Portfolio Return Analysis

1.Single Index Model (SIM)

The Single Index ì (SIM) is an asset pricing model, according to which the returns on a security can be represented as a linear relationship with any economic variable relevant to the security.

In case of stocks, this single factor is the market return.

The SIM for stock returns can be represented as follows:

Where:

Alpha (α) represents the abnormal returns for the stock

β(rm − rf) represents the movement of the market modified by the stock’s beta

ε represents the unsystematic risk of the security due to firm-specific factors.

According to this equation, asset’s returns is influenced by the market (reflected in beta), it has firm specific excess returns (reflected in alpha) and also has firm-specific risk (the residual).

2. Arbitrage Pricing Theory

Arbitrage Pricing Theory (APT) is an alternate version of the Capital Asset Pricing Model (CAPM). This theory, like CAPM, provides investors with an estimated required rate of return on risky securities. APT considers risk premium basis specified set of factors in addition to the correlation of the price of the asset with expected excess return on the market portfolio.

As per assumptions under Arbitrage Pricing Theory, return on an asset is dependent on various macroeconomic factors like inflation, exchange rates, market indices, production measures, market sentiments, changes in interest rates, movement of yield curves etc.

the Arbitrage pricing theory based model aims to do away with the limitations of the one-factor model (CAPM) that different stocks will have different sensitivities to different market factors which may be totally different from any other stock under observation. In layman terms, one can say that not all stocks can be assumed to react to single and same parameter always and hence the need to take multifactor and their sensitivities.

Calculating the Expected Rate of Return of and Asset Using Arbitrage Pricing Theory (APT)

Arbitrage Pricing Theory Formula – E(x) = rf + b1 * (factor 1) +b2 *(factor 2) + ….+ bn *(factor n)

Where,

E(X) = Expected rate of return on the risky asset

Rf = Risk-free interest rate or the interest rate that is expected from a risk-free asset

(Most commonly used in U.S. Treasury bills for the U.S.)

B = Sensitivity of the stock with respect to the factor; also referred to as beta factor 1, 2 …

N = Risk premium associated with respective factor

As the formula shows, the expected return on the asset/stock is a form of linear regression taking into consideration many factors that can affect the price of the asset and the degree to which it can affect it i.e. the asset’s sensitivity to those factors.

If one is able to identify a single factor which singly affects the price, the CAPM model shall be sufficient. If there is more than one factor affecting the price of the asset/stock, one will have to work with a two-factor model or a multi-factor model depending on the number of factors that affect the stock price movement for the company.

3) Capital Asset Pricing Model (CAPM) Definition

Capital Asset Pricing Model (CAPM) is a measure of the relationship between the expected return and the risk of investing in security. This model is used to analyze securities and pricing them given the expected rate of return and cost of capital involved. The CAPM calculation formula and examples are provided below.

CAPM Formula The (capital asset pricing model) CAPM formula is represented as below

Expected Rate of Return = Risk-Free Premium + Beta * (Market Risk Premium)

Ra = Rrf + βa * (Rm – Rrf)

The CAPM calculation works on the existence of the following elements

#1 – Risk-free return (Rrf)

Risk-Free Rate of Return is the value assigned to an investment that guarantees a return with zero risks. Investments in U.S securities are considered to have zero risks since there is a minimal chance of the government defaulting. Generally, the value of the risk-free return is equivalent to the yield on a 10-year U.S government bond.

#2 – Market Risk Premium (Rm – Rrf)

Market Risk Premium is the expected return an investor receives (or expects to receive in the future) from holding a risk-laden portfolio instead of risk-free assets. The premium rate allows the investor to take a decision on if the investment in the securities should take place and if yes, the rate that he will earn beyond the risk-free return offered by government securities.

#3 – Beta (βa)

The Beta is a measure of the volatility of a stock with respect to the market in general. The fluctuations that will be caused in the stock due to a change in market conditions is denoted by Beta. For example, if the Beta of a stock is 1.2, it would cause a 120% change due to any change in the general market. The opposite is the case for Beta less than 1. For Beta which is equal to 1, the stock is in sync with the changes in the market.

4) Portfolio return:

portfolio managers will have many assets in their portfolios in different proportions. The portfolio manager will have to therefore calculate the returns on the entire portfolio of assets. The returns on the portfolio are calculated as the weighted average of the returns on all the assets held in the portfolio.

The formula for portfolio returns is presented below:

w represents the weights of each asset, and r represents the returns on the assets. For example, if an asset constitutes 25% of the portfolio, its weight will be 0.25. Note that sum of all the asset weights will be equal to 1, as it will represent 100% of the investment. The returns here are single period returns with same periods for each asset’s returns.

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Portfolio Return Analysis