The text says that using 2-corner portfolio analysis is mathematically equivalent to the more cumbersome MVO formulas for determining portfolio standard deviation (assuming correlations = 1). It then says that the actual corner portfolios do in fact consider correlation and the results are therefore pretty accurate but will ultimately result in more conservative estimates than if actual correlation were directly factored in (ie not 1).
1.) So how are we assuming correlations of 1 but the actual CPs include specific correlation - how are we not therefore capturing the correct correlation but instead assuming Corr =1 ?
2.) Can someone provide a simple example of how borrowing on short sale leverages up return - tyring to think through it and my mind is dead at the moment.
Foggy as its been awhile, but I assumed that the individual corner portfolios used specific correlations to come up with the allocation mix. The two corner portfolios, however, assume a correlation to each other of 1 to identify a portfolio in between the two corners. Someone can confirm/deny that…
You have $50. You receive $50 in short proceeds (exclude borrow costs), so in your pocket is now $100. You invest it and it goes up 10%, so you now have $110. You close out the short and are left with $60. 60/50=20% return.
If you just invested your $50 and earned 10% you have $55. $55/50=10% return.
I don’t recall reading that the corner portfolios have correlation = 1.0 to each other. It just so happens that the corner portfolios lie on the MVO curve, so if you can figure out the corners, you don’t really have to run MVO. The linear combo of 2 corner portfolios is an approximation to the one that would be exactly on the MVO curve. The portfolio standard deviation using this linear approximation is slightly higher than that from the MVO curve, which results in more conservatism.
Think of interest payable under leverage as a fixed cost. Once you have covered the fixed cost, any return over and above the fixed cost falls straight to your bottom line. That’s where you get the boost to return.
Mr. Leverage, your second point/example was great. Can you take that one step further and show how it actually applies in the risk-free and tangency portfolio context? Say for instance we’ve solved for the weights and the TP is leveraged to get the desired return (8% below). Using your example with actual dollar amounts, how does leveraging the TP allocation tie back perfectly to the 8% total return - so that $100 in your example is magnified enough to achieve the 8% return?
8% = 2%(-20%) + 7%(120%)
Thanks again. Your example is clear, I’m just getting confused how the additional weighting in reality earns a higher return bc doesn’t everything in the TP still just earn 7% ?
Yes your TP still earns 7%, however what you need to look at is the amount of capital you have invested at that level. By levering up, you invest more and therefore even though it only earns the same 7%, its on a much larger amount of money. Ie-if i earn 10% on $1 its a dime, if I earn 10% on $100 its $10. Same return, different level of capital employed.
Therefore, the return number changes when you look at how much you made over the money you actually have. That number does not include the amount you borrowed. To your example it would be:
2% cost to borrow funds on 20 = .40
use the $20 and invest it along with the $100 you have in your pocket, so you invest $120 @ 7% = $8.40
pay the borrow cost of $.40 out of your $8.40, so you are left with $8 in gains
that $8 is divided by the your capital of $100, giving you a 8/100= 8% return
If you actually read the CFAI text, it’s much clearer.
On p 236, the text says you can use the standard formula to calculate the exact stdev with the covariance table given.
Or you can take a short cut to approximate this stdev by a linear combination of 2 adjacent corner portfolios.
In the particular example given, the EF bows out to the left implying that a linear approximation will over-estimate the exact stdev which, in effect, reflect a less-than-one positive correlation between 2 CPs.
You could draw a diagram to visualise this. Think like convexity adjustment to bond duration.
BBUI, But if the 2CPs reflect actual, less than 1 correlations and we are using those 2 CPs in our calculation, then how are we overstating the standard deviation at all? Don’t they inherently include the actual diversification benefits (ie actual correlation)?
Mr. Leverage, thanks again. Your examples were very helpful!
What is important is the shape of the curve connecting the 2 CPs, it’s not straight. We are using the 2 CPs to deduce a point lying on that curve. Linear extrapolation only give an estimate not the exact stddev. Like I said before, we can use the weights and covariance matrix to calculate the exact value but it’s unyieldy, not going to be asked on the exam. You can do it easily with excel though.
just draw a chart, x axis is stddev, y is return. Draw a horizontal line at the level of return required, it will cut the actual curve at a lower x than that of the straight line connecting 2 CPs.
Mr leverage, why does your example seem to breakdown when you use number other than $20 and $100? If I use your same analysis but with borrowing $50, it doesn’t get me back to 8% total return??
50 x .02 = 1
150 x .07 = 10.5
10.5 - 1 = 9.5
9.5 / 100 = 9.5% and not 8%
Do the weights (-20% and 120% as before) represent the capital to borrow/allocate? So that If you wanted to borrow another amount, then you’d have to resolve for (in this case) -50% weighting to Rf and 150% to TP?
Should be obvious I guess. If you borrow/invest a different dollar amount than that will change the weights, so the return will also change. I guess I was thinking that same formula would result in the same return even if the amount borrowed and invested in TP changed, but clear now…that would therein change the weights and thus returnn.