Portfolio Analysis: Market Value, Returns, and Risk Management

Goal of Portfolio Analysis

  • The objective of this analysis is to replicate single-asset analysis at a portfolio level.

  • Key objectives include computing:     * Market Value (MV) of a portfolio.     * Portfolio returns using two distinct approaches.     * Expected (average) portfolio returns.     * Portfolio risk (Standard Deviation/Volatility).     * Understanding the concept and impact of Correlation.

Portfolio Foundational Data and Example Case Study

To illustrate portfolio concepts, a consistent example is used involving three firms: North Air, West Air, and Tex Oil. The following tables provide core data from 2004 to 2010.

Asset Prices and Dividends (2004-2010)

Year

North Air Price

North Air Div

West Air Price

West Air Div

Tex Oil Price

Tex Oil Div

2004

$100.00

-

$50.00

-

$50.00

-

2005

$120.00

$1.00

$54.00

$0.50

$49.00

$0.00

2006

$155.00

$1.00

$64.00

$1.34

$46.00

$0.55

2007

$164.00

$1.85

$67.00

$1.48

$49.00

$1.14

2008

$153.00

$2.80

$64.00

$1.66

$58.00

$1.29

2009

$147.00

$2.94

$59.00

$1.80

$74.00

$1.40

2010

$157.29

$2.94

$74.90

$1.80

$77.78

$1.40

Individual Realized Returns (2005-2010)

  • North Air:     * 2005: 21%21\%; 2006: 30%30\%; 2007: 7%7\%; 2008: −5%-5\%; 2009: −2%-2\%; 2010: 9%9\%     * Average (Expected) Return: 10%10\%     * Risk (Standard Deviation): 13.42%13.42\%

  • West Air:     * 2005: 9%9\%; 2006: 21%21\%; 2007: 7%7\%; 2008: −2%-2\%; 2009: −5%-5\%; 2010: 30%30\%     * Average (Expected) Return: 10%10\%     * Risk (Standard Deviation): 13.42%13.42\%

  • Tex Oil:     * 2005: −2%-2\%; 2006: −5%-5\%; 2007: 9%9\%; 2008: 21%21\%; 2009: 30%30\%; 2010: 7%7\%     * Average (Expected) Return: 10%10\%     * Risk (Standard Deviation): 13.42%13.42\%

Portfolio Market Value (MV)

  • Definition: The Market Value of a portfolio is the amount of money required to create the portfolio, or the amount an investor would receive if they liquidated (sold) the portfolio.

  • Formula: MVp=∑MVi\text{MV}_p = \sum \text{MV}_i (The sum of the market values of each individual asset in the portfolio).

  • Example Calculation: Purchase of 5050 shares of North Air and 100100 shares of West Air at 2004 year-end.     * 2004 Year-End:         * MV North Air: 50×$100.00=$5,00050 \times \$100.00 = \$5,000         * MV West Air: 100×$50.00=$5,000100 \times \$50.00 = \$5,000         * Portfolio MV (2004): $10,000\$10,000     * 2005 Year-End:         * MV North Air: 50×$120.00=$6,00050 \times \$120.00 = \$6,000         * MV West Air: 100×$54.00=$5,400100 \times \$54.00 = \$5,400         * Portfolio MV (2005): $6,000+$5,400=$11,400\$6,000 + \$5,400 = \$11,400

Portfolio Return: Approach 1 (Basic Definition)

  • Definition: Portfolio returns are profits earned on a portfolio for every dollar invested.

  • Sources of Profit:     * Capital Gains: Change in the Market Value of the Portfolio (MV<em>end−MV</em>beginning\text{MV}<em>{end} - \text{MV}</em>{beginning}).     * Income: Total dividends or interest generated by all assets in the portfolio.

  • Investment Basis: Measured by the MV of the portfolio at the beginning of the period. This is consistent with single-asset concepts.     * Constraint: Unlike single-asset calculations, you cannot assume the purchase of exactly one share/unit for a portfolio because it contains multiple assets.

