Portfolio theory exam-style question

Hi

I have a question on the portfolio theory chapter - I'm working on an old set of notes and not sure if this question will be in the current notes so I'll reproduce it below:

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An investor has 2 assets available : A and B. These are independent. Return on asset A is normally distributed with expectation 8% and variance 36%%. Asset B has a discrete probability distribution, where the returns are as follows:
-15% (probability 0.04)
1% (probability 0.2)
10% (probability 0.6)
20% (probability 0.16).

The question asks me to specify the equation of the efficient frontier in sigma-E space for portfolios invested in assets A and B.

It then states that an investor's risk preferences can be adequately described by indifference curves, in E-V space, of the form
E = exp (0.01V + alpha) - 1 for some alpha > 0

The question then asks me to find the expected return of this investor's optimal portfolio.
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My problem is the following. When I see the notation "%%", I immediately convert it into a decimal. So I am assuming the returns on asset A are normally distributed, ~N( 0.08 , 0.0036 ).

When I work the question through, I do not get the same equation for the efficient frontier, and I do not get the same expected return on the optimal portfolio. For example, my equation for the efficient frontier is :
sigma = (143.375E^2 - 23.84E + 0.9932)^0.5

whereas the acted solution is :
sigma = (143.375E^2 - 2384E + 9932)^0.5

I can see where the mistake is coming in, but what I don't understand is why the formulae I'm using do not seem to apply when you convert percentages to decimals. I had previously assumed that as long as the units of E and V are consistent, there shouldn't be a problem - but I don't understand why this doesn't seem to be the case.

For the avoidance of doubt, the formulae I'm using for the efficient frontier calculation are

E = xaEa + xbEb
V = xa^2Va + xb^2Vb

Many thanks in advance for any help.

Kind regards,
Claire

 

Hi,

Although the equations you end up with look slightly different depending on whether you use decimals or percentages, either approach should work fine provided you're careful.

So, in the ActEd solution (working in percentages) you get:

variance, V = 143.375E^2 - 2384E + 9932

So, if you substitute in, for example, E = 2%, you get:

V = 5737.5 and so sigma = 75.746, ie 75.746%.

With your solution:

V = 143.375E^2 - 23.84E + 0.9932

Here, however, you need to substitute in E = 0.02 and get:

V = 0.57375 and so sigma = 0.75746, which corresponds to 75.746%.

 

Hi

Thanks very much for taking the time to reply to my question.

Having discussed it this morning with some other actuarial students at work, I think we've found the answer and it might be a bit more complicated than you've indicated.

I agree with everything you've said in your reply, but I think the problem actually comes in later when we introduce the equation for the investor's indifference curves.

When I was doing the question, I had taken the equation for the indifference curves as it was given in the question without adjustment. However I think we would need to adjust the indifference curve equation to make it apply to E and V when expressed as decimals.

So, where the indifference curve in the question is given as :

E = exp (0.01V + alpha) - 1 for some alpha > 0

In order to convert this equation to apply to E and V expressed as decimals, we would need to write:

100E = exp (0.01*(100^2)V + alpha) - 1 for some alpha > 0

Using this new indifference curve equation and equating it to my formula for the efficient frontier gives us the correct answer.

Thanks again for your help!

Claire

 

Yes, that's right.

So working in decimals in part (v), for the efficient frontier you get:

\(V = 143.375{E^2} - 23.84E + 0.9932,\;\; \Rightarrow \frac{{dV}}{{dE}} = 286.75E - 23.84\)

and the indifference curve equation needs to be:

\(100E = \exp \left[ {0.01 \times {{100}^2}V + \alpha } \right] - 1\)

ie the E is 100 times "smaller" in decimals and so needs to be multiplied by a factor of 100, whilst the variance is 100 squared times "smaller" and so needs the factor of 100 squared in front of it.

Rearranging this equation then gives:

\(V = {0.01^2} \times 100\;\left[ {\ln (100E + 1 - \alpha } \right],\;\;\; \Rightarrow \frac{{dV}}{{dE}} = \frac{{100}}{{100E + 1}} \times 100 \times {0.01^2} = \frac{1}{{100E + 1}}\)

And equating the two derivatives gives:

\(28675{E^2} - 2097.25E - 24.84 = 0\)

From which:

\(E = \frac{{2097.25 \pm \sqrt {{{( - 2097.25)}^2} + 4 \times 28675 \times 24.84} }}{{2 \times 28675}} = 0.08351,\; - 0.01037\)

ie E = 8.351% or -1.037%

I'll make sure that we add this alternative, but equally valid, solution into the Course Notes when they're next updated, which unfortunately will be next year as I've just signed them off for the 2015 exams.

Very well done working this all out.

 

Another question...

Hi

I'm having exactly the same problem again with an even simpler question this time! I'm wondering if you can help...

The question is :
"An investor holds an asset that produces a random rate of return, R, over the course of a year. Calculate the shortfall probability using a benchmark rate of return of 1%, assuming that R follows a lognormal distribution with μ = 5% and σ^2 = (5%)^2 "

I immediately write the following:

R ~ logN ( μ = 0.05 , σ^2 = 0.0025)
log R ~ N (0.05 , 0.0025)

then

P (R < 0.01) = P (log R < log 0.01) = ɸ (( log(0.01) - 0.05)/0.05 ) = ɸ(-93.103)

which is clearly not right.

The answer states the following:

If R ~ logN (5, 25) then
P(R<1) = P(Z < (ln1 -5)/5) = P(Z<-1)

where am I going wrong? It doesn't make sense to me that rewriting a percentage as a decimal makes the whole solution fall apart.

Many thanks in advance for your help.

Claire