Modern Portfolio Theory (MPT) is a theory of investment and portfolio management, which shows how investors can maximize the expected return of the portfolio under a given level of risk by changing the ratio of various assets in the portfolio. Given the expected level of return, investors can change the investment weight of the portfolio to achieve the lowest possible level of risk for that rate of return.

Key points

- Modern Portfolio Theory (MPT) is a theory of investment and portfolio management, which shows how investors can maximize the expected return of the portfolio under a given level of risk by changing the ratio of various assets in the portfolio.
- According to modern portfolio theory (MPT), investors must bear a higher level of risk to obtain higher expected returns.
- By diversifying the types of securities, the overall risk of the investment portfolio may be reduced.

## The key assumptions of modern portfolio theory

The core idea of MPT is that risk and reward are directly related. This means that investors must take a higher level of risk to obtain a higher expected return. Another main idea of MPT is that through diversification across multiple types of securities, the overall risk of the investment portfolio can be reduced. If investors face two investment portfolios with the same expected return, the rational decision is to choose a portfolio with a lower total risk.

In order to conclude that the risk, reward, and diversification relationship is true, some assumptions must be made.

- Given its unique circumstances, investors try to maximize returns
- Asset returns are normally distributed
- Investors are rational and avoid unnecessary risks
- All investors can access the same information
- Investors agree on expected returns
- Does not consider taxes and transaction costs
- The size of a single investor is not large enough to affect market prices
- Ability to borrow unlimited amounts of capital at risk-free interest rates

Some of these assumptions may never be true, but MPT is still very useful.

## Examples of applying modern portfolio theory

An example of the application of MPT relates to the expected return of the investment portfolio. MPT shows that the overall expected return of a portfolio is the weighted average of the expected return of a single asset itself. For example, suppose an investor has a two-asset investment portfolio worth $1 million. The expected return of asset X is 5%, and the expected return of asset Y is 10%. The asset X of the portfolio is 800,000 USD and the asset Y is 200,000 USD. Based on these figures, the expected return of the investment portfolio is:

Portfolio expected return = ((800,000 USD / 1 million USD) x 5%) + ((200,000 USD / 1 million USD) x 10%) = 4% + 2% = 6%

If the investor wants to increase the expected return of the investment portfolio to 7.5%, all the investor needs to do is to transfer the appropriate amount of capital from asset X to asset Y. In this case, the appropriate weight for each asset is 50%:

7.5% expected return = (50% x 5%) + (50% x 10%) = 2.5% + 5% = 7.5%

The same idea applies to risk. A risk statistic from MPT is called beta, which measures the sensitivity of the portfolio to market systemic risks, that is, the vulnerability of the portfolio to a wide range of market events. A Beta value of 1 means that the portfolio faces the same amount of systemic risk as the market. The higher the Beta value, the greater the risk, and the lower the Beta value, the lower the risk. Suppose an investor has a $1 million investment portfolio and invests in the following four assets:

Asset A: Beta is 1, investment of 250,000 USD

Asset B: 1.6 Beta, investment of 250,000 USD

Asset C: Beta is 0.75, investment is 250,000 USD

Asset D: Beta is 0.5, investment is 250,000 USD

The portfolio beta coefficient is:

Beta = (25% x 1) + (25% x 1.6) + (25% x 0.75) + (25% x 0.5) = 0.96

The beta coefficient of 0.96 means that the systematic risk of the portfolio is similar to that of the market as a whole. Suppose the investor wants to take more risks, hope to get more returns, and decides that a beta of 1.2 is ideal. MPT means that by adjusting the weight of these assets in the portfolio, the required beta can be achieved. This can be done in many ways, but here is an example that demonstrates the desired result:

5% is transferred from asset A, and 10% is transferred from asset C and asset D. Invest this capital in asset B:

New beta version = (20% x 1) + (50% x 1.6) + (15% x 0.75) + (15% x 0.5) = 1.19

With some changes in the weight of the portfolio, the required beta can be almost perfectly achieved. This is a key insight of MPT.

.