Handouts CAPM Beta

Systematic Risk & SML

Financial Management

Banikanta Mishra

Ravenshaw University

January – March 2009

Systematic Risk Historically

2/6/2014 Professor Banikanta Mishra 2

A typical NYSE stock has a STDEV of around 49% per year

A portfolio of 500 large NYSE stocks has 20% annual STDEV

Why?

As we keep adding stocks “randomly” to our portfolio,

STDEV reduces

But the larger our existing portfolio,

the lower is the (marginal) reduction in risk

We may loosely call this

The Principle of Diminishing Marginal Risk-Reduction

The Risk That Won’t Go Away


There is a MINIMUM level of risk that cannot be diversified

This is the non-diversifiable risk or the market-risk that affects ALL shares

If I hold only the shares of one computer company (Dell, IBM, or HP), I am exposed to the risk that my company may lose market-share to

the other computer companies

If I hold shares of all US computer companies, I am protected against this.

But, I am exposed to the risk that US computer firms may not do well.

If I hold shares of all industries in USA, I am protected against this risk. But, I am exposed to the risk that US economy may NOT do well.

If I hold shares across different countries, I am protected against this risk. But, what if the Global Equity-Market does not do well?

I CANNOT protect my portfolio against this NON-DIVERSIFIABLE risk.

The Market Portfolio


So, except the risk-free asset,

every asset and every portfolio would have some

Non-Diversifiable or Systematic Risk.

Sub-optimal portfolios would have, in addition,

Diversifiable or Unsystematic Risk

Best-diversified portfolio is Market portfolio,

which has no Diversifiable or Unsystematic Risk

The theoretical Market Portfolio consists of all assets,

with weight on any asset j equal to

the fraction of total market value accounted for by the asset j

In practice, any broad-based Market Index can be taken as the Market Portfolio

assetsallofMVstheofsumtheisand

jAssetofValueMarkettheisMVwhere

MV

MVjn

1kk

j

Non-Diversifiable or Systematic Risk


Repeat: every asset and every portfolio would have some Non-Diversifiable Risk.

The ERR and the RRR on any asset or portfolio would depend ONLY on its own Non-Diversifiable or Systematic Risk

We know we can measure a portfolio’s Systematic Risk by its Variance.

How do we measure the Systematic Risk of an asset, call it Asset-i

Non-Standardized: Covariance with the Market Portfolio = siM

Standardized: Covariance / Variance of the Market PF =

The above “Standardized” Risk is what we call the Beta of Asset-i

So, 2M

MiiM2M

iMi

s

ss

s

s

2M

iM

s

s

How do We Compute eta?


Through regression analysis of historical data

MARKET MODEL

Rit = a + i RMt + eit

Where

Rit is the return on Asset-i in period t,

RMt is the return on Market Portfolio in period t,

a = intercept, = slope, e = error-term

Period is usually taken as daily or monthly

60 monthly returns (last five years) or

250 daily returns (last one year) is

taken as an appropriate large sample sizes

Computing Beta: An Example


Stock (i) Market (M)

Rit RMT

-0.60% -0.21%

2.99% -0.77%

3.55% 0.91%

2.09% -0.09%

-1.46% 1.62%

-2.52% -0.12%

2.58% 1.80%

1.33% -0.47%

-2.98% 1.12%

-1.73% 0.22%

-2.94% -1.47%

-0.12% -2.19%

Mean 0.0174% 0.0297%

Variance (s2) 0.0579% 0.0144%

STDEV (s) 2.4060% 1.1983%

Covariance* (Stock, Market) = si,M = 0.003231%

Correlation (Stock, Market) = i,M = 0.1164

Beta of Stock = i = si,M / s2M = 0.23

Beta of Stock= i = ( i,M si sM) / s2M= 0.23

Beta of Stock = Slope of Regresssion Line= 0.23

Beta from Regression


Return on Market

R

E

T

U

R

N

O

N

S

T

O

C

K

+

+

+

+

+

+

+

+

+

+

Slope = i

What Does this eta Mean?


measures how the stock moves on the average

with the Market

during a particular period (day, week, month, year)

If market moves by 1% during that period,

the stock moves - on the average - by % during that period

For example,

If Market goes up (down) by 1% during a period,

our stock-i is expected to move up (down) by 0.23%

during that period

During any particular period, it can – actually will –

move up (or down) by more or less than 0.23%

If another stock, say j, has a of 1.10,

then it would move up or down – on the average - by 1.10%

when Market moves up or down by 1% during a period

eta: High, Average, Low


As we just saw,

if < 1.0,

then it moves less than the Market,

=> less SYSTEMATIC RISK than Market

DEFENSIVE ASSET

if > 1.0,

then it moves more than the Market

=> more SYSTEMATIC RISK than Market

AGGRESSIVE ASSET

if = 1.0,

then it moves same as the Market,

=> same SYSTEMATIC RISK as Market

NEUTRAL ASSET

Can < 0?

eta and Total Risk


eta does NOT measure Total Risk

Therefore, low eta does NOT mean low Total Risk

But, that does not matter

As we care ONLY about Systematic Risk

Why?