  • Calculation Steps (2005 Example):     1. Calculate Total Profits:         * Capital Gains: $11,400−$10,000=$1,400\$11,400 - \$10,000 = \$1,400         * Income: (50 \times \1.00) + (100 \times \0.50)=$50+$50=$1000.50) = \$50 + \$50 = \$100         * Total Profit: $1,400+$100=$1,500\$1,400 + \$100 = \$1,500     2. Calculate Amount Invested: Beginning MV = $10,000\$10,000     3. Calculate Return: Return=Total ProfitsAmount Invested=$1,500$10,000=0.15 or 15%\text{Return} = \frac{\text{Total Profits}}{\text{Amount Invested}} = \frac{\$1,500}{\$10,000} = 0.15 \text{ or } 15\%

Portfolio Return: Approach 2 (Asset Weights and Returns)

  • Concept: A portfolio return depends on the return of each individual asset and how the investment is distributed among those assets.

  • Portfolio Weights Definition: The fraction of the portfolio's total market value invested in a specific asset at a specific time.

  • Weight Formula: wi=MV of AssetiMV of Portfoliow_i = \frac{\text{MV of Asset}_i}{\text{MV of Portfolio}}     * By definition, the sum of all weights must equal 11 (or 100%100\%).

  • Example (Weight Calculation for 2004 at Start):     * Weight North Air: $5,000$10,000=0.5\frac{\$5,000}{\$10,000} = 0.5     * Weight West Air: $5,000$10,000=0.5\frac{\$5,000}{\$10,000} = 0.5

  • Example (Weight Shift by end of 2005):     * Weight North Air: $6,000$11,400=0.5263\frac{\$6,000}{\$11,400} = 0.5263     * Weight West Air: $5,400$11,400=0.4737\frac{\$5,400}{\$11,400} = 0.4737     * North Air's weight increased because its price appreciation (20\%\, \text{increase from } \100 \text{ to } \120120) exceeded West Air's price appreciation (8\%\, \text{increase from } \50 \text{ to } \5454).     * Rebalancing: To keep weights at original 50/5050/50 levels, an investor must sell shares of the outperforming asset (North Air) and buy shares of the underperforming asset (West Air).

  • Return Formula: For nn assets with returns R1,R2,…,RnR_1, R_2, …, R_n and weights w1,w2,…,wnw_1, w_2, …, w_n:     * Rp=∑i=1nwiRi=w1R1+w2R2+…+wnRnR_p = \sum_{i=1}^{n} w_i R_i = w_1 R_1 + w_2 R_2 + … + w_n R_n

  • Example Calculation (2005 Return via Weights):     * Rp=(0.5×0.21)+(0.5×0.09)=0.105+0.045=0.15 or 15%R_p = (0.5 \times 0.21) + (0.5 \times 0.09) = 0.105 + 0.045 = 0.15 \text{ or } 15\%

Summary of Portfolio Return Methods

  • Both approaches yield the same mathematical result.

  • Why Investors Prefer Approach 2:     * Investors cannot control individual asset returns (market-driven).     * Investors can control asset weights (allocation-driven).     * The formula allows for direct modeling of expected returns and risk based on allocation decisions.

Realized Returns for Multi-Asset Portfolios

Year

North Air

West Air

Tex Oil

Portfolio 1 (50% North, 50% West)

Portfolio 2 (33.33% Each)

2005

21%21\%

9%9\%

−2%-2\%

15%15\%

9.33%9.33\%

2006

30%30\%

21%21\%

−5%-5\%

25.5%25.5\%

15.33%15.33\%

2007

7%7\%

7%7\%

9%9\%

7%7\%

7.67%7.67\%

2008

−5%-5\%

−2%-2\%

21%21\%

−3.5%-3.5\%

4.67%4.67\%

2009

−2%-2\%

−5%-5\%

30%30\%

−3.5%-3.5\%

7.67%7.67\%

2010

9%9\%

30%30\%

7%7\%

19.5%19.5\%

15.33%15.33\%

Avg

10%

10%

10%

10%

10%

Estimating Expected Portfolio Return

  • Definition: The average of all possible future returns, weighted by their probabilities.

