4.4 Investment Return, Risk Metrics, and Portfolio Principles

4.4 Investment Return, Risk Metrics, and Portfolio Principles

Every investment decision in the capital markets embodies a fundamental trade-off: risk versus return. Rational investors demand higher expected returns to compensate for accepting greater uncertainty. For securities professionals preparing for the Philippine SEC Certification Examination Phase 1, understanding how to calculate historical returns, quantify dispersion, decompose systematic and unsystematic risks, and apply portfolio diversification principles is vital for advising retail and institutional clients.

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Investment Return Measurements

Return measures the financial gain or loss generated on an investment relative to the capital committed.

Holding Period Return (HPR)

Holding Period Return (HPR) represents the total return generated over the entire timeframe an asset is held, irrespective of the calendar length of that period. Total return comprises two distinct components:

1. Capital Gain/Loss Yield: Price change relative to initial cost ($[P_1 - P_0] / P_0$).
2. Income Yield (Dividend or Coupon Yield): Cash distributions received ($D / P_0$).

The combined formula is: $HPR = \frac{P_1 - P_0 + D}{P_0} = \frac{P_1 - P_0}{P_0} + \frac{D}{P_0}$

Where:

• $P_0$ = Initial purchase price
• $P_1$ = Ending market price (or sale price)
• $D$ = Total cash dividends or interest coupons received during the holding period

Annualized Return

Because holding periods vary across investments (e.g., 90 days vs. 3 years), returns must be annualized to enable valid comparisons:

• Compound Annual Growth Rate (CAGR): $\text{Annualized Return} = (1 + HPR)^{1/t} - 1 = (1 + HPR)^{\frac{365}{\text{Days}}} - 1$

Arithmetic vs. Geometric Mean Return

When evaluating historical investment performance over multiple consecutive years, analysts use two divergent averaging methodologies:

• Arithmetic Mean: The simple numerical average of annual returns: $\bar{R}_{\text{arithmetic}} = \frac{\sum_{t=1}^n R_t}{n}$
• Geometric Mean (Compound Annual Return): The constant rate that compounds the initial capital to the final terminal wealth: $R_{\text{geometric}} = \left[ \prod_{t=1}^n (1 + R_t) \right]^{1/n} - 1 = \left[ (1 + R_1)(1 + R_2)\cdots(1 + R_n) \right]^{1/n} - 1$

The Mathematical Volatility Drag

Whenever annual returns exhibit volatility, the geometric mean is strictly less than the arithmetic mean ($R_{\text{geom}} \le R_{\text{arith}}$). The difference between them widens as return volatility increases.

Classic Illustrative Example: An investor purchases a PSE-listed mining stock at Php 100:

• In Year 1, the stock surges +50% to Php 150.
• In Year 2, the stock plunges -50% to Php 75.
• Arithmetic Mean: $[(+50\%) + (-50\%)] / 2 = \mathbf{0.0\%}$
• Actual Investor Wealth: The investor started with Php 100 and ended with Php 75, suffering an actual -25% total loss over 2 years.
• Geometric Mean: $\sqrt{(1 + 0.50)(1 - 0.50)} - 1 = \sqrt{1.50 \times 0.50} - 1 = \sqrt{0.75} - 1 = 0.8660 - 1 = \mathbf{-13.40\%}$ per year.

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Quantifying Investment Risk: Variance and Standard Deviation

Risk in modern finance is defined as the dispersion, uncertainty, or volatility of actual outcomes around the expected mean return.

Expected Return

Given a set of $n$ mutually exclusive economic scenarios, each with probability $p_i$ and anticipated return $R_i$, the Expected Return ($E(R)$) is: $E(R) = \sum_{i=1}^n p_i R_i$

Variance and Standard Deviation

• Variance ($\sigma^2$): The probability-weighted sum of squared deviations from the expected return: $\sigma^2 = \sum_{i=1}^n p_i \left[ R_i - E(R) \right]^2$
• Standard Deviation ($\sigma$): The square root of variance, returning risk to the same percentage units as the return: $\sigma = \sqrt{\sigma^2} = \sqrt{\sum_{i=1}^n p_i \left[ R_i - E(R) \right]^2}$

Under a normal distribution curve:

• 68.26% of annual returns fall within $E(R) \pm 1\sigma$
• 95.44% of annual returns fall within $E(R) \pm 2\sigma$
• 99.74% of annual returns fall within $E(R) \pm 3\sigma$

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Systematic vs. Unsystematic Risk

Total risk (measured by standard deviation $\sigma$) consists of two distinct components: $\text{Total Risk} = \text{Systematic (Market) Risk} + \text{Unsystematic (Firm-Specific) Risk}$

Portfolio Risk (σ)
      ▲
      │     Total Risk
      │    .---------------
      │   /                
      │  /  Unsystematic   (Diversifiable Risk - reduced through broad diversification)
      │ /   Risk           
      │/─────────────────── Constant Systematic Risk (Non-diversifiable)
      │
      └────────────────────────► Number of Stocks in Portfolio

Comparison of Risk Dimensions

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Beta ($\beta$) and the Capital Asset Pricing Model (CAPM)

Because investors can diversify much firm-specific risk at comparatively low cost, equilibrium pricing models such as CAPM focus on non-diversifiable systematic risk. Beta ($\beta$) measures the sensitivity of a security's returns relative to movements in the broad market benchmark (such as the Philippine Stock Exchange Index - PSEi).

