BALANCE OF PAYMENTS



CAPITAL ASSET PRICING MODEL

CAPM = rf + β(rm – rf)

where, rf = the risk free rate

rm = the return on the market

(rm – rf) = is the market risk premium

β = is a measure of market risk (see notes below)

Key Assumptions of the CAPM:[1]

• All investors have a single holding period.

• All investor select portfolios on the basis of expected return and standard deviation

• All investors can borrow or lend unlimited capital at the risk free rate

• All investors have identical estimates of expected returns, variances, and covariances.

• All investors are price takers (i.e. buying and selling activity does not affect the stock price).

• All assets are divisible and liquid.

• There are no transaction costs (including taxes).

• There are no restrictions on short sales.

• The number of available assets is fixed.

Note: While these assumptions many be generous, it is the market’s general acceptance of the CAPM model that makes it a critical element for financial analysis. Dr. William F. Sharpe won the Nobel Prize in Economics for his capital asset pricing work.[2]

Expected Rate of Return

[pic]

where Pi, is the probability of return (ri). Note: the sum of Pi to Pn must equal 1.

Table 1: Expected Rates of Return

|U.S. Economic Situation |Probability |Staples |Boston Properties |

| | |(return) |(return) |

|Strong |30% |40% |100% |

|Normal |40% |15% |15% |

|Weak |30% |(10%) |(70%) |

| |100% |[pic]= 15% |[pic]= 15% |

[pic]staples = 0.3(40%)+ 0.4(15%) + 0.3(-10%) = 15%

[pic]boston properties = 0.3(100%) + 0.4(15%) + 0.3(-70%) = 15%

Risk (σ)

Standard deviation (σ) is one measure of risk.

[pic]

Table 2: Calculating Standard Deviation (σ) for Staples

|[pic] |[pic]2 |[pic] |

|40% - 15% = 25% |625% |(625%)0.3 = 187.5% |

|15% - 15% = 0% |0 |(0)0.4 = 0 |

|-10% - 15% = -25% |625% |(625%)0.3 = 187.5% |

| |Variance = σ2 = 375% |

| |Standard Deviation = σ = 19.36% |

Risk (Estimated σ)

[pic]

Where [pic]

Example

Given the following information calculate the standard deviation.

|Year |rt |

|2002 |15% |

|2003 |-5% |

|2004 |20% |

[pic]

[pic]

The calculation for the standard deviation of a portfolio is provided in the formula below:

[pic]

where, N is the total number of assets included in the portfolio.

Risk Coefficient of Variation

The coefficient of variation is the standard deviation (σ) divided by the expected return ([pic]).

[pic]

Expected Return on a Portfolio ([pic])

[pic]

Example: Portfolio of four Stocks

| |Expected Return ([pic]) |Investment |Weight (wi) |(ri x wi) |

|Staples |10% |$25,000 |0.25 |2.50% |

|Boston Properties |9% |$25,000 |0.25 |2.25% |

|Biogen Idec |15% |$25,000 |0.25 |3.75% |

|TJX |12% |$25,000 |0.25 |3.00% |

| | |$100,000 |100% |[pic] 11.5% |

Correlation Coefficient (ρ)

[pic]

The calculation for the standard deviation of a portfolio is provided in the formula below:

[pic]

where, N is the total number of assets included in the portfolio.

The calculation for the expected return of a portfolio is shown below:

[pic]

Example

Calculate the standard deviation and the expected return of a portfolio with 40% invested in the following US equity and 60% invested in the following German equity index.

| |Expected Return |Expected Risk (σ) |

|United States equity index (US) |14% |15% |

|German equity index (GER) |18% |20% |

|Correlation coefficient (ρUS,GER) |0.34 |

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Beta

Beta is a measure of the extent to which the returns of a given stock move with the stock market. (i.e. “market risk”)

[pic] or [pic]

where, the Greek letter rho, (ρim), is the correlation coefficient for the stock “i” and the market “m.”

