Learn About Measuring Risk & Return of Mutual Fund

5.1 Risk Return Relationship

One of the most important principles in finance is that risk and return are linked. Investors expect higher returns only when they take on higher risk. If markets are efficient, every asset should offer a return that matches the level of risk it carries.

Understanding the Tradeoff

• Risk→ The chance that actual returns will differ from expected returns. Statistically, this is measured by standard deviation.
• Return→ The reward investors receive for bearing risk.
• Relationship→ Higher uncertainty means higher potential returns, but also higher chances of losses.

This balance is often described as the “comfort zone test.” Some investors are comfortable with volatile assets like cryptocurrencies, while others prefer the stability of government bonds.

Examples of Risk–Return Levels

Illustration

Suppose the risk-free rate (government bond yield) is 7%.

• An index fund averages 13% over time. The risk premiumhere is 6% (13% – 7%).
• But this 13% is not guaranteed. One year it may be –10%, another year +25%.

This shows that higher returns are possible, but they come with greater volatility.

Market Line Concept

Finance theory uses the Security Market Line (SML) to show the relationship between risk and return.

• The slope of the line represents the return per unit of risk.
• A steep slope indicates investors are highly risk-averse (they demand more return for taking risk).
• A flatter slope suggests investors are more willing to accept risk for lower compensation.

5.2 Portfolio and Security Returns

A portfolio is simply a collection of securities. Investors rarely put all their money into a single stock or bond; instead, they spread investments across multiple assets. This diversification helps balance risk and return. The expected return of a portfolio depends on:

1. The expected return of each security.
2. The proportion of funds invested in each security.

Thus, portfolio return is essentially a weighted average of the returns of its individual securities.

Example Portfolio – Latest Context

Suppose an investor holds shares in five companies:

Total Current Value = ₹4,09,000 Total Expected Value = ₹4,52,500

Value Relative Contribution

Portfolio Value Relative = 1.105 (≈ 10.5% growth)

Holding-Period Returns

Portfolio Expected Return = 10.58%

Insights

• The portfolio’s return is a weighted average of individual returns.
• Securities with higher portfolio weights (like Reliance and HDFC Bank) contribute more to overall return.
• If an investor wanted the highest possible return, they might invest only in Tata Motors (17%). But this would expose them to concentrated risk.

Diversification spreads risk across multiple securities, reducing volatility while still delivering a reasonable expected return.

5.3 Risk and Return Calculation

To understand how risk and return are calculated for a single security, let’s take the case of Infosys Ltd. over a 5-year period. We will consider both capital gains (price changes) and dividends (income) to compute the annual return.

Infosys Share Data

Step 1: Annual Returns

Average Return = (12.73 – 11.53 + 29.24 + 15.83) / 4 = 11.57%

Step 2: Risk (Standard Deviation)

We now assign probabilities to each year’s return to measure variability.

Variance = 205.104

Interpretation

• Average Return = 11.57%→ Infosys delivered a moderate long-term return.
• Standard Deviation = 14.32%→ Returns are volatile, showing significant ups and downs.
• The negative return in 2020 highlights the risk of equity investments, while 2021 shows the potential for high gains.

Comparative Illustration

5.4 Return Calculation of Portfolio ( Two Assets)

Portfolio Return Calculation – Two or More Assets

When an investor holds multiple securities, the portfolio’s expected return is simply the weighted average of the returns of the individual securities. The weight of each security is based on the proportion of funds invested in it.

Where:

• = Expected portfolio return
• = Proportion of investment in each security
• = Expected return of each security
• Sum of weights = 1

Example Portfolio – Latest Context

Suppose Mr. Arjun invests in six companies with the following allocations and expected returns:

Step 1: Weighted Average Return

So, the expected portfolio return = 14.5%.

Step 2: Contribution of Each Security

Insights

• The portfolio return is not dominated by one stock; it is the weighted sum of all.
• Reliance contributes the most (4%) because of its higher weight and decent return.
• Adani Enterprises has the highest individual return (20%), but since only 10% is invested, its contribution is limited to 2%.
• Diversification ensures that even if one stock underperforms, the portfolio return remains balanced.

Portfolio Risk and Return – Two Securities (New Example)

When analyzing a portfolio of two securities, we must consider not only the individual risks (variance or standard deviation of returns) but also the relationship between the securities. This relationship is captured through covariance and the correlation coefficient.

