Risk and Return: Types, Measurement, Portfolio Theory

Key Concepts and Definitions

What is Risk?

Risk in finance refers to the uncertainty about the future returns of an investment. It is the possibility that actual returns will differ from expected returns. Risk can be measured in different ways, such as standard deviation or variance of returns.

What is Return?

Return is the gain or loss an investor earns from an investment over a specific period. It can be expressed as:

• Absolute return: The total gain or loss in monetary terms.
• Percentage return: The gain or loss as a percentage of the initial investment.

Types of Risk

1. Market Risk (Systematic Risk)

Market risk is the risk associated with the entire market or economy. It cannot be diversified away.

• Example: A recession causing stock prices to fall across all sectors.

2. Unsystematic Risk (Idiosyncratic Risk)

This risk is specific to a particular company or industry and can be reduced through diversification.

• Example: A strike at a factory reducing its production.

3. Default Risk

The risk that a borrower will fail to repay a loan or meet a financial obligation.

• Example: A company failing to pay interest on its bonds.

4. Liquidity Risk

The risk that an asset cannot be sold quickly without a significant loss in value.

• Example: Real estate investments are less liquid compared to stocks.

5. Inflation Risk

The risk that inflation will erode the purchasing power of returns.

• Example: If an investment yields 5% but inflation is 6%, the real return is negative.

Measurement of Risk and Return

Measures of Return

1. Arithmetic Mean Return

   • Calculated as the sum of periodic returns divided by the number of periods.
   • Formula: $$\text{Arithmetic Mean}=\frac{{\sum }_{i=1}^n{R}_i}{n}$$
   • Example: If an investment yields returns of 10%, -5%, and 15% over three years, the arithmetic mean is: $$\frac{10+(-5)+15}{3}=\frac{20}{3}=6.67\%$$

2. Geometric Mean Return

   • Accounts for compounding and is more accurate for long-term investments.
   • Formula: $$\text{Geometric Mean}={\left(\prod _{i=1}^n(1+R_i)\right)}^{\frac{1}{n}}-1$$
   • Example: Using the same returns (10%, -5%, 15%), the geometric mean is: $${\left((1+0.10)\times (1-0.05)\times (1+0.15)\right)}^{\frac{1}{3}}-1=6.37\%$$

Measures of Risk

1. Standard Deviation (σ)

   • Measures the dispersion of returns around the mean.
   • Formula: $$\sigma =\sqrt{\frac{{\sum }_{i=1}^n(R_i-\bar{R}{)}^2}{n-1}}$$
   • Example: If returns are 10%, 12%, and 8%, and the mean is 10%, the standard deviation is calculated as: $$\sigma =\sqrt{\frac{(10-10{)}^2+(12-10{)}^2+(8-10{)}^2}{3-1}}=\sqrt{\frac{0+4+4}{2}}=\sqrt{4}=2\%$$

2. Variance

   • The square of standard deviation, representing the squared dispersion of returns.

3. Coefficient of Variation (CV)

   • Measures risk per unit of return.
   • Formula: $$CV=\frac{\sigma }{\bar{R}}$$
   • Example: If an investment has a standard deviation of 10% and a mean return of 20%, the CV is: $$CV=\frac{10}{20}=0.5$$

Risk-Return Tradeoff

Investors generally expect higher returns for taking on more risk. This relationship is visualized in the Risk-Return Tradeoff Curve:

graph LR
    A["Low Risk"] -->|"Investment"| B["Low Return"]
    C["High Risk"] -->|"Investment"| D["High Return"]
    A -->|"Tradeoff"| C
    B -->|"Tradeoff"| D

Example: Risk-Return Tradeoff in Nepal

Consider two investment options in Nepal:

1. Nepal Government Bonds: Low risk, low return (~8%).
2. Nepal Stock Exchange (NEPSE) Index Fund: Higher risk, higher return (historically ~12-15%).

Portfolio Theory and Diversification

Diversification

Diversification reduces unsystematic risk by spreading investments across different assets. The key principle is:

"Don’t put all your eggs in one basket."

Portfolio Return and Risk

• Portfolio Return: The weighted average of individual asset returns. $$R_p=\sum _{i=1}^n{w}_i{R}_i$$ where $w_i$ is the weight of asset $i$ in the portfolio.

• Portfolio Risk: Depends on the covariance between asset returns. $${\sigma }_p=\sqrt{\sum _{i=1}^n\sum _{j=1}^n{w}_i{w}_j{\sigma }_i{\sigma }_j{\rho }_{ij}}$$ where ${\rho }_{ij}$ is the correlation coefficient between assets $i$ and $j$.

