Question

(AAPL Fun) Recent data, collected on a quarterly (3-month) basis suggests, that Apple stock price (AAPL) is equal likely to increase or decrease in every quarter. When AAPL increases at the end of a quarter, it does so by a factor of 1.105, with respect to its value at the beginning of the quarter. When AAPL decreases at the end of a quarter, it does so by a factor of 1/1.105 0.905, with respect to its value at the beginning of the quarter. AAPL closed at $143.17 today. In this exercise, we will do portfolio optimization over a period of six months. The current six-month U.S. Treasury bill yield closed today at 0.97%. The investor we consider, Aversy, has a logarithmic utility. (a) Find the probability mass function of AAPL after six months from now, denoted by S (b) Finance&, formerly Bacel&Co. s selling call options on AAPL with strike price $150 and maturity six months for $5.55 per share. That is, one pays $5.55 now for one share of this call option, and the payoff after six months is (S 150), where max(0,x). Find the probability mass function of the call option payoff after six months from now, denoted by C c) Suppose Aversy has a million dollars to invest in two assets () the AAPL call option in (b) and (ii) the six-month T-bill. What is the optimal investment portfolio for Aversy? (Because Aversy is too risk-averse, he decided to avoid investing in the stock itself.)

Answer 1

1-a) As Jerry wants to invest in the lowest risk portfolio, his main goal is to lower the risk irrespective of returns so he should go with an asset with the lowest standard deviation. as standard deviation is the measure of risk here so Safe asset is the one with the lowest risk of 12%. so he should go with Safe Assets.

1-b) Staurt can earn 25% of returns just by investing in high-end asset. but while earning 25% returns he wants to minimize his risk.

So we need to combine different - different assets

Let's make a portfolio of risky and balances asset portfolio:

let say Stuart invets X in Risky asset, and 1-X in balance portfolio

So, 30*X+15*(1-X) = 25

so, X= 1/3 and 1-X =2/3

here, $\sigma$1 = Standard deviation of risky assets

$\sigma$ 2 = Standard deviation of balance asset

as there is no correlation between assets so correlation =0

and w1 and w2 are the weights of risky asset and balance asset

The risk at the portfolio level can be calculated as $\sigma$(portfolio)^2 =w1^2 $\sigma$1^2+w2^2$\sigma$2^2

$\sigma$ (portfolio)^2 = (2/3)^2 * (40%)^2 + (1/3)^2 * (20%)^2

$\sigma$(Portfolio) = 27.48%

so it is better than high-end portfolio alone.

so Stuart should invest 1/3 in balance portfolio and 2/3 in the risky asset.

1- C) Same as above calculate the portfolio weight of high end and low end asset.

Let W1 = weight of balance asset. W2 = weight of safe asset

$\sigma$1 = Standard deviation of high-end asset   $\sigma$2 = Standard deviation of low-end asset

So to calculate the weight of these two assets:

W1*return on high end asset + W2* return on low end asset = required return

W1*25%+ 10%* (1-W1) = 15%

W1 = 1/3 W2 = 2/3

$\sigma$(portfolio)^2 =W1^2 * $\sigma$1^2+W2^2 * $\sigma$2^2

$\sigma$(portfolio)^2 = (1/3)^2* (30%)^2 + (2/3)^2 * (15%)^2

$\sigma$(portfolio) = 14.14%

so Dave should invest 1/3 and 2/3 in high end and low end asset.

1-D)

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