# Introduction

This is a short write-up that uses Modern Portfolio Theory or Mean-Variance Analysis to give some guidelines on how the Polar Star Limited Fund can be used to create a portfolio with a better risk adjusted return compared to a basket of standard startegies.

Throughout we will use four investible assets

- Cash - returns from libor rates
- Fixed Income - Bloomberg Barclays Global Aggregate Bond Index
- MSCI - Global USD Total Return Index
- Polar Star Ltd

The challenge then becomes how to allocate capital between the different assets. Harry Markowitz introduced Modern Portfolio Theory in the Journal of Finance in 1952. For this work, we was later awarded the Nobel Price in Economics.

The basic idea follows the behaviour of a rational investor. Suppose there are two portfolios with the same expected return, but one has less risk than the other. The rational investor should choose the portfolio that is less risky. A simple way to illustrate the point is to consider a simple game.

Suppose we enter a casino that only has a single game, the flipping of a slighly biased coin. After carefully studying the statistics of the coin tosses we have come to the conclusion that the probability of the coin landing on heads is 0.51. Now suppose the casino has 1 000 coin flipping tables and we have $1 000. We can wager any amount on any number of games. The games all run in parallel and in each case we want to bet that the coin will land with heads facing upward. How do we allocate our capital?

There are two extreme cases that are worth exploring, the first is to wager the entire amount on a single table, and the second is to wager $1 on each of the tables.

Let us denote by \(P(\text{win})\) and \(P(\text{loss})\) the probabily of a win or a loss. The expected return of the single bet can be found using the calculation below

\[ \begin{aligned} E(\text{return}) &= P(\text{win}) \times 1000 - P(\text{loss}) \times 1000 \\ &= (0.51 - 0.49) \times 1000 \\ &= 20 \end{aligned} \]

Similarly, the expected return of entering a $1 wager on each bet can by found by the calculation below

\[ \begin{aligned} E(\text{return}) &= \underbrace{\left[ P(\text{win}) \times 1 - P(\text{loss}) \times 1 \right] + \dots + \left[ P(\text{win}) \times 1 - P(\text{loss}) \right] }_{\text{1000 times}} \\ &= \left[ P(\text{win}) - P(\text{loss}) \right] \times 1000 \\ &= (0.51 - 0.49) \times 1000 \\ &= 20 \end{aligned} \]

So, both types of wager give the same expected return. The difference between the two wager becomes noticible when we consider the risks involved. In the coin flipping casino we can measure risk as the probability of losing all our capital. From our initial statistical analysis we know that the probabilty of losing all our capital on a single bet is 0.49. The probability of losing all our capital in the $1 wagers can be found using the calculation below

\[ \begin{aligned} P(\text{Lose everything}) &= \underbrace{0.49 \times \dots \times 0.49}_{\text{1000 times}} \\ &= 0.49^{1000} \\ &\approx 0 \end{aligned} \]

So, both choices have the same expected return but one is clearly less risky than the other. For this game we have created, a rational gambler should choose to take many small wagers in stead of a single large gamble.

# Risk-Reward Space

In this section we construct a collection of portfolios that include Polar Star Ltd (red) and another that do not (blue). For the portfolios containing Polar Star Ltd we created a weight matrix

\[W = v \otimes v \otimes v \otimes v\]

consisting of the outer product of a vector

\[ v = \left( \begin{align} 0.00 \\ 0.02 \\ 0.04 \\ \vdots \\ 1 \end{align} \right) \]

with itself. We then reduce the weight matrix \(M\) to only include rows that sum up to 1. In this way we create a grid of all possible allocations with 2% jumps in allocation. To get matrices of similar size for the case without Polar Star Ltd we used a vector

\[ v^{\prime} = \left( \begin{align} 0.000 \\ 0.004 \\ 0.008 \\ \vdots \\ 1 \end{align} \right) \]

Each dot in the image below represents a point in risk-reward space. For the purposes of this write-up we will think of annualised standard deviation of monthly returns as a measure of risk and annualised return as a measure of reward. The general upward trend of the dots correspond to the analogy of “greater risk equals greater reward”. On the plot we highlight four points, each of these correspond to the maximum allocation awarded to each of the investible assets, also drawn from the collection of random portfolios.

The plot below can be interpreted from two points of view. Firstly, for a given target annualised return, what are the portfolio weights that had the lowest risk? Secondly, for a given level of risk, what are the portfolio weights that gave the greatest return? This is where the idea of the Sharpe ratio enters the picture. We define the Sharpe ratio as

\[ \text{Sharpe Ratio} = \frac{\text{Annualised Return}}{\text{Annualsed Standard Deviation of Monthly Returns}}. \]

All points lying above a line going through the origin with slope 1 have a Sharpe Ration of 1. This means that they have the same risk and return. All points above this line have a greater risk compared to return. Similarly all points below this line have a smaller return compared to the risk taken. In the plot below we include this line as a point of reference.