  • Calculation Method: Calculated using the expected returns of individual assets (E[Ri]E[R_i]) and the asset weights.

  • Formula: E[Rp]=∑i=1nwiE[Ri]E[R_p] = \sum_{i=1}^{n} w_i E[R_i]

  • Example Calculation (50/50 North/West):     * Expected Return North Air = 10%10\%     * Expected Return West Air = 10%10\%     * E[Rp]=(0.5×10%)+(0.5×10%)=10%E[R_p] = (0.5 \times 10\%) + (0.5 \times 10\%) = 10\%

Portfolio Risk and Correlation

  • Definition of Portfolio Risk: The probability and magnitude by which actual returns differ from expected returns, measured by Standard Deviation (σ\sigma).

  • Factors Determining Portfolio Risk:     1. The risk (Standard Deviation) of individual assets.     2. The weights of each asset.     3. The Correlation between asset returns.

Correlation of Returns

  • Definition: Measures how returns of two assets move relative to each other on average.

  • Range: Correlation (ρ\rho) is always between −1.0-1.0 (−100%-100\%)) and +1.0+1.0 (+100%+100\%)).

  • Sign Significance:     * Positive (+): Returns move together in the same direction.     * Negative (-): Returns move in opposite directions.

  • Magnitude Significance:     * +1+1: Perfectly same direction.     * >+0.5> +0.5: Strong positive relationship.     * +0.5 to −0.5+0.5 \text{ to } -0.5: Weak relationship.     * 00: No linear relationship.     * <−0.5< -0.5: Strong negative relationship.     * −1-1: Perfectly opposite direction.

Correlation Data for Case Study

  • North Air and West Air: 0.620.62 (62%62\%)) — Strong positive. They are the same industry (Air), so they move together.

  • North Air and Tex Oil: −0.9233-0.9233 (−92.33%-92.33\%)) — Extremely strong negative.

  • West Air and Tex Oil: −0.7133-0.7133 (−71.33%-71.33\%)) — Strong negative.

  • Implication: When airline stocks go up, oil stocks tend to go down (and vice-versa), likely due to fuel costs impacting airline profitability.

Portfolio Risk Calculations

  • Formula Note: The general risk formula for nn assets is complex; calculations for two assets depend on σ1,σ2,w1,w2, and ρ1,2\sigma_1, \sigma_2, w_1, w_2, \text{ and } \rho_{1,2}.

  • Portfolio 2 (Equal weights of North Air, West Air, and Tex Oil):     * Weights: 33.33%33.33\% each.     * Expected Return: 10%10\%     * Standard Deviation (Volatility): 4.4%4.4\%     * Insight: Note that while each individual stock had a risk of 13.42%13.42\%, the combined portfolio risk is significantly lower (4.4\%\) despite having the same 10%10\% expected return. This is the benefit of diversification driven by negative correlation.

Questions & Discussion

  • How would you interpret these correlation numbers?     * Interpretation: Positive correlation (0.620.62) between North Air and West Air suggests industry-wide factors affect them similarly. Negative correlations with Tex Oil suggest that factor affecting the energy sector (like fuel prices) have opposite effects on airlines.

  • Is this interpretation consistent with the returns you computed? Explain.     * Analysis: Yes. Looking at the realized returns table, in 2008 and 2009, North and West Air had negative returns while Tex Oil had very high positive returns (21%21\% and 30%30\%). In 2005 and 2006, the reverse occurred.

  • What is the missing information for calculating the standard deviation of Portfolio 2?     * Requirement: To calculate the volatility of a 3-asset portfolio, one needs the standard deviation of each asset and the correlations between all pairs (North/West, North/Tex, West/Tex).