Solving for a Portfolio with a Guaranteed Payoff Across States
Summary
The problem gives four assets, their initial prices, and their values in three possible future states. It asks for holdings that produce a fixed payoff of £30 in every state, then asks for the portfolio’s profit or loss. The answer recommends writing one payoff equation per state and solving the resulting linear system. Since there are more assets than states, it says there are multiple solutions and suggests setting some holdings to zero to find one.
The response also sketches a different task: maximizing expected payoff subject to spending a fixed budget, using the stated state probabilities. That optimization is not the same as constructing a portfolio with a guaranteed payoff, and the proposed all-in guess is not derived or verified. The document therefore illustrates how payoff constraints can be represented as equations, but it does not actually provide the requested holdings or establish the guaranteed profit or loss. Its final optimization discussion is only a suggestion, not a solution to the original question.
Key ideas
- A guaranteed payoff requires the portfolio to have the target value in every possible state.
- The state-by-state portfolio values can be written as a system of linear equations in the asset holdings.
- More assets than states can leave multiple portfolios satisfying the same payoff constraints.
- Expected-payoff maximization under a budget constraint differs from guaranteeing a fixed payoff in each state.
- The answer does not solve the requested system or verify its suggested portfolio.
Tags
Full text
# Portfolio of Assets
# Portfolio of Assets
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 by jaamor (score 1)
https://quant.stackexchange.com/a/17417
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 $w_a$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.