What is Market model or asset pricing model?

 Single index Model: 

It is an asset pricing model which is developed by William Sharpe in 1963. This model is also known as Sharpe index model and Market Model. With the help of this model optimum portfolio is calculated. It measures the expected return and portfolio risk. This model helps to determine how changes in the market affect the expected return of securities in a portfolio. Only 3 n parameters are needed in this model that is expected return, beta, firm specific variance and two special parameters is market return and market variance.

Risk: It is a difference between actual return and expected return. There are two type of risk: Systematic risk (market risk affects all company or industry in market) and unsystematic risk (specific risk related to company or industry).

Systematic risk = β_i² *σ_m² (variance of market index)

Unsystematic risk (e_i²) = σ_i² - β_i²σ_m²

Total risk = β_i² σ_m²+ e_i²

Expected return of a portfolio:

E(r)_p = Ʃⁿ _i=1  wi Ri

Where,

w_i = weighted average of security i

 Single index model Formula:

The co-relation between securities in a portfolio and market index is shown with the help of formula:

R_i = α_i + β_iR_m + e_i

Where,

R_i = Expected return of stock

α_i = An independent or insensitive variable or alpha co-efficient

β_i = sensitive variable or beta co-efficient

R_m = return on market index

e_i = Error term(random variable)

The formula divided into two parts one shows independent factor α_i which is insensitive to market return it means it does not change with change in market condition and the second is β_i which is dependent factor that changes with market condition. Or we can say sensitive to market return.

Variance of security i:

σ_i² = β_i² σ_m² +σ_ei²

Variance of security i and j:

σ_ij= β_i β_j σ_m²

Standard deviation of given portfolio:

σ_p²= β_p² σ_m²

Correlation of security i:

R² = β_i² σ_m²/ σ_i²

Beta of portfolio:

β_p = Ʃ_i=1ⁿw_i β_i

Alpha of portfolio:

α_p = Ʃ_i=1ⁿ w_i α _i

Example: Suppose the market index return is 8%. The stock of company X is sensitive to market index. After analysing the five years data it is assumed that the market index increases to 3% in next weeks. Find out the expected return of X’s stock with the help of single index model. Assume alpha and beta value is -0.05% and 1.28 respectively.

Solution: R_i = α_i + β_iR_m + e_i             (Assumed standard error = 0)

= -0.05 + 1.28 * 3

= 3.79%

Example: Suppose a portfolio consist 2 stocks one company Y’s stock and another is company Z’s stock with the help of given data find out how the portfolio stocks return changes by changes in market index return with the help of market model. Standard deviation of market index is 2.23%.

Particulars	Security Y	Security Z
Alpha (α)	-0.06%	0.03%
Beta (β)	1.69	1.20
Standard deviation of error	2.95	3.21
Standard error	0.30	0.51
Market index return (increase)	3%	3%

Solution:

R_y = α_y + β_yR_m + e_y           

= -0.06 + 1.69 * 3.00 + 0.30

= 5.31%

R_z = α_z + β_zR_m + e_z             

= 0.03 + 1.20 * 3 + 0.51

= 4.14%

Variance of security Y:
 σ_y² = β_y² σ_m² +σ_ey²

= 1.69²*2.23² + 2.95²

= 2.8561* 4.9729 + 8.7025

= 22.90

Variance of security Z:
σ_z² = β_z² σ_m² +σ_ez²

= 1.20² * 2.23² + 3.21²

= 1.44 * 4.9729 + 10.3041

= 17.47

Correlation coefficient of security Y:

R² = β_y² σ_m² / σ_y²

= 1.69²*2.23² / 22.90

= 0.62

Correlation coefficient of security Z:

R² = β_z² σ_m² / σ_z²

= 1.20² * 2.23² / 17.47

= 0.41

Co-variance of security Y and Z:

σ _yz =β_y β_z σ _m²

= 1.69 * 1.20 * 2.23²

= 10

Correlation of security Y and Z:

= σ_yz / √ σ_y² * σ_z²

= 10 / √22.90*17.47

= 0.50

The result of R² shows that security Y is more market sensitive than security Z. But the correlation between these two securities is higher than co-variance.