ONLY the Systematic Risk determines the Risk Premium and thus influences an asset’s RRR and ERR

Security STDEV Beta

LB 25% 0.80

HB 18% 1.20

Which asset is more risky? Which asset will have higher RRR?

Portfolio Beta


Suppose that you have $5,000 to invest NOW

You put in 25% ($1,250) in LB and 75% ($3,750) in HB

What is your Portfolio now?

Suppose, by the end of the year,

your LB holding would increase in value to $1,800

and your HB holding would increase in value to $4,200

What would then be your Portfolio at the year-end (t=1)?

because, at year-end, LB would account for 30% (=1800/6000) of Portfolio VALUE

10.120.1x%7580.0x%25HBHBwLBLBwp,here,So

assetiiwn

1ip

i)byforaccountedVALUEtotaloffractiontheisiw(where

08.120.1x%7080.0x%30HBHBwLBLBwp

Portfolio Beta with RF Asset


Suppose that you have $5,500 NOW which you invest as follows

RF $500 Risky Assets $5,000 = 0 = 1.10

LB $1,250 HB $3,750

= 0.80 = 1.20

We have seen that the of the above Risky Assets portfolio is1.10.

So, of the Overall $5,500 Portfolio is

Would this equal

00.110.1x11

1000.0x

11

1ww RiskyRiskyRiskfreeRiskfreep

1/11 10/11

25% 75%

= 1.00

?20.15500

375080.0

5500

12500

5500

500

s and RRRs


As we have already said,

ERR (or RRR) of an asset depends ONLY on its Systematic Risk

Since Beta measures the Systematic Risk,

ERR (or RRR) depends only on

Suppose we compute the of each asset

and plot it against their Rs (ERRs or RRRs)

How would that relationship look?

A famous theory – Capital Asset Pricing Model (CAPM) - says:

IT WOULD BE A STRIAGHT LINE

Ri = Rf + i (RM - Rf)

CAPM



Ri is the RRR or ERR on asset-i

Rf is the Pure Time Value of Money component

i is the Amount of Systematic Risk of the Asset

RM - Rf Is the per-unit Reward for Bearing the Systematic Risk (per-unit here refers to per 1.0 unit of )

RM - Rf is called the Market Risk Premium (MRP), since it is the Risk Premium on the Market Portfolio

SML: Security Market Line


R

Intercept = Rf

Slope = RM - Rf

If an asset has a of ZERO, its R would be Rf

As increases from ZERO onwards, R increases by RM - Rf

for every 1.0 increase in

SML with Numbers


R

Intercept = 5%

Slope = 2%

If an asset has a of ZERO, its R would be 5%

As increases from ZERO onwards, R increases by 2%

for every 1.0 increase in

So, if an asset has a of 0.80, its R would be 5% + (0.80 x 2%) = 6.60%

An Implication of SML


If asset-x’s is d more than asset-y’s

then asset-x’s RRR is d (RM - Rf) more than asset-y’s

and, therefore, x’s ERR should also be d (RM - Rf) more than y’s

Example

Rf = 5% RM - Rf = 2%

LB = 0.8 HB = 1.20

RRRLB = 6.60% RRRHB = 7.40%

Check: HB’s is 0.40 more than LB’s

and, as expected, HB’s RRR is 0.40 x 2% = 0.80% more than LB’s RRR

Another Implication of SML


Reward to Risk Ratio (the Risk Premium per Unit Risk or Risk Premium Per Unit Beta)

is the same across all assets

and is equal to the Market Risk Premium

Example

Rf = 5% RM - Rf = 2%

LB = 0.8 HB = 1.20

RRRLB = 6.60% RRRHB = 7.40%

RPLB = RRRLB - Rf = 6.60% - 5% = 1.60% RPHB = 7.40% - 5% = 2.40%

RP per Unit Beta = 1.60% / 0.80 = 2% RP per Unit Beta = 2.40% / 1.20 = 2%

Reward-to-Risk Ratio & Selection


Reward to Risk Ratio = (Ri – Rf) / i

is used in choosing between assets

Example

Rf = 5.00% RM - Rf = 2.00%

LB = 0.8 HB = 1.20

ERRLB = 7.10% ERRHB = 8.00%

RP per Unit Beta = RP per Unit Beta = (Reward-to-Risk Ratio for Asset-LB) (Reward-to-Risk Ratio for Asset-HB)

(7.10% - 5.00%) / 0.80 = 2.625% (8.00% - 5.00%) / 1.20 = 2.500%

So, CHOOSE LB.

(But, which one gives higher Excess Return = ERR – RRR ?)

Mispriced Assets


If an asset lies above the SML,

it has a higher Reward-to-Risk Ratio

than the Market Risk Premium

and is thus underpriced or undervalued

If an asset lies below the SML,

than the Market Risk Premium

it has a lower Reward-to-Risk Ratio

and is thus overpriced or overvalued

Solving for Unknowns


One-Asset Case


or

Ri = Rf + i MRP

Four variables: Ri, Rf, i, AND (RM - Rf) or MRP or RM

Given any THREE, we can solve for the FOURTH one

Two-Assets Case


Rj = Rf + j (RM - Rf)

Six variables: Ri, Rj, i, j, Rf, AND (RM - Rf) or MRP or RM

Given any FOUR, we can solve for the OTHER TWO

CALCing


Asset-i Market2% 3%

-1.90% -1.20%

0.75% 1.10%

0.92% 0.70%

-0.05% 0.00%

i = 0.94

What Have We Learnt So Far?


