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RETURN, RISK AND EQUILIBRIUM

BY

PROF. SANJAY SEHGALDEPARTMENT OF FINANCIAL STUDIESUNIVERSITY OF DELHI SOUTH CAMPUS

NEW DELHI-110021

PH.: 0091-11-24111552Email: sanjayfin15@yahoo.co.in

K2

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PROMINENT INVESTMENT OBJECTIVES

• Regular income

• Income on Income

• Capital appreciation

• Safety of capital

• Liquidity

• Tax considerations

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THE TWO PARAMETER FRAMEWORK

Maximize Expected Utility E(U) = f(Return, Risk)

Pre-conditions for the Two-parameter to work in financial markets

• Historical distribution of returns exhibit normality

• Investor utility functions are quadratic in nature, i.e.

E(U) f(mean, variance)

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EX-ANTE RETURN AND RISK ON SECURITY i

Expected Return

E(Ri) = PiOi

Standard Deviation of return

i = Pi[Oi - E(Ri) ]2

Variance of returns

Var(Ri) = 2

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AN EXAMPLE

Economic Probability Outcome (%) .

scenario Security 1 Security 2

Good 0.3 25 40

Average 0.5 20 10

Poor 0.2 15 - 10

E(R1) = .3 x 25 + .5 x 20 + .2 x 15 = 20.5%

E(R2) = .3 x 40 + .5 x 10 + .2 x -10 = 15%

1 = [.3(25 - 20.5)2 + .5(20 - 20.5)2 + .2(15 - 20.5)2]1/2 = 3.5%

2 = [.3(40 - 15)2 + .5(10 - 15)2 + .2(-10 - 15)2]1/2 = 18% approx.

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EX-POST RETURN ON SECURITY i

One- period return

P1 – P0 D1

R1 = ----------- + ----------

P0 P0

Capital Dividend

gains Yield

yield

NOTE: 1. Ignoring dividend yield one can compute approximate one-period return as

P1 – P0

R1 = -----------

P0

2. The square of the period returns can be used to estimate average returns.

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CAPITALISATION CHANGES AND STOCK PRICES

1. Stock Dividends

If a firm issues a stock dividend on n:m, then ex-stock dividend price = m/(n+m), where n = new share issued, m = old shares with shareholders.

Example: A firm issues tock dividend of 2:3, The ex-stock dividend price = 3/5.

2. Rights Issues nP1 + mP2

The ex-rights price of a firm = ---------------

n + m

where n = rights shares, m is old shares, n = issue price, m = last cum-rights selling price

Example: A firm offers a rights price is 2:3. The issue price is IOE, while the cum-rights selling price is 25 Euros.

2 x 10 + 3 x 25 95

Ex-rights price = -------------------- = ----- = 19 Euros

2 + 3 5

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Ex-rights price 19

Rights adjustments factor = ------------------------------------- = ----

Last cum-rights selling price 30

3. Stock Splits

If a firm announces a stock split of n:m, ten ex-stock split price is n/m.

Example: If stock split is 1:10, then ex-stock split price is 1/10.

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MEASURING AVERAGE RETURNS

Geometric average returns

Using one-period gross returns, we estimate Ex-post GM returns

RGM = [(1 + R1) (1 + R2) … (1 + RK)1/K] - 1

Arithmetic average returns

Using one period net return we estimate Ex-post arithmetic average returns as

w

E(R) or AM = Ri/N

i=1

Arithmetic average is preferred because

• Easier to compute• It has more desirable mathematical properties• A better proxy for forward looking returns.

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ESTIMATING EX-POST RETURN: AN ILLUSTRATION

Months Monthly Return (1%)

1 6

2 2

3 4

E(Rt) = Ri/3 = 4%

6 + 2 + 4

1. Ex-post return based on arithmetic mean = ------------- = 4%

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2. Ex-post return base don geometric mean = cube root of [(1.06) (1.02) (1.04)] – 1

= [(1.06) (1.02) (1.04)]1/3 – 1 =

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EX-POST MEASURES OF RISK

Total risk measure standard deviation of returns

i = 1/N-1 (Ri - E(Ri))2

An Example

Months Monthly Return (1%)

1 6

2 2

3 4

E(Rt) = Ri/3 = 4%

i = ½ [(6 - 4)2 + (2 - 4)2 + (4 - 4)2]

= 2%

NOTE: Standard deviation as a risk measure assumes that historical returns follow a normal distribution

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DECOMPOSING TOTAL RISK

Total Risk = systematic risk + unsystematic risk

2i = 2

i 2M + 2

ei

Systematic risk variations in stock returns is owing to shifts in common macro-economic factors.

