An investor puts 2,000 into a deposit account with a fixed...

An investor puts $$\$ 2,000$$ into a deposit account with a fixed rate of return of $10 \%$ per year. A second sum of $$\$ 1,000$$ is invested in a fund with an expected rate of return of $16 \%$ and a standard deviation of $8 \%$ per year. a. Find the expected value of the total amount of money this investor will have after a year. b. Find the standard deviation of the total amount after a year.

An investor puts $$\$ 2,000$$ into a deposit account with a fixed rate of return of $10 \%$ per year. A second sum of $$\$ 1,000$$ is invested in a fund with an expected rate of return of $16 \%$ and a standard deviation of $8 \%$ per year.
a. Find the expected value of the total amount of money this investor will have after a year.
b. Find the standard deviation of the total amount after a year.

Key Concepts

Expected Value

Expected value is the average outcome of a random variable over a large number of trials. In finance, it represents the mean return of an investment, calculated by weighting each possible outcome by its probability. This concept helps investors estimate long?term returns of their portfolios.

Linearity of Expectation

The principle of linearity of expectation states that the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of whether the variables are independent. This property is essential in finance when evaluating the combined expected returns from different investments.

Standard Deviation

Standard deviation is a measure of how spread out values are around the mean, quantifying the uncertainty or variability in the outcomes of a financial asset. In investment analysis, it is used to assess the risk associated with returns, indicating how much the actual returns might deviate from the expected return.

Variance Aggregation in Portfolios

When assessing a portfolio composed of multiple investments, the overall risk (variance) is determined by how the variances of individual investments combine. For independent investments, the total variance is the sum of the individual variances, which helps in quantifying the portfolio's overall risk. This concept is fundamental in financial risk management and diversification.

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