Fixed Income Practice – Question 8 Credit Analysis Models

CFA Level II Fixed Income
#1

Hello,

Could you help me on this please ?

It concerns a question (Question 8) at the end of the reading 31 of Fixed Income Level II _ 2022 editions.

I would like to know how we find the first up/down move of the interest rates in the arbitrage Tree.

We have 20% Vol but I don’t know how we find rate (up) of 2.1180% and rate (down) of 1.4197%.

Someone could help me ?

Many Thanks

Best

J

The following information relates to Questions 1–15
Daniela Ibarra is a senior analyst in the fixed-income department of a large wealth
management firm. Marten Koning is a junior analyst in the same department, and
David Lok is a member of the credit research team.
The firm invests in a variety of bonds. Ibarra is presently analyzing a set of bonds
with some similar characteristics, such as four years until maturity and a par value of
€1,000. Exhibit 1 includes details of these bonds.

Bond 1 : A zero-coupon, four-year corporate bond with a par value of €1,000. The
wealth management firm’s research team has estimated that the risk-neutral
probability of default for each date for the bond is 1.50%, and the recovery
rate is 30%

Bond 2 : bond similar to B1, except that it has a fixed annual coupon rate of 6%
paid annually

Ibarra asks Koning to assist her with analyzing the bonds. She wants him to perform the analysis with the assumptions that there is no interest rate volatility and that
the government bond yield curve is flat at 3%.
Ibarra performs the analysis assuming an upward-sloping yield curve and volatile
interest rates. Exhibit 2 provides the data on annual payment benchmark government
bonds.
She uses these data to construct a binomial interest rate tree based on an assumption of future interest rate volatility of 20%.

Maturity Coupon Rate Price Discount Factor Spot Rate Forward Rate
1 -0,25% 100 1,002506 -0,25%
2 0,75% 100 0,985093 0,7538% 1,7677%
3 1,50% 100 0,955848 1,5166% 3,0596%
4 2,25% 100 0,913225 2,2953% 4,6674%

Answer the first five questions (1–5) based on the assumptions made by Marten
Koning, the junior analyst. Answer Questions 8–12 based on the assumptions made
by Daniela Ibarra, the senior analyst.
Note: All calculations in this problem set are carried out on spreadsheets to preserve precision. The rounded results are reported in the solution

question 8: As previously mentioned, Ibarra is considering a future interest rate volatility of 20% and an upward-sloping yield curve, as shown in Exhibit 2. Based on her analysis, the fair value of Bond B2 is closest to:

A. 1,101.24

B. 1,141.76

C. 1,144.63

#2

For straight bonds, you needn’t build a binomial interest rate tree. Use the spot rates and Bob’s your uncle.

#3

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#4

Many Thanks for the response, but in the correction, a binomial tree is built in order to find the Price.

#5

how do you even build that binomial? Surely they need to give us one of the one year forward rate in time =1 first? then we can we the relation of i_u = (i_L)e^(2vol)

#6

The tree is found by iteration.

1yr rate comes from spot rate

We have have to pick the 2 forward rates, where when we work out the value of an option free bond it will equal the market price

We can pick a starting point form the calculated 1yr forward rate starting in 1yrs time.
Then we work out the lower rate based on the volatility assumption
The upper rate is then calculated using the volaitility assumption

Price the bond.
We change the lower rate, which gives us a new upper rate.
We continue this until we have the price of a 2 year bond

And then move on to year 3

You can’t do this with at least excel but preferably some computer progarmming.

You can’t be expected to do this from scratch in an exam.

1 Like
#7

This question in the CFA question bank didnt even gave us the binomial tree…

Also, for T=1, from what you say the forward rate is just found by iteration. But in the exam let say if they gave us the forward rate at the up node at T=1, we can just use volatility assumption to find the volatility in the down node

In year 2, the middle rate is just the implied one year forward rate 2 years from now right? (i.e., in this case the middle rate would be 2y1y forward rate in T=2)

1 Like
#8

You can use the forward rate from the foward rate curve and adjust downwards and upwards to get as your “first guess” as the lower and upper rates. But these are unlikley to give you the current bond price.

You then have to keep changing the lower rate (also adjust upper via formula) until your calcualted valuation matches the market price of a 2 year bond.

This process can be done using solver in excel or is quite easy to programme in python but with a lot of painstaking work it is not really a calcualtor job.

You could be given the correct lower or upper rate one year and have to work out the other but I think determining a bi-nomial interest rate tree from scratch is too complex.

Here is a copy of post I made previously on a different forum

Time 0 = year 1 spot rate

Time 1

  • We have a 0.75% coupon 2 year bond
  • Select a lower rate for time 1. (it will be lower than 1 year fwd)
    • Good first guess = 1.7677/exp(0.2) = 1.4473
    • Calculate upper rate = lower rate x exp(2 x 0.2)
  • Price bond using two rates.
  • Is price too high/too low?
    • Change lower rate, recalc upper rate continue until you get price via model = price in market.

You can check this by pricing 2 year 0.75% with rates from tree for 1st 2 years.

100

= 50% x [7.5/0.9975 = 1007.5/(1.02118 x 0.9975) ] - upper path

  • 50% x [7.5/0.9975 = 1007.5/(1.01497 x 0.9975) ] - lower path

Process is explained LM2 section 4. You can’t be expected to do this in exam!

2 Likes
#9

Thankss!