# Portfolio theory

Portfolio theory provides the foundation for estimating the return required by investors for different assets. Through diversification the exposure to risk could be minimized, which implies that portfolio risk is less than the average of the risk: of the individual stocks. To illustrate this consider Figure e, which shows how the expected return and standard deviation change as the portfolio is comprised by different combinations of the Nokia and Nestlé stock.

Figure 3: Portfolio diversification

If the portfolio invested 100% in Nestlé the expected return would be 10% with a standard deviation of 20%. Similarly, if the portfolio invested 100% in Nokia the expected return would be 15% with a standard deviation of 30%. However, a portfolio investing 50% in Nokia and 50% in Nestlé would have an expected return of 12.5% with a standard deviation of 21.1%. Note that the standard deviation of 21.1% is less than the average of the standard deviation of the two stocks (0.5 • 20% + 0.5 • 30% = 25%). This is due to the benefit of diversification.

In similar vein, every possible asset combination can be plotted in risk-return space. The outcome of this plot is the collection of all such possible portfolios, which defines a region in the risk-return space. As the objective is to minimize the risk for a given expected return and maximize the expected return for a given risk, it is preferred to move up and to the left in Figure 4.

Figure 4: Portfolio theory and the efficient frontier

The solid line along the upper edge of this region is known as the efficient frontier. Combinations along this line represent portfolios for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolio lying on the efficient frontier represents the combination offering the best possible return. Thus, the efficient frontier is a collection of portfolios, each one optimal for a given amount of risk.

The Sharpe-ratio measures the amount of return above the risk-free rate a portfolio provides compared to the risk it carries.

Where r is the return on portfolio i, rf is the risk free rate and a is the standard deviation on portfolio i's return. Thus, the Sharpe-ratio measures the risk premium on the portfolio per unit of risk.

In a well-functioning capital market investors can borrow and lend at the same rate. Consider an investor who borrows and invests fraction of the funds in a portfolio of stocks and the rest in short-term government bonds. In this case the investor can obtain an expected return from such an allocation along the line from the risk free rate rf through the tangent portfolio in Figure 5. As lending is the opposite of borrowing the line continues to the right of the tangent portfolio, where the investor is borrowing additional funds to invest in the tangent portfolio. This line is known as the capital allocation line and plots the expected return against risk (standard deviation).

Figure 5: Portfolio theory

The tangent portfolio is called the market portfolio. The market portfolio is the portfolio on the efficient frontier with the highest Sharpe-ratio. Investors can therefore obtain the best possible risk return tradeoff by holding a mixture of the market portfolio and borrowing or lending. Thus, by combining a risk-free asset with risky assets, it is possible to construct portfolios whose risk-return profiles are superior to those on the efficient frontier.

# Capital assets pricing model (CAPM)

The Capital Assets Pricing Model (CAPM) derives the expected return on an assets in a market, given the risk-free rate available to investors and the compensation for market risk. CAPM specifies that the expected return on an asset is a linear function of its beta and the market risk premium:

Where rf is the risk-free rate, Pi is stock i's sensitivity to movements in the overall stock market, whereas (r m - r f ) is the market risk premium per unit of risk. Thus, the expected return is equal to the risk free-rate plus compensation for the exposure to market risk. As Pi is measuring stock i's exposure to market risk in units of risk, and the market risk premium is the compensations to investors per unit of risk, the compensation for market risk of stock i is equal to the Pi (r m - r f ).

Figure 6 illustrates CAPM:

Figure 6: Portfolio expected return

The relationship between (3 and required return is plotted on the securities market line, which shows expected return as a function of (3. Thus, the security market line essentially graphs the results from the CAPM theory. The x-axis represents the risk (beta), and the y-axis represents the expected return. The intercept is the risk-free rate available for the market, while the slope is the market risk premium (rm - rf)

CAPM is a simple but powerful model. Moreover it takes into account the basic principles of portfolio selection:

1. Efficient portfolios (Maximize expected return subject to risk)

2. Highest ratio of risk premium to standard deviation is a combination of the market portfolio and the risk-free asset

3. Individual stocks should be selected based on their contribution to portfolio risk

4. Beta measures the marginal contribution of a stock to the risk of the market portfolio

CAPM theory predicts that all assets should be priced such that they fit along the security market line one way or the other. If a stock is priced such that it offers a higher return than what is predicted by CAPM, investors will rush to buy the stock. The increased demand will be reflected in a higher stock price and subsequently in lower return. This will occur until the stock fits on the security market line. Similarly, if a stock is priced such that it offers a lower return than the return implied by CAPM, investor would hesitate to buy the stock. This will provide a negative impact on the stock price and increase the return until it equals the expected value from CAPM.

# Alternative asset pricing models

## Arbitrage pricing theory

Arbitrage pricing theory (APT) assumes that the return on a stock depends partly on macroeconomic factors and partly on noise, which are company specific events. Thus, under APT the expected stock return depends on an unspecified number of macroeconomic factors plus noise:

33) Expected return = a+hx + r factor: +b2 + r factor 2 +K +bn-r factor n + noise

Where bp b2,.. .,bn is the sensitivity to each of the factors. As such the theory does not specify what the factors are except for the notion of pervasive macroeconomic conditions. Examples of factors that might be included are return on the market portfolio, an interest rate factor, GDP, exchange rates, oil prices, etc.

Similarly, the expected risk premium on each stock depends on the sensitivity to each factor (bt, b2,.. .,bn) and the expected risk premium associated with the factors:

34) Expected risk premium = hx + (r factor, - rf) + b2 + (r factor 2- r{) +... + bn + (r factor n - rf)

In the special case where the expected risk premium is proportional only to the portfolio's market beta, APT and CAPM are essentially identical.

APT theory has two central statements:

1. A diversified portfolio designed to eliminate the macroeconomic risk (i.e. have zero sensitivity to each factor) is essentially risk-free and will therefore be priced such that it offers the risk-free rate as interest.

2. A diversified portfolio designed to be exposed to e.g. factor 1, will offer a risk premium that varies in proportion to the portfolio's sensitivity to factor 1.

## Consumption beta

If investors are concerned about an investment's impact on future consumption rather than wealth, a security's risk is related to its sensitivity to changes in the investor's consumption rather than wealth. In this case the expected return is a function of the stock's consumption beta rather than its market beta. Thus, under the consumption CAPM the most important risks to investors are those the might cutback future consumption.

## Three-Factor Model

The three factor model is a variation of the arbitrage pricing theory that explicitly states that the risk premium on securities depends on three common risk factors: a market factor, a size factor, and a book-to-market factor:

Where the three factors are measured in the following way:

- Market factor is the return on market portfolio minus the risk-free rate

- Size factor is the return on small-firm stocks minus the return on large-firm stocks (small minus big)

- Book-to-market factor is measured by the return on high book-to-market value stocks minus the return on low book-value stocks (high minus low)

As the three factor model was suggested by Fama and French, the model is commonly known as the Fama-French three-factor model.