# Modern Portfolio Theory

Modern Portfolio Theory (MPT) is one of the most influential theories in investment. It explains how risk and return are related and how diversification can minimize risk and improve return. Although counter intuitive, an investment portfolio consisting of only the highest return asset gives a lower expected return than if another lower return asset is added to the portfolio. This however requires that the two assets do not always move in the same direction. MPT describes how to design an optimal portfolio to provide the maximum return for a given risk level.

Modern portfolio theory is attributed to Harry Markowitz, who proposed the theory in 1952. He later received a Nobel Prize for his work.

## Putting a number on risk

Modern portfolio theory quantifies risk using statistical measures of the amount of variation that can be expected in the return of assets. If an asset's return does not vary, the asset is essentially risk-free. For example, if the asset pays guaranteed interest payments with no risk of default, then the asset can be viewed as a risk-free asset, in terms of the variance of its return. On the other hand, if the asset's return can vary (for example the price of a stock), then the asset would have some risk. One criticism of modern portfolio theory is that when defining risk as simply the variance in return, there is no distinction between losses and gains.

The risk (variance in return) of a portfolio of assets can be derived mathematically using the risks of each asset and the correlations between the assets.

Suppose that we have a portfolio of two assets with equal splits, and they both have the same variance and expected return. The expected return of the portfolio is the same as the expected return of each asset. If the assets are perfectly correlated (always move in the same direction), then the portfolio variance will be the same as the variance of the assets (i.e. the risk has not been reduced). However, if the assets have no correlation, then the variance of the portfolio is less than the variance of the two assets. In fact, the variance of the portfolio will be half the variance of the assets (for the mathematically inclined, refer to the formula for the variance of the addition of two random variables.

## How to construct a portfolio (simple example)

Suppose that you have a choice of different risky assets. For every level of return, there is one asset that offers the lowest possible risk, and for every level of risk, there is a portfolio that offers the highest return. If risk and return of these assets are plotted, we obtain the so called efficient frontier.

### The Efficient Frontier

There is always a tradeoff between risk and return for investments. In modern portfolio theory, an efficient frontier, is a set of portfolios that maximize expected returns for each level of risk. A typical efficient frontier is shown below:

The efficient frontier represents the best tradeoff between risk and reward for risky assets. The points represent different individual assets. Assets in the upper half region should be preferred over the assets in the lower half region because they provide higher returns for the same risk (for example the blue points are preferable to the red points). The efficient frontier is obtained from the assets that provide a specific return for the lowest risk. These assets are shown in green. Different points along the curve are tradeoffs between risk and reward.

If you have a choice of selecting only one risky asset, you can decide which asset to go with using the upper half of the efficient frontier, considering the amount of risk you are willing to tolerate.

### Combining a risky asset with a risk-free asset

What if you also have a choice to split your investment between two assets? Modern portfolio theory provides a conceptual framework to decide the optimal portfolio that combines several assets. To keep the example simple, we will look into combining a risky asset with a risk-free asset. The principles of MPT can of course also be applied to choose the best portfolio of multiple risky assets.

#### What is the correlation between a risk-free asset and a risky asset?

The correlation between a risk-free asset and a risky asset is zero. This is because the return on the risk-free asset is fixed and is not influenced by the risky asset. A key idea in modern portfolio theory is that diversification reduces risk when the asset returns do not always move in the same directions.

When two assets are uncorrelated, there should be a combination of the two that provides a given return for less risk than with only one of the assets. This combination is given by the capital allocation line.

#### The Capital Allocation Line

The capital allocation line (CAL) represents the best tradeoff between risk and return when combining a risky asset with a risk-free asset in a portfolio. Here is the CAL for our example:

You can obtain the capital allocation line by plotting a line that starts from the risk-free return point and touches the efficient frontier curve. The risk-free return point is simply the point that has risk level (or variance) zero, and the return of your risk-free asset. This point represents investing 100% in the risk-free asset and 0% in the risky asset. The point where the line touches the curve, on the other hand, represents investing 100% in the risky asset and 0% in the risk-free asset.

For a given return, combining a risky asset with a risk-free asset reduces the risk.

Conversely, for a given risk level, combining a risky asset with a risk-free asset improves return.

You can find the optimal portfolio for your needs by choosing the point along the capital asset allocation line according to your desired risk level or return. The percentage split between the two asset is simply the proportion of the line from the risk-free point until your chosen point.

## Criticisms of MPT

The ideas of modern portfolio theory are very useful and there are many investment professionals who rely on MPT to design portfolios. However, there are some shortcomings with MPT to be aware of, due to the way risks are quantified, and the reliance on historical data.

## Modern portfolio theory and ETFs

Exchange Traded Funds (ETFs) are investment vehicles that make it possible to easily diversify with a wide variety of different assets within a market. This goes in line with the core idea in MPT of diversification. However, the percentage split in the holdings of an ETF are typically found according simply the market capitalization of companies, rather than via the mathematical approach of MPT.

A well-diversified portfolio provides reasonable protection under normal market conditions. See our investment risks article for a more in depth discussion of the types of investment risks and how they can arise.

## Takeaways

Modern portfolio theory focuses on the relationship between assets in a portfolio and shows how you can take advantage of the fact that when there are different uncorrelated assets in a portfolio, gains from one asset can offset losses from the another asset. This results in a reduction of risk for a given level of return.

MPT also provides a framework for finding out the optimal portfolio of a combination of assets.