Portfolio of Assets

Question

The following represents a model for an economy.

At time $t=0$, four assets have the value $X_1= £5$, $X_2=£5$, $X_3=£10$ and $X_4=£4$.

Three possible states of the world exist ($\alpha_1$, $\alpha_2$ and $\alpha_3$) at time $t=1$, with this being viewed at $t=0$, and these states can occur with probabilities $p_1=0.25$, $p_2=0.5$ and $p_3=0.25$ (respectively).

In state $\alpha_1$, the values of asset at $t=1$ are $X_1=£6$, $X_2=£3$, $X_3=£12$ and $X_4=£9$.

In state $\alpha_2$, the values of asset at $t=1$ are $X_1=£9$, $X_2=£6$, $X_3=£12$ and $X_4=£3$.

In state $\alpha_3$, the values of asset at $t=1$ are $X_1=£12$, $X_2=£6$, $X_3=£9$ and $X_4=£3$.

Assume it is possible to own a part of an asset but it is not possible sell an asset which you do not own.

Show that it is possible to set up a portfolio of assets at $t=0$ which will definitely have a value of $£30$ at $t=1$, no matter which state of the world occurs at time $t=1$.

What is the guaranteed profit/loss of this portfolio?

I'll be honest, I'm at a total lose with this topic. A point in the right direction would be great. Maybe published text or a similar solution on this website? (If one exists?). Thank you.

Answer 1

Usually you would solve a system of equations to get the answer to textbook problems like this.

Here you have 4 assets and 3 states of the world, so your system will have infinite solutions.

If you want your asset to have exactly $ 30 of value at time 1, then just set one of the weights (3 or 4) as zero and solve the system.

Otherwise, to solve a more realistic problem, you can solve an optimization problem to maximize your expected payoff, given a $30 budget.

Let $ w_a, w_b, w_c, w_d$ be the amount of dollars of each asset purchased at $t_0 $ and $P_i$ the value of your portfolio in the state of the world $i\in \{1,2,3\}$

$$ max \,\,\, 0.25 * P_{1} + 0.5 * P_{2} + 0.25 * P_{3} $$ $$ s.t. \,\,\, 5w_a+5w_b+10w_c+4w_d=30 $$ Where $$ P_1 = 6 w_a + 3 w_b + 12 w_c + 9 w_d $$ $$ P_2 = 9 w_a + 3 w_b + 12 w_c + 3 w_d $$ $$ P_3 = 12 w_a + 6 w_b + 9 w_c + 3 w_d $$

My guess is that the best portfolio consists entirely of $w_a$.