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I have a quarterly return in quarter i for an asset which is typically held for 10 years. Which maturity Treasury yield should I use as a risk free rate in this context, and from what period? I initially think quarter i-1, but that thinking may be flawed.

My thinking is I use the 10Y Treasury yield and divide by 4 to make quarterly. I’ve also been told to use the 3M Treasury yield as it’s most often used in literature.

There are mixed answers out there, so I’m hoping to answer this. Typically I’m reading one should use a Treasury yield as a proxy for the RFR with maturity roughly equal to the typical holding period of the asset, but others have different answers. Also getting different answers about whether to use Treasury yield from same quarter, or previous quarter, as RFR rate for quarterly return in quarter i.

Edit: I’ll be using the resulting excess returns for an arithmetic Sharpe ratio (simple excess returns then Sharpe ratio scaled by sqrt(4)) so I am effectively ignoring the effects of compounding for the sake of using the formal Sharpe ratio.

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  • $\begingroup$ Don't divide the yield by 4 - don't forget about compounding. $\endgroup$ Sep 16 at 18:37
  • $\begingroup$ I’ll be using for Sharpe ratio where I use simple returns and multiply by sqrt(4) to “annualize.” So I’m actually assuming linear returns for the sake of sticking with the arithmetic Sharpe. Does that change your comment? $\endgroup$
    – Jason008
    Sep 16 at 18:38
  • $\begingroup$ Yes, but when you subtract your portfolio return from the risk free rate $r_f$, you should use the quarterly risk free rate, which isn't the same as the arithmetically compounded (divided by 4) yield. $\endgroup$ Sep 16 at 19:25
  • $\begingroup$ I think I’m looking for a more detailed answer than that but thank you $\endgroup$
    – Jason008
    Sep 16 at 19:36
  • $\begingroup$ One issue with long term bond yields is that they are IRRs not necessarily actual returns. $\endgroup$
    – fesman
    Sep 17 at 5:39
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This is a very deep and interesting question. Actually I think the answer depends on the cashflows for your asset and how they will be reinvested.

For example, if your asset is held for 10 years but will be returned to you at one lump sum with all the principal and interest at the end of the 10 years, then a different benchmark rate should be used than an asset which is held for 10 years, but returns its principal and interest in a self-amortizing way, so that by year 10 there is no principal left to return.

In both cases, you will need to figure out how you expect the cash flows to come back, and calculate its duration. Then you need to find a similar duration instrument as its benchmark.

The next step is once you have this benchmark, and you roll forward in time, are you getting closer to your 10 year horizon? In other words, a year from now, are you 9 years from the end of your holding period? If so, then you should not be using the yield but rather the total return of your benchmark. This is important because 10 year fixed income instruments, especially a zero coupon one to match a lumpsum payment, will have a lot of fluctuations due to interest rate changes.

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