# Optimal portfolio standard deviation

This is from the 2014 June mock PM, set 9, question 4: if Donner wants to construct an optimal portfolio with an expected S.D of 12%, he should combine his proposed fully diversified portfolio with which of the following: borrow 20% assets; lend 20% assets; lend 80% assets?

The fully diversified portfolio has expected return =13% and S.D=15%, risk free=4% Can someone please explain how they got lend 20% of assets?

So the CAL formula…

=Rf + [(Rt - Rf)/std dev(t)] x personal std dev

His tangent portfolio has a higher std dev than what is prefered, so he has to reduce his risk. He does so by buying Rf securities, essentially lending at the risk free rate.

The 20% is arbitrary -> no calculation is needed given our available answers

Thanks, but what is the calculation for this?

there is none, it is an intuitive problem.

Just figure out the portfolio weights. You know the stdev of Rf is zero and target is 12% so you must have 80% in the risky asset and 20 in the risk free asset.

Hey, one way is to do it like JSobes said, the other way is:

Given that you have the target sd -> 12%

You have from a previous answer in that item set the s.d. of your portfolio (I dont remember very well wich was it).

After that you know that

sd(optimal portfolio)= weight of risky portfolio * sd of risky portfolio,

therefore-> weight=(sd(optimalportfolio)/sd(riskyportfolio))

and that’s how you can get the weight of Risky assets (which is 80%), so in order to “complete” the investment, you lend (invest) 20% of the assets in the RFR

The 20% is not remotely arbitrary. His portfolio has σ = 15% and he wants σ = 12% = 80% × 15%. So he wants 80% of his portfolio and 20% (= 100% – 80%) of the risk-free asset.