**Diversifying Your Portfolio In Montreal’s Financial Landscape With Forex** – After completing this reading, you should be able to: Explain mathematics and summary statistics in portfolios. Calculate the risk and return of an asset, given appropriate inputs. Calculate the risk and expected return of a portfolio of multiple risky assets, given the expected return, correlation and return correlation of individual assets. Perform mean-variance analysis. Understand the mean-standard deviation diagram and the resulting efficient market frontier. Calculate the optimal portfolio and determine the location of the capital market curve. Understand how portfolio risk can be reduced by diversifying across multiple securities or across multiple asset classes. The Challenge Facing Rational Investors

In a given market, an investor is presented with different assets/securities with different levels of return based on the underlying risk.

## Diversifying Your Portfolio In Montreal’s Financial Landscape With Forex

A practical consideration faced by investors is the best choice of asset or combination of assets or security investments that show an optimal balance between risk and investment return and maximize their expected use.

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Harry Markowitz conceptualized the Mean-Variance Portfolio Theory, also known as The Modern Portfolio Theory, in 1952. Through the concepts presented in the theory, investors can draw practical guidelines for building investment portfolios that maximize of their expected return based on a given level of risk.

We define some of the basic terms that we will use in the context of Modern Portfolio Theory:

Risk: In the context of MPT, risk can be defined as the difference or deviation of the investment return from the expected level.

Opportunity set: This is the set of available portfolios that an investor can choose based on their combinations of risk and return.

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Diversification: Diversification is the process of mixing different assets within a portfolio to ensure that unsystematic risk is smoothed out. In this case, the negative performance of a given asset/security within the portfolio is offset by the positive performance of other assets within the portfolio.

The principle theory behind the concept of diversification is that investors should hold portfolios and focus on the relationship between individual securities within the portfolio.

The assumption made in the theory is that investment decisions are made only with respect to the mean and variance of investment returns.

Thus, when applying the MPT framework to the selection of investment portfolios, the investor must consider the properties (ie, the risk and return) of the available investment portfolios, the

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In general, MPT attempts to explain a method of constructing a portfolio that will generate a maximum return for a given level of risk or a minimum risk for a stated return using the investor’s utility, which is assumed to be known. The investor tries to find the best balance between the return and the risk associated with the investment.

Although MPT has some limitations, it continues to be a standard for portfolio managers by using statistical methods to show investors the benefits of diversification.

Since a portfolio is a collection of assets or securities, it is clear that in order to find the expected return of a portfolio, we need to know the mean and the variance and covariance of each security included in the collection. This brings us to the mathematics and statistics of portfolios.

In choosing a set of portfolios, where we are determined to find efficient portfolios, it is assumed that investors make their decisions based on the expected return and the difference in returns per unit of time. So, we can stick to these two steps. One point to note here, however, is that the variance of any portfolio equals its risk.

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Let (R_i) represent the return generated by asset (A) in state (i) and (p_i) represent the probability that state (i) will occur.

In the next 1 year, the return on stock J is expected to be 8% with probability 0.7 and 10% with probability 0.3.

The risk of an asset is measured by the standard deviation. Let (R_i) represent the return generated by asset (A) in state (i) and let (p_i) represent the probability that state (i) will occur. The standard deviation of the asset, can be calculated as below:

In the next 1 year, the return on stock J is expected to be 8% with probability 0.7 and 10% with probability 0.3.

## Portfolio Risk: Analytical Methods

Suppose an investor has $1 to invest in any available securities in the market and that (w_j) represents the proportion of available funds invested in asset (j) where (j = 1, 2 , ddots, N).

If we define the return of portfolios by (R_p), then the return of the portfolio is given by:

Now denote the expected return of a portfolio by (E_p), the expected return of a portfolio is given by:

Then it is clear that the expected return of a portfolio is the weighted mean of the expected returns of individual securities.

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An investor builds a portfolio of two assets, Asset A and asset B. The expected return on asset A is 8%, and the Expected return on asset B is 10%.

Calculate the Expected return of the portfolio if 65% of the funds are invested in asset A and the rest in asset B:

The portfolio standard deviation or variance, which is simply the square of the standard deviation, consists of two key components: the difference in the underlying assets and the covariance of each underlying asset pair as shown above.

Also, due to the portfolio variance equation above, we can know that the lower the covariance between the assets/securities returns, the lower the total variance of the portfolios. This means that investing in assets/securities whose returns are uncorrelated will reduce the variance of a portfolio, which is the goal of diversification.

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Consider a two-asset portfolio where asset A has an allocation of 80% and a standard deviation of 16%, and asset B has an allocation of 20% and a standard deviation of 25%. The correlation coefficient between assets A and B is 0.6.

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When presented with a range of returns with their respective probabilities, the expected return can be calculated using the following formula where (r_i) and (p_i) represent the individual returns and their probabilities, respectively.

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Note: The risk of the portfolio is less than the risk of each individual asset which promotes the benefit of diversification.

We have examined so far how we calculate the expected return and variance of a portfolio given a collection of assets. Next is understanding how an investor can select an efficient portfolio from a given opportunity set.

In theory, a portfolio is made up of all investable assets. However, this is not practical, and thus, we need to find a way to filter the investable universe.

Ie, an investor seeks to find the combination of portfolio assets that minimizes risk for a given level of return, or, maximizes return for a given level of risk.

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Let’s consider an investor who is presented with securities A and B below and has to make an investment decision based on their return and risk. How does the investor choose a portfolio of these two securities?

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Assume that 40% of the total fund is invested in security A and 60% is invested in security B. The expected return on the portfolio of these two assets can be calculated as:

Assuming that the two securities are uncorrelated, i.e., with a correlation coefficient of zero, we can calculate the variance of the portfolio as:

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We observed that investing in two securities reduces the volatility of the return of security B. However, the allocation above seems inefficient because it is possible to build a portfolio of two assets in a way that leads to -ot in better terms for the investor in terms of risk and return. An investor gets the best return for the least risk.

Specifically, we are trying to create an efficient portfolio. For example, we can consider investing 60% of the total fund in security A and 40% in security B, resulting in a total portfolio standard deviation of 18.44% and an expected return of 12.40%. Thus, we can obtain an investment opportunity set of different portfolio compositions that match the expected utility of the investor by changing the allocation of the underlying assets. Here the different blue lines represent different investor’s utility functions:

To find the optimal portfolio, we use the investor’s indifference curve. It consists of different combinations of risky assets that lead to specific portfolio risk-return characteristics, graphically plotted with portfolio expected return on the y-axis and portfolio standard deviation on the x-axis.

For each level of return, the portfolio with the least risk will be chosen by a risk-averse investor. This risk reduction for each level of return creates a minimum-variance frontier – a collection of all minimum

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