Calculating Beta

$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \frac{\rho_{i,m} \sigma_i \sigma_m}{\sigma_m^2} = \rho_{i,m} \times \frac{\sigma_i}{\sigma_m}$

Where:

• $\text{Cov}(R_i, R_m)$ = Covariance between security $i$ and market $m$
• $\sigma_m^2$ = Variance of the market benchmark
• $\rho_{i,m}$ = Correlation coefficient between security $i$ and the market
• $\sigma_i, \sigma_m$ = Standard deviations of security $i$ and the market

Interpreting Beta Values:

• $\mathbf{\beta = 1.0}$: The stock possesses average market risk; moves in tandem with the PSEi.
• $\mathbf{\beta > 1.0}$: Aggressive Stock. More volatile than the market (e.g., property developers, tech startups with $\beta = 1.4$).
• $\mathbf{\beta < 1.0}$: Defensive Stock. Less volatile than the market (e.g., regulated electric utilities like Meralco, consumer staple food manufacturers with $\beta = 0.7$).
• $\mathbf{\beta = 0.0}$: Asset is completely uncorrelated with market movements (e.g., default-free Philippine Treasury bills).
• $\mathbf{\beta < 0.0}$: Rare asset moving inversely to the broad market (e.g., gold mining equities or inverse hedging instruments).

The CAPM Equation

Formulated by William Sharpe, John Lintner, and Jan Mossin, the Capital Asset Pricing Model (CAPM) determines the required rate of return for any risky asset based on its systematic risk: $E(R_i) = R_f + \beta_i \times \left[ E(R_m) - R_f \right]$

Where:

• $E(R_i)$ = Required or expected rate of return on security $i$
• $R_f$ = Risk-free rate of return (conventionally proxied in the Philippines by benchmark sovereign debt, such as 91-day T-Bills or 10-year FXTNs)
• $E(R_m)$ = Expected return on the market portfolio (represented by the PSEi)
• $\left[ E(R_m) - R_f \right]$ = Market Risk Premium (MRP), representing the excess compensation investors demand for holding equities over riskless sovereign debt
• $\beta_i \times \left[ E(R_m) - R_f \right]$ = Asset's individual risk premium

Jensen's Alpha ($\alpha$)

Alpha ($\alpha$) measures excess return earned beyond what is predicted by the CAPM: $\alpha = R_{\text{actual}} - E(R_i)$

• $\alpha > 0$: Security generated superior risk-adjusted return (undervalued, beat the benchmark).
• $\alpha < 0$: Security underperformed given its systematic risk level.

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Risk-Adjusted Performance: The Sharpe Ratio

Evaluating performance purely on nominal returns is dangerous; a portfolio manager who generates 15% return by taking triple the risk of the market may be destroying risk-adjusted value.

The Sharpe Ratio Formula

Developed by Nobel laureate William Sharpe, the Sharpe Ratio measures the excess return generated per unit of total risk (standard deviation): $\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}$

Where:

• $R_p$ = Realized or expected portfolio return
• $R_f$ = Risk-free rate
• $\sigma_p$ = Standard deviation of portfolio returns (total risk)

Sharpe Ratio vs. Treynor Ratio

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Modern Portfolio Theory (MPT) and Diversification

Introduced by Harry Markowitz in 1952, Modern Portfolio Theory (MPT) mathematically proved that investors can reduce portfolio risk without sacrificing expected return by combining assets with imperfect correlations.

Two-Asset Portfolio Expected Return and Variance

• Expected Return of a Two-Asset Portfolio: $E(R_p) = w_1 E(R_1) + w_2 E(R_2)$
• Variance of a Two-Asset Portfolio: $\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \text{Cov}_{1,2}$ $\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_1 \sigma_2 \rho_{1,2}$

Where $w_1, w_2$ are portfolio weights ($w_1 + w_2 = 1$), and $\rho_{1,2}$ is the correlation coefficient between Asset 1 and Asset 2 (ranging from $-1.0$ to $+1.0$).

The Power of Correlation ($\rho$)

• Perfect Positive Correlation ($\rho = +1.0$): No risk reduction occurs; portfolio standard deviation is simply the weighted average of individual asset standard deviations.
• Imperfect Correlation ($\rho < +1.0$): Diversification benefits occur. Portfolio standard deviation is strictly less than the weighted average of individual standard deviations.
• Perfect Negative Correlation ($\rho = -1.0$): All risk can theoretically be completely eliminated, creating a riskless portfolio.

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Practical Exam Traps & Regulatory Context