We've  learned about some economical concepts, asset classes and risk management. We also know how to compute returns and volatility. However we only scratched the surface of portfolio management and (modern) portfolio theory.

Imagine you want to invest in Amazon, Apple and IBM. How do you allocate your capital? A trivial answer is 1/3 in each. Yet is it optimal? It is not. Indeed, a well managed portfolio accounts for return, volatility and correlations.

In this article we will learn how to allocate our capital on different assets, and how to take your risk aversion and return expectation into account. This process is called Modern Portfolio Theory (MPT).

### Key takeaways:

• A portfolio is a combination of assets such as stocks, bonds, or cash.
• Modern Portfolio Theory (MPT) helps investors allocate capital in a portfolio that maximizes expected returns while exposing themselves to a sustainable level of risk.
• MPT is an approximation of reality

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## What is a portfolio?

The term portfolio refers to the combination of assets such as stocks, bonds, or cash. The monetary value of each asset drives the risk/reward ratio of the entire portfolio. As a portfolio manager, your job is twofold:

• Picking investments: decide which assets you will be part of your portfolio
• Allocate capital: decide on how much money you will allocate to each asset

Your job is to maximize the expected return and cut the risk of your portfolio. It is a multi-objective optimization problem: many efficient solutions are available. The trade-off between risk and return helps decide on the preferred one.

Thus an optimal portfolio is not the set of securities with the highest expected returns or lowest risk.

Imagine that you have \$100k and suppose you already selected a portfolio of 30 assets. How would you build your portfolio? i.e., how would you distribute your cash in the different assets?

A simple, yet not optimized, solution would be an equally weighted portfolio.

## Portfolio selection

Harry Markowitz was awarded a Nobel Prize in Economics for its Modern Portfolio Theory (MPT) he introduced in his 1952 paper "Portfolio Selection".

MPT forms the foundation for all subsequent theories on how risk is quantified. It still influences the way we invest today.

MPT assumes investors are risk-averse (for a given level of return, investors favor a less risky portfolio). Therefore, it helps investors allocate capital and construct a portfolio that maximizes expected returns while exposing themselves to a sustainable risk level. Indeed, an investment's risk and return characteristics should not be viewed alone but evaluated by how the investment affects the overall portfolio.

MPT assumes the capital markets are perfect:

• no transaction costs
• no taxes
• the same level of available information for every market participant
• perfect competition in the market

It also assumes the portfolios' returns to be normally distributed. Of course, none of these hypotheses is true. MPT is an approximation of reality.

Risk-return trade-off is one of the basic concepts of Modern Portfolio Theory. An optimal portfolio does not include securities with the highest potential returns or with the lowest risk. It is a delicate balance between the two: a mix of (i) securities with the greatest potential returns and of (ii) securities with the lowest degree of risk.

To illustrate let's consider 3 fictive assets (A, B and C) with different expected returns and volatility:

Assets A and C have the same volatility, while assets B and C have the same returns. If you were to choose between A and C, you would go for A as it offers better returns at the same risk level. And if you were to choose between B and C, you would go for B as it has lower volatility for the same returns.

However, what would you do if you were to choose between A and B? Or a composition of the three?

That's what Markowitz tried to answer in 1952. The best option is not to take one and not the other, but often a composition of the two. The same applies to many (10, 100, or even 1000) assets.

## An illustration with two fictive assets

Let's consider the case of two assets, where you have to decide how much you want to invest in assets A and B.

What are the expected returns and volatility of a portfolio containing the two assets? A linear combination of the two? Not necessarily. The values are a function of the correlation between the returns of the assets. Possible values (dashed line) of a portfolio given a certain correlation between two assets (red points) returns series.

Correlation is determinant in Markowitz theory. The more correlated the assets, the less you benefit from diversification. A portfolio's returns of correlated assets (correlation = 1) is a linear combination of its assets (top left figure). Whereas with a perfect negative correlation (correlation = -1), we could diversify all risk away.

Of course, this reduction of volatility always comes at a price: you reduce your returns (hence the name: risk-return tradeoff). Where does the optimum portfolio stand? Well, it depends on you and the level of risk you are ready to take.

Let's take a specific example: Gradient of possible portfolios: from {100% Asset A, 0% Asset B} (light blue) to {0% Asset A, 100% Asset B} (dark blue).

This technique helps you decide on a portfolio allocation given a maximum level of risk you are willing to take.

From there we can define two portfolios:

### Minimum Variance Portfolio

It is the portfolio with the smallest volatility. A risk averse investor would go for this one. In this example it stands at about 13% of asset A, and 87% of asset B.

### Tangency Portfolio

The Tangency Portfolio is the most efficient portfolio from a risk-reward standpoint, i.e., with the highest Sharpe ratio. The Sharpe ratio is the expected returns divided by the volatility.

In our example, it corresponds to 75% of asset A and 25% of asset B.

## Asset allocation for two real assets

Let's sharpen our intuition with real assets portfolios. Suppose you have three favorite stocks (Amazon, JPMorgan, and UPS), and you want to invest in two of them. How would you pick which stocks you should invest in? And how much capital should you allocate to each?

Let's plot each stocks' growth since 2010 (i.e. how much you would own today if you invested \$1 in 2010). We also plot the expected returns as a function of the annualized volatility for each stock:

Which pair should we choose? AMZN-UPS, AMZN-JPM, or UPS-JPM?

AMZN drives the expected return of our portfolio, while UPS reduces the volatility. Let's plot the two-assets portfolios of each pair to have a clearer view: Gradient of possible portfolios (for 2 out of 3 assets). Each line represents a two-asset portfolio.

From the above graph, we have:

• AMZN-JPM is the worst pair: it has the highest volatility for the same expected returns.
• UPS-JPM reaches low volatility, which is excellent, but the expected returns are also low.
• AMZN-UPS yields the best results. The Minimum Variance Portfolio (22% AMZN, 78% UPS) and the Tangency Portfolio (65% AMZN, 35% UPS) are on that line.

If we were to build a portfolio with these weights: Growth from \$1 of the Minimum Variance Portfolio (P.MVP, green line), and the Tangency Portfolio (P.TP, blue line).

The Minimum Variance Portfolio (P.MVP) has a slightly better performance than holding the UPS stocks alone while also having less volatility.

The Tangency Portfolio (P.TP) has excellent performance while being less volatile than holding AMZN only.

## Asset allocation for three real assets

Finally, let's build a portfolio of 3 assets. Can we improve our expected returns, or reduce our risk, or both?

Again, we can compute the universe of possible portfolios. Since we are building a 3-assets portfolio, the result is now a plane (and not a line):

Again we can define two different portfolios:

• The Minimum Variance Portfolio (minimum volatility) corresponds to 19% AMZN, 65% UPS, 16% JPM
• And the Tangency Portfolio (maximum Sharpe ratio) to 66% AMZN, 34% UPS, 0% JPM (JPM is not useful to increase the Sharpe ratio).

We plot the performance of these portfolios with respect to the other assets:

Again the Minimum Variance Portfolio (P.MVP) has a slightly better return than holding the UPS or JPM alone while also having less volatility. The Tangency Portfolio (P.TP) stays the same as before (adding JPM would not improve the Sharpe ratio).

That's all for today,