# CFA Morning 2016 Q4 B & C

Hi all

A couple of questions on the 2016 Morning mock and the answers Schweser covers in its online videos. Here are the questions:

On B, I got this wrong because I added inflation to the risk-free rate and solved for the leverage based on 2.0% risk-free (nominal) rather than 0.5%. Just to check I understand this, is it the case that we should assume that the stated risk-free rate is nominal and therefore already incorporates the inflation of 1.5% (i.e. the real risk-free rate is -1.0%) or is it that the risk free rate is only ever stated as the real-risk free rate and it is the real risk-free rate that should be used for e.g. Sharpe ratio calculations?

On C, I got this right based on the theory but one thing that confuses me is that the maths doesn’t bear it out when I try to prove it based on the portfolio variance formula. Assuming the 8.0% return is accomplished through either: weights of 38.84% for Portfolio 3 and 63.16% for Portfolio 4 (from Part A) or weights of 106.2% for Portfolio 4 and -6.2% for the risk-free asset (NB: using my erroneous 2.0%):

Lowest possible unlevered (i.e. combo of Portfolo 3 and Portfolio 4) standard deviation assuming -1 correlation between P3 and P4 = square root of 36.84%^2 x 13.8%^2 + 63.16%^2 x 11.2%^2 + 2 x 36.84% x 63.16% x -1 x 13.8% x 11.2% = 0.002585 + 0.005 + - 0.007193 = 1.98%

Highest possible unlevered (i.e. combo of Portfolo 3 and Portfolio 4) standard deviation assuming +1 correlation between P3 and P4 = square root of 36.84%^2 x 13.8%^2 + 63.16%^2 x 11.2%^2 + 2 x 36.84% x 63.16% x 1 x 13.8% x 11.2% = 0.002585 + 0.005 + 0.007193 = 12.16%

So a range of 1.98-12.16% depending on the correlation.

Levered standard deviations assuming standard deviation of risk free asset is zero and correlation between Portfolio 4 and the risk free asset is also zero = square root of 1.06195^2 x 11.2%^2 = 11.89%.

So the levered position standard deviation could be higher e.g. if the corner portfolios are perfectly negatively correlated.

Is it the maths that wrong or the reasoning? (i.e. should we always assume corner portfolios are perfectly correlated?)

Thanks and sorry for wordiness!

Risk-free rate = 0.5% (if nominal or real is not stated, then it is always nominal)

Nominal risk-free rate = 0.5% (stated, so it is direct)

Real risk-free rate = -1.0%

Assuming perfect correlation is to set the maximum standard deviation of the combined portfolio (i.e. ceiling), so the standard deviation of the combination of corner portfolios is <= 12.16%

Thanks - is it the default assumption that the corner portfolios are perfectly positively correlated (i.e. is that ‘by definition’?). Because if not, then there is a world in which the bracketing corner portfolio could have a lower standard deviation than the tangential portfolio, but that wouldn’t make sense as the tangential portfolio should be the most efficient in terms of return per unit of risk…

For corner portfolios, assuming perfect correlation is the most conservative thing to do (i.e. forming a straight line between one corner portfolio to the next)

For the same level of return, the standard deviation of the (tangential portfolio + risk-free asset/borrowing) will be lower than the standard deviation of the combination of corner portfolios.

See the image attached. Notice how the standard deviation compare between Portfolio 1 and 2.