May 2001 Exam Course 6 - #4

[w4c71101]

Does anyone have any idea how to do question 4 from the May 2001 Course 6 Exam? Thanks for the help I really appreciate it.

[Exam Slave]

The written answer? This has been discussed in another thread. I'll see if I can find it for you.

http://www.actuary.ca/phpBB/viewtopic.php?t=2350

[w4c71101]

thanks for the help on this I appreciate it

[Exam Slave]

No problem.
I went over it (and that thread) yesterday.
1. Determine cost of a two-year BIG bond at 5.5% coupon. We're given the Treasury YTM and the spread. Add together to get YTM of BIG bond.
2. Discount cash flows to get price. Price will equal one of the prices given for one of the r-sub-L.
3. Determine cost of a three-year BIG bond at 6% coupon. We're given the Treasury YTM and the spread. Add together to get YTM of BIG bond.
4. Discount cash flows to get price. Price will equal one of the prices given for one of the r-sub-L. I'll do this one for you: there's no need to discount cash flows; YTM(BIG3)=coupon(BIG3), so Price = 100.
5. Now you have a tree/lattice, assuming you know how to get the other branches.
6. Discount the cash flows of the 6.75% bond, average the choices, etc.
7. For part b, take the prices above 100 after the third year -- and remember the present is the second year -- and lock them at 100.
8. Determine price difference to get price of call option.

If anyone has an official answer for part b, please disclose it. My answers do not contain an answer for part b. Perhaps this "representative" answer was good enough for the SOA, but it doesn't help us much.

[LB]

Does anyone know how the call level is determined? I know they are calling at par, but why? Do we assume this? It was not given to us in question.

[Exam Slave]

A thousand pardons, and a few points off for me.
Call premium is 1%, so the bond will be called at 101. Find nodes after 3.5 years (second set of branches) that are above 101.

[Exam Slave]

I thought I was lost, but I did receive a 5. I guess I tried just hard enough for that score. I remember being totally lost on it.

[LB]

I thought I was lost, but I did receive a 5. I guess I tried just hard enough for that score. I remember being totally lost on it.

This is my first time taking this. How was the grading? By that I mean when you got a five how many did you think you got correct (or how many points). How did you feel after the exam?

Any input would be helpful.

[Exam Slave]

I was referring to this particular problem (the 5), but I also received a 5 on the exam. I thought I did pretty well, but I've been thinking this for years now.
I shall be working on non-mathematical lists from the boring sections, as they seem to be my weakness, which mainly consists of my stopping when I'm satisfied with my answer.

[LB]

Sorry, I am not familiar with the scoring. So you get a breakdown of each question? Is the 5 for that question the same as a 5 for entire exam? (ie if there were not multiple choice questions and you recieved a 6 on all written answer questions you would recieve a 6 for exam).

[Exam Slave]

If you fail the exam, you get a breakdown of each question, scaled 0 to 10.
"Score of 6 or higher means performance met or exceeded the adjusted standard. (See enclosure for more explanation of adjusted standard.)"
Silly me: I threw out the further explanation about 8 months ago.

There could be many explanations, and if someone has the actual text, please join in.
I assume the question's deciles are determined by partitioning the question's marking points the same way that the exam points are partitioned. (See http://www.soa.org/eande/view.pdf , page 21 of 30, for explanation of marking points.)

[Exam Slave]

And anyone have an answer to part b? Please?

Eveidently the SOA makes up for missing answers to 4b by giving us TWO answers for #2, which I believe I aced. This time.

[Retired Bookworm]

(Lesson #1: never go "back" from message preview) Trying again.

I agree with the calibration techniques offered in this discussion but I have two monkey wrenches to throw in the solution:

1) "the logarithm of the 1-year rate obeys a binomial distribution" Not the rates themselves. This means this model is not "multiplicative binomial."

2) The 13% volatility y(1) given does not equal the volatility between r(L) and r(LL). I calculate y(2) to be 22.8%

My solution:

First, I adjusted the branch interest rate formula to use the logarithm on the 1-year rates, specifically from:

r [*,/] (1 + y) to r*e^([+,-]y)

And got the following rate paths:
...............[0,1].........[1,2]........[2,3]
r(UU)........7.25%......8.26%......10.37%
r(UL)........7.25%......8.26%.......6.58%
r(LU)........7.25%......6.37%.......8.00%
r(LL).........7.25%......6.37%.......5.07%

Likewise, I got this price tree:
T=0........T=1.......T=2
..........................96.72
..............97.16....100.16
98.47.........................
..............100.56..98.84
..........................101.60

So for (a) I get $98.47. For (b) I take the PV of $.60 to get $.13.

Could someone look at my solution and see what I might be missing, as these numbers clearly differ from others posted. It certainly was not easy to get this far. :D

[Exam Slave]

Sorry Bookie, but that is very incorrect.
Page 777 of HOFIS states the assumptions word for word. Thus, we use this chapter's notation and directions for constructing the tree.
One period's forward rate does not directly affect the next one in this construction.

[Retired Bookworm]

Ok, I was using VOI, not FIS, but I see how the definition of Volatility works in FIS (didn't notice until now that they are different). I'll look up the assumptions you speak of (looking at JAM right now), but I still don't see how to get the first rate from these formulas. I assumed it was 4%, but VOI method doesn't work that way.

BTW, what in the world is "VERY incorrect"? You get 0 or 1 for each point they are looking for.

1) I used the other definition of "Volatility" for the other "Binomial tree" definition (Wrong due to interpretation). Very likely used wrong initial rate.
2) recognized and adjusted for logarithmic requirement. (r(H) is the same for both, r(HH) is different).
3) recognized under VOI, that volatility was not constant and calculated second volatility
4) built tree correctly with assumptions used. Pricing formula correct.
5) Correctly applied strike rate and priced call.
6) Showed all work

Looks like all I missed was #1. Assuming I knew what a warrant was and calibrated correctly, I'd say this looks like a 7 or 8 out of 10. I haven't taken a written answer exam for the SOA yet, but I know from other institutions that one wrong assumption in the beginning does not hurt as much as not finishing the problem.

[Exam Slave]

You're assuming that all actuarial institutions are alike. I'm not in that camp.
Looks to be a score of about 2 to 5. Wrong book and mathematically linking one-year forward with 2nd-year forward rate shows that you don't have a grasp of the material. Well, at least the particular material in question.

My post on 4/22 provides an outline of how to do this problem.
Again:
1. Price BIG's 2-year bond by using YTM(BIG) = YTM(T)+credit spread
2. Turns out to be 99.632. Hey, that's the price of a 5.5% coupon bond in the tree when r-L= 6.369%! Therefore, r-L = 6.369%.
3. Q: What is current rate for BIG? A: 4.2% (I think my outline doesn't mention this.) This can be checked by taking the expected value of BIG 2-year bond at year 1 and dividing by the current price. (103.81/99.632)
4. Same for the 3-year bond. this one's easier, since the price of 3-year bond = 100, since the coupon (6%) = YTM.
5. Tree's done. Price the bond in question. My answer is 102.196.

Part b. Anyone with an SOA answer to this? I find it hard to believe that a representative answer completely ignores part of the question.
Q's for part b:
1. Call strike is 101? (If so, then my previous outline is very incorrect.)
2. Any nodes above 101? I don't recall any. If none, then call value = 0?