# Risk and Return

Risk-averse investors attempt to maximize the return they earn per unit of risk. Ratios such as Sharpe ratio, Treynor’s ratio, Sortino ratio, etc. and coefficient of variation measure return per unit of investment risk.

It is important to analyze and attempt to quantify the relationship between risk and return. Modern portfolio theory and the capital asset pricing model offers a framework for analysis of risk-return trade-off.

Risk in equity investments is broadly classified into unique risk (also called unsystematic risk) and systematic risk.

Total Risk = Unique Risk + Systematic Risk

## Unique risk vs systematic risk

Unique risk is the risk that arises from investment-specific factors. By adding 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.

Since an individual investment is not diversified, it carries total risk and the appropriate measure of risk is the standard deviation. On the other hand, the risk of a fully diversified portfolio is best measured by its beta coefficient.

## Individual investment risk and return

### 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:

$$\text{E}(\text{R})=\text{r} _ \text{1}\times \text{p} _ \text{1}+\text{r} _ \text{2}\times \text{p} _ \text{2}+\text{...}+\text{r} _ \text{n}\times \text{p} _ \text{n}$$

Where,
E(R) is the expected return on individual asset,
r1, r2 and rn are the first, second and nth return outcomes, and
p1, p2 and pn are the associated probabilities.

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

$$\text{r}=\frac{\text{P} _ \text{t}-\text{P} _ {\text{t}-\text{1}}+\text{i}}{\text{P} _ {\text{t}-\text{1}}}$$

Where,
Pt is the ending value of investment,
Pt-1 is the beginning value of investment, and
i is its income.

### Variance, standard deviation, and beta of individual investment

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

$$\sigma=\sqrt{\frac{\sum{(\text{x} _ \text{i}\ -\ \mu)}^\text{2}}{\text{n}}}$$

Where,
σ is standard deviation,
xi 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:

$$\sigma=\sqrt{\sum{(\text{r} _ \text{n}\ -\ \text{E}(\text{R}))}^\text{2}\times \text{p} _ {\ \text{n}}}$$

Where,
σ is standard deviation,
rn is the nth return outcome,
E(R) is the expected return i.e. the probability-weighted average return, and
pn is the associated probability.

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.

## Portfolio risk and return

### Expected return of a portfolio of investments

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

$$\text{E} _ \text{r}=\text{w} _ \text{1}\times \text{R} _ \text{1}+\text{w} _ \text{2}\times \text{R} _ \text{2}+\text{....}+\text{w} _ \text{n}\times \text{R} _ \text{n}$$

Where,
Er is the portfolio expected return,
w1 is the weight of first asset in the portfolio,
R1 is the expected return on the first asset,
w2 is the weight of second asset, and
R2 is the expected return on the second asset and so on.

### Portfolio risk: variance, standard deviation & beta

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:

$$\sigma _ \text{P}=\sqrt{{\text{w} _ \text{A}}^\text{2}{\sigma _ \text{A}}^\text{2}{+\text{w} _ \text{A}}^\text{2}{\sigma _ \text{A}}^\text{2}+\text{2}\times \text{w} _ \text{A} \text{w} _ \text{B}\sigma _ \text{A}\sigma _ \text{B}\rho}$$

Where,
σP = portfolio standard deviation,
wA = weight of asset A in the portfolio,
wA = weight of asset B in the portfolio,
σA = standard deviation of asset A,
σB = standard deviation of asset B, and
ρ = correlation coefficient between returns on asset A and asset B.

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

Portfolio beta is the weighted average beta coefficient of the portfolio's constituent securities.

## Diversification ratio

Diversification ratio ® is the extent of diversification of an investment portfolio. It is calculated by dividing the weighted average volatility (standard deviation) of the constituent investments divided by portfolio standard deviation.

$$\text{Diversification Ratio}=\frac{\text{w} _ \text{A}\times\sigma _ \text{A}+\text{w} _ \text{B}\times\sigma _ \text{B}+\text{...}+\text{w} _ \text{n}\times\sigma _ \text{n}}{\sigma _ \text{P}}$$

Since the portfolio standard deviation in a diversified portfolio is lower than the weighted average of individual investment standard deviations, the ratio is greater than 1. A higher ratio is better.