[pic]

therefore, the formula for the covariance of the stock with the market “covim” is:

[pic]

Diversifiable risk (a.k.a. company-specific risk or unsystematic risk) is risk caused by random events that are specific to the company (i.e. lawsuits, strikes, and acquisitions).

Market risk (a.k.a. nondiversifiable or systematic risk) is risk that affects the entire market and cannot be eliminated (i.e. war, inflation, and recessions).

Remember, Beta measures only systematic risk, while standard deviation is a measure of total risk (systematic or market risk, and unsystematic risk, the risk of the security).

Calculating Beta (β) with Excel

Example: Find βPQU given the following information.

|Year |Market |PQU |

|1 |25.70% |40.00% |

|2 |8.00% |-15.00% |

|3 |-11.00% |-15.00% |

|4 |15.00% |35.00% |

|5 |32.50% |10.00% |

|6 |13.70% |30.00% |

|7 |40.00% |42.00% |

|8 |10.00% |-10.00% |

|9 |-10.80% |-25.00% |

|10 |-13.10% |25.00% |

|First: Make sure that your computer has the Analysis Took Pack, you can |Click on “Analyst ToolPak” and “Analyst ToolPak –VBA” and then click |

|download it by clicking on “Tools” and then “Add-Ins” |“OK.” |

|[pic] |[pic] |

Once you have the Analyst ToolPak added, you can select it by clicking “Tools” and then “Data Analysis” (Note: If you already had the Analyst ToolPak downloaded, this would already appear under “Tools”)

A pop up box will appear, select “Regression” from the list.

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Then enter the data for your analysis. Note: Your “Y” variable is the stock returns and the “X” variable is the market returns.

[pic]

From your regression output (shown below) you can find the calculated Beta (in this case it is 0.830833.

The intercept is 0.025608.

[pic]

Note: The R square measures the percent of a stock’s variance that is explained by the market. In this example 35% of the stock’s variance is explained by the market. The 95% confidence interval shows the range in which we are 95% sure that the true value of beta lies. In this example β falls somewhere between -0.082983 and 1.744648 (a wide range is typical for most stocks).

Other Asset Pricing Models

1. Arbitrage Pricing Theory (APT) - The APT proposes that the relationship between risk and return is more complex and may be due to multiple factors such as GDP growth, expected inflation, tax rate changes, and dividend yield.

Note: The model does not specify what factors influence stock returns.

[pic]

where, rj = required rate of return on a portfolio sensitive only to economic Factor j and bij = sensitivity of Stock i to economic Factor j.

2. Fama-French 3-Factor Model – The Fama-French 3 – Factor Model attempts to explain the relationship between risk and return using the following 3 factors:

1. The excess market return, rM-rF.

2. The return on, S, a portfolio of small firms (where size is based on the market value of equity) minus the return on B, a portfolio of big firms. This return is called rSMB, for S minus B.

3. The return on, H, a portfolio of firms with high book-to-market ratios (using market equity and book equity) minus the return on L, a portfolio of firms with low book-to-market ratios. This return is called rHML, for H minus L.

[pic]

where, bi = sensitivity of Stock i to the market return; cj = sensitivity of Stock i to the size factor; and dj = sensitivity of Stock i to the book-to-market factor.

Example: Fama French vs. CAPM

Inputs: bi=0.9; rRF=6.8%; market risk premium = 6.3%; ci=-0.5; expected value for the size factor is 4%; di=-0.3; expected value for the book-to-market factor is 5%.

ri = rRF + (rM - rRF)bi + (rSMB)ci + (rHMB)di

ri = 6.8% + (6.3%)(0.9) + (4%)(-0.5) + (5%)(-0.3)

Fama-French = 8.97%

CAPM:

ri = rRF + (rM - rRF)bi

ri = 6.8% + (6.3%)(0.9)

CAPM = 12.47%

Note: Dr. Kenneth R. French lists the Fama-French factors on his website:

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[1] This list is modified from Financial Management, 12th edition, Chapter 7, p. 250.

[2]

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