Let us suppose that you currently have Rs$.10,00,000,$ which you are ready to invest in three commodity assets. You are considering three options $$ buy into asset ABC, PQR and XYZ$.$ Each is currently priced at Rs $750/$unit$,$ Rs $820/$unit and Rs $675/$unit$,$ respectively. You also have the option of investing into risk free securities which guarantees an annual return of $6\%.$ There exists a commodity index called India$-$Shine whose value varies randomly over time following a Geometric Brownian motion. The average annual percentage growth rate of India$-$Shine is $18\%,$ while its standard deviation is $30\%.$ The prices of ABC, PQR and XYZ are related to the price of India$-$Shine in the following way: y$($ABC$)=800\mathrm{.}00\mathrm{.}01*$x$($IS$)+1,$ y$($PQR$)=750\mathrm{.}0+0\mathrm{.}0075*$x$($IS$)+2,$ y$($XYZ$)=600\mathrm{.}0+0\mathrm{.}009*$x$($IS$)+3.$ Here y$($ABC$),$ y$($PQR$),$ y$($XYZ$)$ and x$($IS$),$ refer to the prices of ABC, PQR$,$ XYZ and India$-$Shine, respectively. Further, $1,2$ and $3$ are three random variables which are normally distributed and statistically independent. Their means are zero and standard deviations are $20,15$ and $25,$ respectively. The current price of India$-$Shine is Rs $800/$unit$.$ Of course, you are considering an investment portfolio that is some combination of all three plus risk$-$free securities$.$ Your plan of action is to invest now and liquidate your portfolio after six months.

a$.$

You wish to determine a portfolio that minimizes the variance of your portfolio value at the end of six months $($because you are risk averse$)$ with the constraint that the average value after six months be at least Rs$.11,00,000.$ Remember that your portfolio may also include investing in risk$-$free securities$,$ which will earn a guaranteed $6\%.$ In your portfolio, you can allow for negative quantities for any of the three stocks. Describe carefully how to obtain such a portfolio using the binomial lattice approach. That is$,$ you can only use binomial lattice and other standard tools and functions that excel offers and not Monte Carlo simulation. Implement your methodology of obtaining the portfolio on the spreadsheet and report the results. $[$Hint: You may use $1-$week intervals to setup the lattices. As an approximation, you may use $26$ weeks to represent six months for both a$.$ and b$.$ You will

$4$

need a price lattice for India

$-$Shine and an $$average$$ price lattice for the three commodities.$]$ b$.$

Using the same portfolio obtained in $2$ a$.,$ explain and then determine the probability of reaching and exceeding a portfolio value of Rs $11,50,000.$ Next, assuming that the portfolio consists on non$-$negative quantities of ABC, PQR and XYZ$,$ describe how you will determine a portfolio that maximizes the probability of reaching a portfolio value of Rs $11,50,000.$ Again, your approach must use the binomial lattice. $[$Note: The cumulative probability for a normal distribution in excel can be obtained using the Normdist$(..)$ function. Hint: As well, you can use maximizing z function as a surrogate for maximizing probability.$]$