Unsystematic risk: variation in stock returns is owing to micro-economic shocks.

• Unsystematic risk is diversifiable in a large portfolio.

• Sources of unsystematic risk

• Industry factors

• Group factors

• Common factors

• Firm specific factors.

BETA AS A MEASURE OF SYSTEMATIC RISK

• Beta measures the sensitivity of stock returns to market index returns.

• It is estimated as the slope of the regression of stock returns on market returns.

Rit = α + β RMt + eit

• Mathematically

Cov RiRM

β = ----------------

Var RM 13

STOCK CLASSIFICATION ON THE BASIS OF BETAS

β > 1 Aggressive stocks

β = 1 Average stocks

β < 1 but > 0 Defensive stocks

β = 0 Risk-free asset

β < 0 Hedging stocks

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BETA AND STOCK CLASSIFICATION

Estimating Beta of a listed company - An Example

Problem

Month Return on Return on

security i (%) Market Index

1 10 12

2 6 5

3 13 18

4 -4 -8

5 13 10

6 14 7

7 4 15

8 18 30

9 24 25

10 22

Solution

Cov RiRM = .778

Var (RM) 1022

Cov RiRM

I = ------------- = 0.76

Var RM

Time Series Regression: Estimating Beta for HDFC Bank stock

Time Period: 2005-2007

Market Index : CNX S&P 500

The Model: Rit= a + bRmt

Output:

Rit = 0.142 + 0.704Rmt

(1.126) (3.393)

R2= 0.313

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Regression Results using Excel: Estimating Beta for HDFC Bank stock

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Regression StatisticsMultiple R 0.5598034R Square 0.3133798Adjusted R Square 0.2931851Standard Error 0.0670837Observations 36

ANOVA df SS MS F Significance F

Regression 1 0.069834096 0.07 15.52 0.00038525Residual 34 0.15300763 0.005Total 35 0.222841726

Coefficients Standard Error t Stat P-value Lower 95%Intercept 0.0142604 0.012654992 1.127 0.268 -0.01145766Rm 0.7040934 0.178736648 3.939 4E-04 0.340856844

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PRECAUTIONS WHILE ESTIMATING BETA

• Number of observations: 48 - 60

• Observation frequency: monthly or weekly data

• Market index: broad-based and value-weighted

• Trading frequency: Active trading record.

UNLEVERED AND LEVERED BETAS

• Unlevered Beta is a measure of operating risk of the firm.

• Levered Beta is a measure of both operating and as well as financial risk of the firm.

βL = βU [1 + (1 – TC)] D/E

Or

βU = βL/[1 + (1 – TC)] D/E

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THE RELATIONSHIP BETWEEN RISK AND RETURN

Estimating required returns: THE CAPITAL ASSET PRICING MODEL or CAPM approach

R(R) = RF + (ERM – RF) βi

Estimating expected returns: DIVIDEND CAPITALISATION MODEL APPROACH

D1 D1

P0 = ------------- or ER = ----- + g

ER – g P0

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AN EXAMPLE

Risk-free rate of return = 10%

Return on market index = 14%

Beta of IBM stock = 1.25

Dividend paid last year by IBM = 1.70 E

Growth rate of IBM = 6%

Current price of IBM stock = 22E

Estimating R(R)

R(R) = 10 + (14 – 10) 1.25 = 15%

Estimating E(R)

1.70 (1.06)

E(R) = -------------- + .06 = 14.18%

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Since, ER < RR, IBM stock is overvalued. This is a sell signal 21

Equilibrium Value of IBM stock

D1

Eqn. P0 = -------------

R (R) – g

1.80

= ----------- = 20 E

.15 - .06

Since P0 (22E) > Eqm. P0 (20E), the stock is overvalued. It is a sell signal.

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CHANGE IN EQUILIBRIUM PRICE

Change in response to changes in underlying variables.

Variable Old Value Revised value

RF 10% 9%

ERM - RF 4% 3%

1.25 1.33

g 6% 8%

Revised R(R)

= 9 + 3 (1.33) = 13%

1.70 (1.08)

Eqm Price = --------------

.13 - .08

= 36.80 E