Suppose a stock return follows a normal distribution. It has a mean yearly return of $0\mathrm{.}05.$ Its return std is $0\mathrm{.}50.$ What are the total expected returns, volatility, variance, and the $\backslash (1\backslash \%\backslash$mathrm$\{$VaR$\}\backslash )$ of the total holding period return for holding the stock for the short run like $1$ year or long run like $10$ or $30$ years? What are the cumulative return given the loss defined by VaR? What are the Sharpe ratio of these two holding strategies, as well as the mean$/$variance ratio? What you conclude from this comparison.

VaR measures "what is the maximum expected loss over a certain period, at a specific confidence level?"

Suppose $\backslash (1\backslash \%\backslash$operatorname$\{$VaR$\}=$X $\backslash ).$ It means that there's a $\backslash (1\backslash \%\backslash )$ chance that the annual return will be worse than X $.$ The cumulative return $\backslash (=\backslash$mathrm$\{$e$\}^\{\backslash$wedge$\}(-\backslash$mathrm$\{$X$\})\backslash )$ assuming returns are continuously compounded.

Note there can be more than one answers that are correct.

Select $4$ correct answer$($s$)$

Sharpe ratio of buy and hold for the longer period $(30$ years$)$ looks better.

Therefore, based on the Sharpe ratio alone, buy$-$and$-$hold for the long term looks like a better strategy the short term $(1$ year$)$ strategy.

Sharpe ratio of both the short$-$term investing $(1-$year$)$ and long$-$term $(30-$year$)$ investing in this case is the same.

Mean$/$Variance ratio is the same for the short term and long term investing strategies. Therefore, if an investor only look at Mean$/$Variance ratio, which is also a risk$-$reward tradeoff measure, the investor would not see the difference of these two strategies.

The $\backslash (1\backslash \%\backslash$mathrm$\{$VaR$\}\backslash )$ of the $30-$year investment is much more negative than that of the $1$ year investment. This is because after compounding returns over time, the magnitude of the potential loss as measured by VaR is much larger in magnitude than the total expected return.

Warren Buffett once said buy$-$and$-$hold is safer. If investors such as hedge funds only care about the magnitude of potential losses as measured by VaR, then Warren Buffett's statement is wrong. What Buffett missed is the fact that for most stocks, one$-$year volatility is much larger than one year return. As the holding periods n increase, although return grows by a factor of n while volatility grows by a factor of the square root of $\backslash ($ n $\backslash ),$ volatility is a larger number to begin with. For example, $\backslash (50^\{*\}10=500\backslash ),$ which represents an increase of $450$ from $50.$ In contrast, $\backslash (5^\{*\}10=50\backslash ),$ which is only an increase of $45$ from $5.$ Therefore, the smaller a number is to begin with, the smaller the magnitude of the increase is$,$ even though the growth rate is the same.