Re: Finance homework help - probability of share price
- From: "Inquisitive" <s.witage@xxxxxxxxxx>
- Date: 30 Sep 2005 04:20:19 -0700
Hello,
Many thanks for your reply and insight. I have the proper answers now
and they make me think that the question itself was not as clear as it
could have been.
For part a, the annualised yearly return was indeed 18% - the question
was looking for the annualised *continuously compounded* rate which
differs from the annual equivalent rate. An annual continuously
compounded rate of 18% equates to an annually compounded rate of 19.56%
as you stated, i.e. 1*e^0.18 = 1.1956. For the record, the annual
continuously compounded standard deviation was indeed about 4.15%.
For part b, the probability that the share price is > £35 is
apparently the equivalent of the probability of being greater than
ln(35/30). So once you've found ln(35/30), you need to convert this
number to a *standard* normal distribution probability, i.e. a z-score,
and then look up the z-score in a standard normal distribution table or
alternatively use Excel's NormSDist function.
To convert to a z-score, you subtract the annualised continuously
compounded mean and then divide by the annualised continuously
compounded standard deviation, so in this case as ln(35/30) = 0.1542,
you convert this to a z-score as follows: (0.1542-0.18)/0.0415 and then
apply NormSDist to this figure.
However, this isn't the end of the story! The figure obtained, 0.26 is
actually the probability of being less than £35, so the actual ability
we want is 1-0.26 = 0.74, i.e. 74%.
How convoluted is that!
Ronald Raygun wrote:
> Inquisitive wrote:
>
> > ABC's share price is currently £30, and has a monthly,
> > continuously-compounded return which has a mean of 1.5% and a standard
> > deviation of 1.2%. ABC pays no dividends.
> > Assume that each month's continuously-compounded return (=
> > ln(P1)-ln(P0)) is normally distributed and that returns from
> > different months are independent.
> >
> > (a) What are the expected value and the standard deviation of the
> > yearly return (annualised and continuously-compounded) of ABC's
> > shares?
> >
> > (b) What is the probability that ABC's stock price will be at or
> > above £35 in one year?
> >
> > I have done part (a) and believe that the answers are that it has an
> > annual expected return of (1.5 * 12 =) 18% and a standard deviation of
> > (1.2 * 12^0.5 =) 4.15%. Is this correct?
>
> No. If the standard deviation were zero, and the return were therefore
> exactly 1.5% each month, the fund value would grow by 1.5% each month
> and therefore in a year it would grow by 19.56% (1.015^12 - 1).
>
> I'm not really into all this standard deviation stuff, but I reckon
> if the SD is 1.2% it means that 50% (or some other proportion, depending
> on how the data are in fact distributed) of the time the return will
> be between 1.5%-1.2% and 1.5%+1.2%. So the worst case annual return
> for this middle range of returns should be 1.003^12-1 or 3.66% and
> the best case 1.027^12-1 or 37.67%. That kind of suggests that the
> annual SD would be half the difference, i.e. 17.005%, and the mean
> half way between the extremes, i.e. 20.665%.
>
> This is probably all wrong, but I'm no statistician. Tim?
>
> > I'm not sure how to complete part (b). I believe that if the share
> > price has an annual expected return of 18%, then the shares should be
> > worth (30 * 1.18) = £35.40 at the end of the year. But how do you work
> > out what the probability is of the share price being at least £35?
>
> Well, £35 represents a growth from £30 of 16.67%. If we know that the
> expected annual mean is some 20.67%, and if we can assume the distribution
> to be symmetric, then we know the probability is 50% that the observed
> value will be greater than the mean. We want to know what the probability
> is of the value being between 16.67% and the mean, i.e. a 4%-wide band
> which therefore corresponds to 4/17 = 0.235 SDs. Your tables should
> be able to help you out there. The final probability required is then
> the sum of this and the aforementioned 50%.
.
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