Choosing a Risk-Free Rate for Historical Excess Returns and Portfolio Optimization

# T-bond of what maturity to use as risk-free rate when calculating excess return?


# T-bond of what maturity to use as risk-free rate when calculating excess return?












I am comparing different asset classes in order to estimate the expected return and the risk of a portfolio. I have historical data (adjusted close price) for my asset classes for the last 5 years with monthly frequency. The goal is to calculate the efficient frontier for my data at original frequency (monthly) and for annualized data.

I firstly need to calculate the excess return of each asset class, so I need to get 5Y average of monthly excess returns and 5Y average of yearly excess returns.

What bothers me is what risk-free rate to use in the calculation of excess return. The only thing clear to me here is that I need to use US Treasury Bonds yields, but of what maturity and why to use a certain maturity (I assume that there is more than one right answer for my question)?

- Do I use T-bonds with 5Y maturity in both cases, because I am doing optimization on historical data for the last 5Y?

- Do I use T-bonds with 1Y maturity for the annualized data and 1 month maturity for the monthly data?

- Is it more reasonable to use the same yield (I assume current yield) for calculating all historical excess returns over the course of 5Y, or to take historical changes in bond yield into account (so, use historical yields for every year to calculate yearly excess returns for the last 5Y)?

## Answer by Mild_Thornberry (score 3, accepted)

https://quant.stackexchange.com/a/49819

There are a few different views out there in choosing your reference rate. Literature normally uses a T-Bill rate (1 month or 3 months). That's what Fama & French do in their online library. In practice, you're on the right track with matching the maturity of your investment horizon with the maturity of your risk free rate. In theory, you could go a step further an have a term structure, as mentioned here:

http://people.stern.nyu.edu/adamodar/pdfiles/papers/riskfree.pdf

I think that's a bit overkill though. I like to use the 3 month t-bill rate. Bernake shows in figure 5 how the impact of changes in the Fed Funds Rate on equity returns are nicely predicted by CAPM (see Figure 5). The Fed Funds rate is very correlated with the 3 month t-bill rate, not so for longer maturities. My reasoning: if changes in the Fed Funds Rate move stock returns linearly, the market is probably using rates highly correlated with monetary policy (short-term maturities) when doing their own valuations.

https://www.federalreserve.gov/pubs/feds/2004/200416/200416pap.pdf

Choosing your risk free reference rate ultimately comes down to being able to justify your decision. In the end, it should have a small impact on your efficient frontier, since it is so overwhelmed by the risk premium.

## Answer by user42108 (score 1)

https://quant.stackexchange.com/a/49801

Use a rate that's representative of what you'd get on excess cash if you weren't fully invested. For most institutional investors this will be something like either Fed Funds or 1 month LIBOR less a spread (set by your PB / FCM). If you were a corporate treasurer it would likely be roughly equivalent to a money market fund rate.

EDIT: LIBOR is unequivocally the wrong rate to use as it's not what you can earn. LIBID is somewhat closer and will probably be around (most likely above) the 1mL-spread your PB/clearer gives.

## Answer by VanillaCall (score 0)

https://quant.stackexchange.com/a/49821

I would use 3-month bills as a measure of the risk free rate.

## Answer by demully (score 0)

https://quant.stackexchange.com/a/49832

There is no single right answer for this one, there are good reasons for and against using cash (either bills or LIBOR widely used) versus terming out the duration to match your equity/risk asset horizons.

Whatever would be your default when not invested can never really be too far wrong.

Strictly speaking, a treasury bond is not “riskless”. There may be no credit risk, but there is a “term premium” over the expected path of short rates. It’s notoriously hard to measure, but it is baked into yields. Note that if you term out thus with say a 5 year bond, your excess returns should be returns less the return on the 5 year. You can’t just subtract the yield.

The counter-argument goes that only the Treasury is riskless on a 5 year view. Yes, there is some TP; but the yield is nevertheless certain on a buy-and-hold basis. From this perspective, cash or bills can even be seen as “risky”, by virtue of uncertainty over the future path of short rates!

You really just have to pick a side, be consistent and stick with it. Whatever you choose will have its pros and cons. Cash is currently the more fashionable baseline in the asset allocation world, given (1) simplicity, (2] consistency with the excess returns from eg futures, and (3) the relative absence of volatility that you get with bonds.