Efficient Frontier

The Efficient Frontier is a graph showing the combination of assets that offer the highest expected return for a given level of risk. It is derived from portfolio theory.

graph TD
    A["Minimum Variance Portfolio"] -->|"Risk"| B["Efficient Frontier"]
    C["Highest Return for Given Risk"] --> B
    D["Risk-Free Asset"] -->|"Combination"| E["Capital Market Line"]

Capital Asset Pricing Model (CAPM)

CAPM explains the relationship between risk and expected return for assets in financial markets. It introduces the concept of beta (β), which measures an asset's sensitivity to market movements.

CAPM Formula

$$E(R_i)=R_f+{\beta }_i(E(R_m)-R_f)$$

• $E(R_i)$: Expected return of the asset.
• $R_f$: Risk-free rate (e.g., government bond yield).
• ${\beta }_i$: Beta of the asset.
• $E(R_m)$: Expected return of the market.

Example: Calculating Expected Return Using CAPM

Assume:

• Risk-free rate ($R_f$) = 5%
• Market return ($E(R_m)$) = 10%
• Beta of a stock (${\beta }_i$) = 1.2

The expected return of the stock is: $$E(R_i)=5\%+1.2\times (10\%-5\%)=5\%+6\%=11\%$$

In the Real World

1. eSewa and Khalti (Digital Payments)

   • Risk: Cybersecurity threats (e.g., hacking, fraud) and regulatory risks (e.g., changes in government policies).
   • Return: Investors in fintech companies like Khalti or eSewa expect high returns due to the growing digital payment market in Nepal, but they must manage risks like data breaches.

2. Nepal Stock Exchange (NEPSE)

   • Risk: Market risk (e.g., economic downturns) and unsystematic risk (e.g., poor performance of individual companies like NMB Bank or Global IME).
   • Return: Investors in NEPSE stocks earn higher returns compared to fixed deposits but face volatility.

3. Pathao (Ride-Hailing App)

   • Risk: Operational risk (e.g., driver shortages, fuel price fluctuations) and competition.
   • Return: Pathao’s investors expect high returns due to its market dominance, but they must diversify to mitigate risks like regulatory changes.

Worked Example: Risk and Return for a Nepali Retail Shop

Consider Kathmandu Retail Shop, which invests in two assets:

1. Fixed Deposit (FD): 8% return, 0% risk (risk-free).
2. NEPSE Index Fund: 12% expected return, 15% standard deviation.

Step 1: Calculate Portfolio Return

Assume the shop invests 60% in FD and 40% in the NEPSE Index Fund. $$R_p=(0.60\times 8\%)+(0.40\times 12\%)=4.8\%+4.8\%=9.6\%$$

Step 2: Calculate Portfolio Risk (Assuming Zero Correlation)

If the correlation between FD and the index fund is 0: $${\sigma }_p=\sqrt{(0.60^2\times 0^2)+(0.40^2\times 15^2)+2\times 0.60\times 0.40\times 0\times 15\times 0}=0.40\times 15=6\%$$

Step 3: Compare with Undiversified Portfolio

If the shop invests 100% in the NEPSE Index Fund:

• Return = 12%
• Risk = 15%

Diversification reduces risk from 15% to 6% while lowering the return slightly from 12% to 9.6%.

Comparison Table: Risk and Return Metrics

Advantages and Disadvantages of Diversification

Advantages

• Reduces Unsystematic Risk: Spreading investments across assets lowers company-specific risks.
• Smoother Returns: Portfolios with diversified assets tend to have more stable returns.
• Better Risk-Return Tradeoff: Investors can achieve higher returns for a given level of risk.

Disadvantages

• Complexity: Managing multiple assets can be time-consuming and complex.
• Diversification Does Not Eliminate All Risk: Systematic risk (market risk) remains.
• Transaction Costs: Buying and selling multiple assets can incur fees.

Exam Tip

1. Understand Definitions: Clearly define risk, return, and key terms like beta, standard deviation, and diversification.
2. Formulas: Memorize and apply formulas for arithmetic/geometric mean, standard deviation, and CAPM.
3. Real-World Application: Relate concepts to Nepali examples (e.g., NEPSE, banks, fintech companies).
4. Diagrams: Draw the Efficient Frontier and Risk-Return Tradeoff Curve in exams.
5. Numerical Problems: Practice calculating portfolio returns and risks using given weights and correlations.
6. CAPM: Know how to use CAPM to calculate expected returns for stocks or projects.