In this article I want to study if the longer the investmnet time horizon, the hgher the Sharpe ratio. This is based on the assumption that the standard deviation does not increase as fast as the ecpected returrn does and the risk free rates increase linearly.
Growth of the Expected Return and Standard Deviation
First, based on what most finanicl mathematics, the expected return grows
linearly with investment time horizen while the standard deviation grows
linearly with the squared root of the imvestment time horizon. Making the
value of mean over standard deviation bigger with longer investment time
horizon.
Check Real World Data with SPY ETF
This is the data I got when I was doing experiment on the risk of SPY in terms of
imvestment time horizon.
Expected Return
The data from the real world SPY data appears to follow the assumption that the expected return grows linearly with time.
Return Standard Deviation
The data from the real world SPY data appears to be higher than the theoretical data.
Return Standard Deviation Compared with its Growth Linear to Time
The data from the real world SPY data appears to be lwoer than the theoretical data. I will find other time and apply machince learnting to find which paramenters fit the real world data. Now let's just keep going with the assumption that return standard deviation increases linearly with the squared root of time, which is what most financial engineering is based on anyway.
Sharpe Ratio Grows with Investment Time Horizon
Now, we have the foundations that the expected return grows with time and the return standard deviation does with the squared root of time. Assume that the risk free rate stays the same at 2% in all time horizons, such as 2% for 1 year, 2% for 2 year, 2% for 3 years and so on. We see the Sharpe ratio grows.
The Risk Free Rate Needs to Increase to have a near constant Sharpe ratio
Let's assume that the risk free rate grows linearly with time just like the expected return. We see the Sharpe ratio still increases with time. But, that risk free rate grows with time linearly does not make sense under compounding rules.
The Risk Free Rate Increases Eepenentially
Now, let's take compounding into consideration. Assume yield stays at 2% with all time horizon, such as 2% for 1 year, 2 years, 3 years and so on. So if we invest in a risk free rate asset for 1 year, at the end of 1 year, our asset value would be origianal value times 1 + 2%. Reinvesting in the 2nd year, the end value would be original value times (1 + 2%) raised to the power of 2. The risk free rate for 1 year is 2% and for 2 years it (1+0.02)^2 -1, which represents a constant yield for 1 year and 2 years of imvestment time horizen. Applying this concenpt, we get increasing Sharpe ratio with time.
Conclusion
If expected return and return standard deviation follow the assumption of increasing with time for the prior and with squared root of time for the latter, and if risk free rate is constant in all time horizons, the Sharpe ratio would increase with time.
However, if the market tries to maintain a constant Shape ratio in all investment horizons, which I would call it the constant Sharpe ratio assumption, risk annualized free rate will increase with longer investment time horizon, which appears in a normal bond market.
When we see the yield curve is either flat or inverted, this would mean that longer term bond yields are traded cheaper - investors would rather lock in lower yields for longer terms. This could mean investors are not expecting good stock returns long-term.
Under the constant Sharpe ratio assumption, a flat or inverted yield curve means either a lower expected return or higher return standard deviation(recession) from the investment portfolio.
Looking at this differently, this could also mean that stock is cheap, because longer term investment's Sharpe ratio is higher when time increases.
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