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I am conducting out of sample backtests of the MV framework. But how exactly do I derive the Maximum Sharpe Ratio portfolio for this? The standard forumula of the Sharpe Ratio is given by:

$$\frac{(w r - r_f)}{\sqrt{w Σ w'}}$$

Lets say I have an estimation window of 60 monthly returns on whose basis I derive the optimal portfolio weights. Which risk free rate $r_f$ would I have to use here or would I have to use any at all?

I'd appreciate any help!

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Within the context of portfolio theory, the risk-free rate is the interest rate at which investors may lend and borrow capital. This is generally not true in practice, but if you are willing to make this assumption then a common proxy used in industry is the US 3-Month T-Bill rates.

To impose a more realistic borrowing constraint, you could follow the approach in Black (1972) by constructing a mean-variance efficient portfolio uncorrelated with the market portfolio. Let's call the returns on this portfolio by $r_z$, then $\mathbb{E} r_z$ represents the shadow cost of borrowing and you can simply replace $r_f$ with $\mathbb{E} r_z$.

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  • $\begingroup$ Thanks for the answer! First of all, in my study I used so far the 1-month LIBOR for generating the out-of-sample excess returns, since I rebalance my portfolios on a monthly base. Would you suggest to use here the US 3-Month T-Bill rate as well and if yes, why would be the LIBOR 1-month a bad proxy? Second, for deriving the max Sharpe Ratio portfolio on the basis of lets say the 3-Month T-Bill, which exact rate would I have to use? Specifically, for a estimation period of 60 months, would I use the T-Bill of the 60th month or a average of the 60 monthly rates or any other? $\endgroup$ – Dirty Dan Mar 5 at 22:56
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    $\begingroup$ I see, so you are using the past 60 observations of monthly returns for the optimization? In that case, a 1-Month LIBOR/Treasury Yield can work just fine. In general, the time series for the risk-free rate should be the same frequency as the other assets - so you should have 60 observations of risk-free bonds with 1-month maturities. $\endgroup$ – Adam Mar 6 at 12:28
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    $\begingroup$ The purpose of the risk-free rate/excess returns is for your own optimization purposes (i.e. to factor in risk-free opportunity cost). The LIBOR is a good measure of risk-free rates, but is generally not possible for investors to borrow/lend at inter-bank rates so excess returns will be overinflated. If you don't plan on including a risk-free asset in your investment strategy, just set $r_f = 0$. $\endgroup$ – Adam Mar 6 at 12:31
  • $\begingroup$ So I'd have to use the excess return of each month if I'd stick at using a risk free rate? Because the "actual" formula seems like I have to use some specific point value of the risk free rate which I substract from the return. Furthermore, would it be really the max sharpe ratio portfolio if I just neglect the risk free rate? Another issue that I noticed, is that it appears in my backtests that there is kind of a big difference in out-of-sample sharpe ratios when using the risk frate in the estimation process and when not. $\endgroup$ – Dirty Dan Mar 6 at 18:29

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