• Covariance (CovAB)measures how two securities move together.
• Correlation (ρAB)ranges between –1 and +1, showing the strength and direction of the relationship.
• Portfolio risk is reduced when securities are not perfectly correlated.

Example: Reliance Industries & HDFC Bank

Returns Data (5 Years)

Step 1: Mean Return & Standard Deviation

Reliance

• Mean Return = (12 + 15 – 5 + 20 + 18) / 5 = 12%
• Variance = [(0² + 3² + (–17)² + 8² + 6²)] / 5 = 338 / 5 = 67.6
• Standard Deviation = √67.6 = 8. 22%

HDFC Bank

• Mean Return = (8 + 10 – 2 + 14 + 12) / 5 = 8.4%
• Variance = [(–0.4² + 1.6² + (–10.4)² + 5.6² + 3.6²)] / 5 = 146 / 5 = 29.2
• Standard Deviation = √29.2 = 5.40%

Step 2: Covariance & Correlation

• Covariance = (248 / 5) = 49.6
• Correlation (ρAB) = CovAB / (σA × σB) = 49.6 / (8.22 × 5.40) =0.11

Interpretation: Reliance and HDFC Bank have a low positive correlation, meaning they don’t move exactly together. This allows diversification benefits.

Step 3: Portfolio Return & Risk

Suppose the portfolio weights are:

• Reliance = 60%
• HDFC Bank = 40%

Portfolio Return

R_p =(0.60 *12)+(0.40*8.4)=7.2+3.36 =10.56%

Portfolio Risk

σ²ₚ = (0.60² × 8.22²) + (0.40² × 5.40²) + (2 × 0.60 × 0.40 × 8.22 × 5.40 × 0.11)

σ²ₚ = (0.36 × 67.6) + (0.16 × 29.2) + (2 × 0.24 × 44.2 × 0.11)

σ²ₚ = 24.34 + 4.67 + 2.33 = 31.34

σₚ = √31.34 = 5.60%

Insights

• Reliance alone has a risk of 8.22%, HDFC Bank alone has 5.40%.
• The combined portfolio risk is 5.60%, lower than Reliance’s individual risk.
• Diversification reduces risk because the two securities are not perfectly correlated.

5.5 Return Calculation of Portfolio ( Two Assets)

Understanding Portfolio Risk

• Individual Security Risk: Measured by the varianceor standard deviation of its returns.
• Portfolio Risk: Not just a weighted average of individual risks. It depends on how securities move together, captured by covarianceand correlation.
• Covariance: Shows whether two securities’ returns move in the same direction.
• Correlation (rAB): Standardized measure of co-movement, ranging from –1 to +1.

Interpretation:

• rAB = 1 → No diversification benefit.
• rAB = –1 → Perfect diversification (unsystematic risk eliminated).
• rAB = 0 → No relationship between returns.

Example: Reliance Industries & ICICI Bank Returns

Mean & Standard Deviation of Reliance

• Mean Return = (14 – 2 + 8 + 18 + 22) / 5 = 12%
• Variance = (0 + 256 + 36 + 16 + 64) / 5 = 74.4
• Standard Deviation = √74.4 = 8.62%

Mean & Standard Deviation of ICICI Bank

• Mean Return = (9 + 6 + 11 + 13 + 17) / 5 = 11. 2%
• Variance = (4 + 25 + 0 + 4 + 36) / 5 = 13.8
• Standard Deviation = √13.8 = 3.71%

Covariance & Correlation

• Covariance = (0 + 80 + 0 + 8 + 48) / 5 = 27.2
• Correlation = CovAB / (σA × σB) = 27.2 / (8.62 × 3.71) = 0.85

Portfolio Return & Risk

Weights:

• Reliance = 60%
• ICICI Bank = 40%

Portfolio Return (Rp): = (0.60 × 12) + (0.40 × 11.2) = 11.68%

Portfolio Risk (σp):

σ²ₚ = (w²ₐσ²ₐ) + (w²ᵦσ²ᵦ) + (2wₐwᵦₐᵦrₐᵦ)

= (0.36 × 74.4) + (0.16 × 13.8) + (2 × 0.60 × 0.40 × 8.62 × 3.71 × 0.85) = 26.78 + 2.21 + 26.25 = 55.24

σp = √55.24 = 7.43%