# Risk and Return

Investments that pose higher risk (as measured by standard deviation and beta coefficient) have higher expected return and vice versa. Investments with higher Sharpe ratio or Treynor’s ratio i.e. high return per unit of risk are preferred by a risk-averse investor.

It is important to analyze and attempt to quantity the relationship between risk and return. Modern portfolio theory and capital asset pricing model offer a framework for analysis of risk-return tradeoff.

Risk in equity investments is broadly classified into unique risk (also called unsystematic risk) and systematic risk. Unsystematic risk i.e. unique risk is the risk that arises from investment-specific factors. By adding more and more investments to a portfolio, unsystematic risk can be eliminated, hence, it is also called diversifiable risk. Unsystematic risk can be further classified into business risk and financial risk. Business risk is the risk of loss in business while financial risk is the risk of default due to the company taking on too much debt. The systematic risk, on the other hand, is the risk of the whole economy and financial market performing poorly due to economy-wide factors. Its component risks are interest rate risk, inflation risk, regulatory risk, exchange rate risk, etc.

## Expected Return, Standard Deviation and Variance of Individual Assets

### Expected Return of Individual Investment

Expected return of an individual investment can be estimated by creating a distribution of likely return and their associated likelihood using the following formula:

$$ E(R)=r_1\times p_1+r_2\times p_2+...+r_n\times p_n $$

Where E(R) is the expected return on individual asset, r_{1}, r_{2} and r_{n} are the first, second and nth return outcomes and p_{1}, p_{2} and p_{n} are the associated probabilities.

The percentage return on an individual investment can be calculating using the following holding period return formula:

$$ r=\frac{P_t-P_{t-1}+i}{P_{t-1}} $$

Where P_{t} is the ending value of investment, P_{t-1} is the beginning value of investment and i is its income.

### Standard Deviation of Returns of Individual Investment

Historical standard deviation of an investment on a standalone basis can be estimated by calculating its standard deviation using the following general formula:

$$ \sigma=\sqrt{\frac{\sum{(x_i\ -\ \mu)}^2}{n}} $$

Where σ is standard deviation, x_{i} is return value, µ is the mean return value and n is the total number of observations.

Forward-looking standard deviation is usually calculated based on the expected return and their associated probability distribution using the following formula:

$$ \sigma=\sqrt{\sum{(r_n\ -\ E(R))}^2\times p_{\ n}} $$

Where σ is standard deviation, r_{n} is the nth return outcome, E(R) is the expected return i.e. the probably-weighted average return and p_{n} is the associated probability.

### Variance of Returns of Individual Investment

Variance of an individual investment simply equals squared standard deviation. In most cases standard deviation is a better measure.

The ratio of standard deviation of investment to its return is called coefficient of variation and it gives us an indication of total risk per unit of return.

## Expected Return, Standard Deviation and Variance of a Portfolio

### Portfolio Expected Return

Expected return of a portfolio is calculated as the weighted average of the expected return on individual investments using the following formula:

$$ E_r=w_1\times R_1+w_2\times R_2+....+w_n\times R_n $$

Where E_{r} is the portfolio expected return, w_{1} is the weight of first asset in the portfolio, R_{1} is the expected return on the first asset, w_{2} is the weight of second asset and R_{2} is the expected return on the second asset and so on.

### Portfolio Standard Deviation

Standard deviation of a portfolio depends on the weight of each asset in the portfolio, standard deviation of individual investments and their mutual correlation coefficient. If two and more assets have correlation of less than 1, the portfolio standard deviation is lower than the weighted average standard deviation of the individual investments. This is because less than perfect correlation causes certain unique company-specific risks to cancel out each other. Portfolio standard deviation is calculated as follows:

$$ \sigma_P=\sqrt{{w_A}^2{\sigma_A}^2{+w_A}^2{\sigma_A}^2+2\times w_Aw_B\sigma_A\sigma_B\rho} $$

σ_{P} = portfolio standard deviation

w_{A} = weight of asset A in the portfolio

w_{A} = weight of asset B in the portfolio

σ_{A} = standard deviation of asset A

σ_{B} = standard deviation of asset B

ρ = correlation coefficient between returns on asset A and asset B.

### Portfolio Variance

Portfolio variance simply standard deviation raised to the power of 2.