Fast and easy formula for finding optimal portfolio...Will it be allowed?

[Edwin]

It appears as if whatever route you take

1. Langrange Multipliers
2. Maximising MPR via differentiation

the following formula can be used to get the final answer (derived by optimising the Sharpe ratio/MPR);

Xa = ( [Ea - rf]Vb- [Eb - rf] Cab)/([Eb - rf]Va+ [Ea - rf]Vb - [Eb - rf + Ea - rf]Cab)

notice similarity with

Xa= (Vb - Cab)/(Va+ Vb- 2Cab ) for minimum variance portfolio.

for question 4 of April 2009, we get

Xa = 18^2/(18^2+ 5*8^2 ) = 81/161...fast and easy!!!!!!

Tutors, will one loose marks for applying this formula, especially because the April 2009 question 4 didn't state how the efficient frontier should be solved?


You may view attached PDF if the algebraic spaghetti above is confusing.

 

[John Potter]

Wow, what a fantastic formula - I think this is beyond the call of duty when it comes to studying!

There's no way you could get 81/161 unless you knew what you were doing so, unless the question word was "Derive", I would have thought you would surely get full marks for quoting your formula and giving the final answer. If it's "Derive" or a "Show that" question, I would feel a bit uneasy just giving one formula - in this case, why not show off your derivation in the exam?

Good luck!
John

 

[Edwin]

Thanks a lot John.

I think it will depend on what the question says.

 

[stylz]

Hi, Edwin,
Thanks for this useful formula. Just a couple of things:

You've said

for question 4 of April 2009, we get

Xa = 18^2/(18^2+ 5*8^2 ) = 81/161...fast and easy!!!!!!

That seems to be Va/(Va + (Ea-rf)Vb) which is quite different from your formula above. (Allowing for the fact that Cab = 0)

Also the answer in the exam report for the proportion xa invested in Asset A is 80/161 not 81/161. i.e. (1 - 81/161)
Are you able to provide some assistance please?
Thanks

 

[Mike Lewry]

The formula Edwin states for that exam question gives Xb rather than Xa, ie:

Xb = 18^2/(18^2+ 5*8^2 ) = 81/161

This does use the formula he stated since E-r = 1 for this exam question. Since Cab=0, we have:

Xa = 5*8^2/(1*18^2+ 5*8^2 ) = 80/161
Xb = 1*18^2/(1*18^2+ 5*8^2 ) = 81/161

The formula Edwin states comes from maximising the gradient of a straight line through the risk-free asset and a portfolio of the 2 risky assets. (This gradient is the market price of risk.)

The Examiners' Report gets the answer by deriving this formula, so stating it would be a, presumably acceptable, shortcut to the answer.

For the formula to work, we need there to be both a risk-free asset and precisely 2 risky assets. For other cases, a more general approach would be needed.