But be very careful, because this was a defective question! What they
should have asked was, "If you wanted to use the binomial (risk neutral)
method to value an option on this stock and wanted to use quarterly
intervals, what would you use for the downside change?" That's a
perfectly fair question and that's actually the question you answered.
However, the question clearly asked for "the probability that the
business value would fall in the next 3 months". this confuses the p in
the risk neutral method with the probability the business value will
rise. The p that is used in the risk neutral method is NOT the
probability that the business value will go up. It is an artificial
probability that is used under the assumption of risk neutrality and is
not equal to the real probability of the business value changing. This
is one of the most important points in the entire textbook - it was the
reason Scholes & Merton won the Nobel prize! Unfortunately the book
glosses over this a bit and therefore a lot of the test writers don't
understand this.
Second, the question is illogical on the surface. If the business can
only have one of 2 values at the end of a year, then it cannot change
every 3 months. If so, it would have 5 possible values at the end of
the year.
And a minor point, but your answer assumed that 16% was the risk free
rate. A safe assumption, but that's an awfully high risk free rate and
the question wasn't clear about this. If they really wanted to ask the
real probabilities, then they'd have to give you the risk adjusted rate
and the current value of the stock.
I hope this helps.
Richard Goldfarb, FCAS, MBA
Instructor for the MAF Part 5B Seminar
Note - There are a lot of defective questions on the last several exams,
especially those relating to options, capital structure and tax adjusted
cost of capital. Be very careful. I do not recommend that you use old
exam questions to study from - at least not entirely.
For those interested, I will be compiling practice questions and
detailed solutions which will be available by mid April. If you are
interested in purchasing a packet of this information, please notify me
by email. And for those who did not attend my seminar, you can also
purchase the roughly 150 questions I have written for the seminar.
> -----Original Message-----
> From: Lupica, Paul [SMTP:Paul.Lupica@atlantacasualty.com]
> Sent: Wednesday, March 31, 1999 1:29 PM
> To: studygroup5b@lists.casact.org
> Subject: 1997 Exam Question 16
>
> I sat for this exam and I have a copy of it, but I don't have the
> published
> answers. I know I didn't work it out right when I took the exam
> because I
> didn't use the fact that interest was compounded continuously. (Plus
> it was
> marked wrong on my analysis.)
>
> The way this problem is worked out in the CSM study manual is to just
> use
> the shortcut formula for the probability of a rise p:
>
> p = (annual interest rate - downward change) / (upward change -
> downward
> change)
>
> = (16 - (-22.1)) / (28.4 - (-22.1)) = .754
>
> So their answer is 1 - .754 = .246
>
> Not only does this solution not use any continuous compounding, it
> didn't
> answer the question: "What is the probability of a decrease over the
> next 3
> months?"
>
> After doing some preliminary studying, here is how I would now work
> out the
> problem:
>
> First, calculate the standard deviation d of annual returns:
>
> 1 + upward change = e^(d * sqrt(t)) ===> 1.284 = e^(d*1) for annual
> figures ===> d = .25
>
> Now calculate the 3-month upward and downward changes:
>
> 1 + upward change = e^(.25 * sqrt(.25)) ===> upward change = .1331
> 1 + downward change = 1/1.1331 ===> downward change = -.1175
>
> quarterly interest rate = .04
>
> So the probability of a decrease in the next 3 months is 1 - p, where
> p =
> (.04 - (-.1175)) / (.1331 - (-.1175)) = .628.
>
> Ans: .372
>
> Am I overlooking something?
>
> Thanks,
> Paul Lupica
> paul.lupica@atlantacasualty.com