the end of the rst period?
(e) At what annual spread (in basis points) will a two-year CDS trade?
Assume that payments on default are made at the end of each year and premiums
are paid at the start of each year.
18. This question deals with a reduced-form model of risky debt.
Suppose we can depict the risk-free interest rates on a tree (each period is
one year), for which the current interest rate is 10%. Rates can change after
one year and will become 12% or 9% with equal probability. The recovery rate is
constant at 40% (recovery of Treasury at the end of the period). Use continuous
compounding.
(a) Find the price of a risk-free Treasury bill that pays o 100 at the
end of two years (it has no coupon).
(b) There is a defaultable bond that we want to value using a
reduced-form model. You are given the following default probability function at
each node of the tree that depends on the risk-free interest rate:
= 1 exp( a
r)
Here r is the risk-free interest rate, and a
is the hazard rate parameter. Suppose the credit spread on the bond is 20 basis
points. Find the price of the defaultable zero-coupon bond, which also pays o
100 at the end of two years, and the value of the hazard function parameter
“a.” (You will need a spreadsheet and solver to work this out.)
Sundaram
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19. This question refers to the model of Litterman and Iben presented
in the chapter. The model takes as input the prices of risk-less and risk-free
bonds and generates forward probabilities of default. At the end of Section
33.4 is Table 33.4 presenting the forward default probabilities. This question
relates to reverse engineering the spreads in the Litterman-Iben model.
Suppose all forward probabilities of default
increased by 1% per annum. What should the new credit spreads in the model be
to be consistent with the revised forward default probabilities?
