Using mean variance, we know that if Sharpe ratio of new stock is greater than Sharpe ratio of portfolio * correlation of the stock with the portfolio, then we add, that is: If S_new > S_portfolio * Corr(i,portfilo) then you should add and there is benefit to that. Fine. We also know that if Corr(i,portfilo) = +1.0 (perfectly correlated), then there is NO benefit to diversification, i.e., it’s no good to add the stock, it won’t raise your return or lower your standard deviation. Now, go back to the first decision, and set Corr(i,portfilo) = +1.0, then it becomes: if S_new > S_portfolio then you should add and there is benefit to that. What gives?
There is still benefits to diversification, the sharpe ratio for the total portfolio will increase when you add the new stock.
Yeah, but how do you reconcile that with the fact that Corr(i,portfilo) =+1.0, there is no benefit in adding the stock?
Dreary Wrote: ------------------------------------------------------- > Yeah, but how do you reconcile that with the fact > that Corr(i,portfilo) =+1.0, there is no benefit > in adding the stock? I don’t know, I’m not a statistician. The way I think of it is: If my portfolio sharpe is 1.5 And the stock’s sharpe is 2.0 There should be some sort of benefit even if they’re perfect correlated.
There is no ADDITIONAL diversification benefit. that doesn’t necessarily mean it shouldn’t be added to the porfolio.
well maybe, but in this context, you add an asset only if it adds benefit (benefit is stirctly a risk/return concept), so I don’t think you should add for any other reason. I think we need a portfolio management guru to help out.
Stock A and Stock B are perfectly correlated. Stock A has an excess return of 2% (i.e. on average returns 2% more than RFR). Stock B has an excess return of 6% (so returns 6% more than RFR on average). To make it more concrete, let’s suppose RFR is 2%. E® of Stock A = 4% E® of Stock B = 8% Unfortunately your portfolio consists only of Stock A. Do you decide not to invest in Stock B because it is perfectly correlated? Of course not, Stock B has a higher expected return. In fact, probably you eliminate stock A fro the portfolio and only invest in Stock B. In reality, your decision can be one of several, and it depends mostly on how much risk you are willing to have in the portfolio. You can: 1) Combine Stock A with the RF Asset for low levels of risk. 2) Combine Stock B with the RF Asset for low and medium levels of risk 3) Combine Stock A with Stock B for medium levels of risk (between SD(A) and SD(B)) 4) Borrow at RFR and invest in Stock A for medium or high levels of risk. 5) Borrow at RFR and invest in Stock B for high levels of risk. 6) Some mixture of RFR, A, and B (possibly to control for market impact of trades) Lots of choices here. In practice, there’s extra risk in borrowing, and you usually can’t borrow at RFR anyway, so you are unlikely to do #4, unless you are high risk and for some reason stock B is less liquid. The big lesson here is that adding something to the portfolio can be done for two reasons: 1) improved diversification (so watch the correlations) 2) higher expected returns (so watch E® )
Oh, and another thing, you never want to add something with negative expected returns to the portfolio, even if it is uncorrelated. So for example, you might think that adding a short position to IBM might make the risk characteristics of the portfolio better, because when the market goes up, the IBM short goes down. But we do expect that IBM will be profitable over time, so if you have a short position in something that is profitable, its expected returns are gong to be less than RFR. So in those cases, the negative correlation is not sufficient reason to add it. The formula you mentioned will pick this up, because (assuming normal times), a short position in IBM should have a negative Sharpe ratio over the long term.
great to hear from you bchadwick. On your last two lines, is it correct then to say, that if the correlation=+1.0, you will not get any less std deviation by adding the stock? And if the stock offers a higher sharpe ratio than your portfolio, then adding it gives a higher return but may increase your portfolio’s std deviation?
Dreary, I think I may have said things too strongly, partly because in the real world there are two big constraints. So on the CFA exam, it probably is true that you don’t need to bother with adding perfectly correlated assets. The real-world niggles are these: 1) Things are never truly perfectly correlated, unless they are specifically engineered to be that way (and even then, transactions costs usually spoil the fun: look at 2x ETF tracking error, for example). 2) You can’t borrow at RFR unless you are the US Governemnt, so leveraging up on a low SD() asset isn’t really as viable as they make it sound. In the stylized world of CFA finance theory, however, if you do have two perfectly correlated assets, then there is no real reason to add the second one to the portfolio, because you could mix one of them with the RF asset and get the same return as the second one that way. In real life, however, you usually don’t want to leverage unless you absolutely have to, so you would probably just take whichever asset has the highest expected return, and only start leveraging up once you’d gone to 100% of the higher E® asset.
Dreary Wrote: ------------------------------------------------------- > great to hear from you bchadwick. > > On your last two lines, is it correct then to say, > that if the correlation=+1.0, you will not get any > less std deviation by adding the stock? And if > the stock offers a higher sharpe ratio than your > portfolio, then adding it gives a higher return > but may increase your portfolio’s std deviation? I realize I didn’t answer your question head on. If the asset’s return is perfectly correlated to the portfolio, then adding it could increase or decrease the portfolio SD, depending on whether it is a high-volatility asset or a low one. (Remember that correlation is about whether they move up and down together consistently, not whether they move up and down the exact same amount, so two assets can be perfectly correlated, but one is more volatile than the other). However, adding a perfectly correlated asset will not change your portfolio’s Sharpe Ratio. So you won’t get any additional return that you couldn’t get by just changing the mix of your original portfolio and the RF asset.
Great stuff.
yeah awesome explanation. dreary, how are you are you feeling about this thing? i have a bad, bad feeling for myself. with 40% of stuff still to go, i do know anymore what to do.
overwhelmed.