Suppose you are an investment manager, and you are explaining the risk of holding just a single stock $($ex$.$ NVIDIA Corp, NASDAQ. Ticker: NVDA$)$ to your client.

a$.$ Suppose NVDA's historical monthly returns were $5\%$ on average and the standard deviation of the returns is $10\%.$ Assuming the returns are normally distributed, how likely would the monthly be less than or equal to $-5\%,$ which is one standard deviation below average? Hint: use the characteristics of normal distribution discussed in class to answer.

b$.$ Using the normal distribution characteristics, you find that the fifth percentile value at risk of NVDA is $-11\mathrm{.}45\%.$ How would you explain this to your client in a non$-$technical manner?

c$.$ Suppose you recommend to your client to invest in a risk$-$free asset, such as Treasury Bills $($T$-$bills$).$ Assume T$-$bills pay a $0\mathrm{.}5\%$ yield $($i$.$e$.,$ the risk$-$free rate is $0\mathrm{.}5\%),$ what is your client's

expected return if your client chooses to hold $50\%$ NVDA and $50\%$ Treasury Bills in his portfolio? What about the standard deviation of this portfolio?

d$.$ Suppose your client wants to increase his holdings of NVDA from $50\%$ to $80\%.$ Explain the risk and the reward of doing so$.$

e$.$ Continue from d$,$ show your client the risk$-$reward tradeoff by drawing a capital allocation line $($CAL$)$ of NVDA stock and T$-$bills. You can do this on either Excel or by hand.

f$.$ Your client also brought up whether he should invest in an index fund instead. Suppose the S&P $500$ index fund has a $1\%$ monthly risk premium and a standard deviation of $4\%,$ what is the Sharpe ratio of the index fund? Should the client make the switch based on the Sharpe ratios for NVDA and the index fund?