The two attributes of a security that investors focus on:

1. Expected Rate of Return (ERR) or Average Return or Mean Return

2. Risk

The risk that actually matters is NOT the

Total risk = Systematic risk + Unsystematic risk

but the Systematic or Non-diversifiable Risk

(since unsystematic risk is diversifiable)

For a well-diversified portfolio,

that has no diversifiable or unsystematic risk,

Variance measures both Total and Systematic Risk

For individual assets or non-diversified portfolios, measure

Non-standardized Risk by Covariance (with Market)

Standardized Risk by Beta (with respect to Market)

Example-1a


Suppose an asset with of 1.25 has an RRR of 12%.

What are Rf and Rm?

(Or, what is MRP?)

Only TWO variables given. Need ONE MORE variable.

a. If Rf = 7%, then Rm=?

12% = 7% + 1.25 (Rm– 7%) => Rm = 11%

Example-1a



What are Rf and Rm?

(Or, what is MRP?)



12% = 7% + 1.25 (Rm– 7%) => Rm = 11%

b. If MRP = 4%, then Rf =? [=> Rm=?]

12% = Rf + (1.25 x 4%) => Rf = 7% [ => Rm = 7% + 4% = 11%]

c. If Rm = 11%, then Rf =?

12% = Rf + 1.25 (11% – Rf ) => Rf = 7%

Example-1b


Suppose Rf = 8%.

What is the RRR of an asset with of 1.50?


a. If Rm = 13%, then?

RRR = 8% + 1.50 (13% - 8%) = 15.50%

b. If MRP = 5%, then?

RRR = 8% + (1.50 x 5%) = 15.50%

Example-2a



What should be the RRR for an asset with of 0.75?

Only THREE variables given. Need ONE MORE variable.

a. If Rf = 7%, then?

12% = 7% + (1.25 x MRP) => MRP = 4%

=> RRR another = 7% + (0.75 x 4%) = 10%


12% = Rf + (1.25 x 4%) => Rf = 7%

RRR another = 7% + (0.75 x 4%) = 10%

c. If Rm = 11%, then?

12% = Rf + [1.25 x (11% - Rf) ] => Rf = 7%

=> RRR another = 7% + [0.75 x (11% - 7% ] = 10%

Example-2b


Suppose an asset with of 1.50 has an RRR of 15.50%.

AND an asset with of 0.80 has an RRR of 12.00%.

a. What is the RRR on an asset with = 1.20?

15.50% = Rf + (1.50 x MRP)

12.00% = Rf + (0.80 x MRP)

MRP = 5%, Rf = 7%

So, RRRanother = 5% + (1.20 x 7%) = 13.40%

b. What is Rm=?

As shown above, MRP = 5% and Rf = 7% => Rm = 7% + 5% = 12%

c. What is Rf=?

As shown above, Rf = 7%

Example-1a



What are Rf and Rm?

(Or, what is MRP?)



12% = 7% + 1.25 (Rm– 7%) => Rm = 11%

b. If MRP = 4%, then Rf =? [=> Rm=?]

12% = Rf + (1.25 x 4%) => Rf = 7% [ => Rm = 7% + 4% = 11%]

c. If Rm = 11%, then Rf =?

12% = Rf + 1.25 (11% – Rf ) => Rf = 7%

Example-1b


Suppose Rf = 8%.

What is the RRR of an asset with of 1.50?


a. If Rm = 13%, then?

RRR = 8% + 1.50 (13% - 8%) = 15.50%


RRR = 8% + (1.50 x 5%) = 15.50%

Example-2a



What should be the RRR for an asset with of 0.75?

Only THREE variables given. Need ONE MORE variable.

a. If Rf = 7%, then?

12% = 7% + (1.25 x MRP) => MRP = 4%

=> RRR another = 7% + (0.75 x 4%) = 10%


12% = Rf + (1.25 x 4%) => Rf = 7%

RRR another = 7% + (0.75 x 4%) = 10%

c. If Rm = 11%, then?

12% = Rf + [1.25 x (11% - Rf) ] => Rf = 7%

=> RRR another = 7% + [0.75 x (11% - 7% ] = 10%

Example-2b


Suppose an asset with of 1.50 has an RRR of 15.50%.

AND an asset with of 0.80 has an RRR of 12.00%.

a. What is the RRR on an asset with = 1.20?

15.50% = Rf + (1.50 x MRP)

12.00% = Rf + (0.80 x MRP)

MRP = 5%, Rf = 8%

So, RRRanother = 8% + (1.20 x 5%) = 14.00%

b. What is Rm=?

As shown above, MRP = 5% and Rf = 8% => Rm = 8% + 5% = 13%

c. What is Rf=?

As shown above, Rf = 8%

Documents

Handouts CAPM Beta