Why are calculations typically performed on excess returns vs returns

@maratikus
Sorry for the (relatively) late reply. I’m not sure what you mean by high condition number.
I assume that since the risk-free rate has a standard deviation of 0 and 0 correlation with the rest of the portfolio that the covariance matrix is the same whether you have the risk-free rates in there or not. Assuming you use standard MVO with weights constrained to 1, the optimal portfolio is wmin+(1/lambda)*sig-1*(I-1*wmin’)*u, where lambda is a risk aversion coefficient, sig-1 is the inverse of the covariance matrix, I is a nxn idenity matrix, 1 is a vector of 1s, u is the expected return vector, and wmin is the minimum variance portfolio (sig-1*1/1’*sig-1*1). I know it may seem counter-intuitive, but because (1wmin’)*rf=rf, then (I-1*wmin’)*u=(I-1*wmin’)*(u-rf), and it doesn’t matter what the risk-free rate is or whether you subtract it out. The portfolio weights are the same regardless.
 
jmh, I see what you are saying. As RFR goes up, Efficient Frontier shifts down and since indifference curves are parallel, the same weights would be optimal.
I didn’t quite communicate what I was trying to. In my head when I think of mean-variance optimization, I automatically jump to Sharpe maximization. For a given efficient frontier, you draw a tangent line from (0 vol, RFR - ret). Then because of leveraging (borrowing or lending), you get your optimal portfolio. As a result, you really want to maximize for Sharpe (given borrowing and lending are possible). As RFR goes up, the tangent line changes and portfolio weights change. Optimal weights become (sig^-1)*(u-rf)/(1’*(sig^-1)*(u-rf)). Those weights are sensitive to RFR. It’s evident in the example that I showed earlier (RFR changes from 0 to 2%). When you start performing numerical calculations, condition number shows up. http://en.wikipedia.org/wiki/Condition_number
 
@maratiksu
I see what you’re saying. I guess the way I do things is different. For instance, I tend to generate expected return forecasts on excess return data. If you did the forecast in terms of the normal returns, then you’re right, of course.
So if I were doing Sharpe ratio optimization I would add back the risk-free rate to my mean return forecast and the formula in your post would essentially simplify to (sig^-1)*(ux)/(1’*(sig^-1)*(ux)), where ux is my initial excess return forecast. So in this sense, it wouldn’t matter what the RFR is.
Another interesting point is whether the weights from mean variance optimization are more sensitive than the Sharpe ratio optimization to a change in the return. If I’m not mistaken, the mean-variance weights are more sensitive.
 
jmh, I see what you are saying. Of course, if you estimate excess returns, you don’t care about RFR. The initial post was about adjusting returns for rfr -> I talked about that. No disagreements here.
In regards with sensitivity of weights to the vector of mean returns -> that’s a relatively simple calculation (you have a formula for x, just differentiate with respect to u and you will get a matrix of sensitivities). It becomes more complicated when you start using Bayesian methods and introduce constraints but I don’t want to get into that because that would be work-related.
 
Remember also that if you are using historical returns, the risk-free rate is seldom constant in reality. The standard deviation in the Sharpe ratio should really be SD(Port_R - Rfr). For equities, this may not be a huge issue because equity volatility is typically so much larger that the contribution of RFR volatility to risk premium volatility is negligible. But for fixed income investments, it may be more important.
 
I mostly agree bchadwick, but I think the historical standard deviation of the risk-free rate (in the US at least) is so much lower than all but very low duration fixed income instruments that it shouldn’t matter (it’s normally like 0.1% annualized when long-term bonds might be 8%).
It might be better to think about it in terms of expected volatility of the risk-free asset, rather than the historical fact that the risk-free asset hasn’t had 0 volatility. If you expect the risk-free asset to have 0 volatility, you can exclude it from consideration. There might be times when it doesn’t have 0 volatility, like if the Fed changed rates sharply over the course of the next 1 or 2 months when you might want to consider it, but its a reasonable assumption.
 
jmh, have you done any work on estimating betas for international stocks?
 
Though I work on the international side of my firm, I haven’t really needed to estimate betas on individual international stocks. I recall there being some snags when doing this, such as making sure you use the right risk-free rate for the investor. My group focuses more on global strategy rather than doing bottom-up research, so I done the regression for sectors internationally, but we assume the investor has hedged their currency exposure so I can just use the US$ returns with the US risk-free rate.
 
I don’t have experience with that but I’d like to tackle this problem at some point.
I can think of a few problems with calculating betas (or running regressions) on international stocks or indices. One - different settlement time. Daily correlation between S&P 500 and Dax would be understated if I used settlement prices. If I only deal with European and US markets, I could just measure all prices at the same time (for example, 10:30 am Eastern time). However, if I deal with US, Europe and Asia, I can’t find an overlapping time interval when all markets are open. Two - correctly adjusting for currencies but that doesn’t seem to be a problem for you as you deal with hedged investments. Three - choice of benchmark for beta calculations. For example, S&P 500 is easy to deal with but that’s probably not the best benchmark.
There would be similar problems with calculating duration of global portfolios.
I’d appreciate advise.
 
I like to do weekly returns to avoid the issue of closing hours; it makes the periods overlap better.
S&P is not so bad, particularly with developed country benchmarks. I’ve used the MSCI All-Cap World Index, which is 96% correlated to the S&P on a weekly basis.
I probably ought to be using currency returns in there too. I haven’t because I’ve been doing my return analysis as hedged dollar returns, but technically the currency should matter anyway. Since I write and have a word budget, I don’t have the space to go into all the complexities that introduces, but I’ve been thinking about how to include that.
 
You must log in or register to reply here.
Back
Top