Week 5 - Single Index Model

Question 1

What does the performance of the Mean-Variance Model (week 1-4 model) depend on?

Answer 1

Quality of inputs
I.E. 
- n expected returns (E(ri))
- n return of St Dev (δi)
- n (n-1)/2 Correlation or CoVariance

Question 2

How is the Single Index model different from the Mean-Variance Model?

Answer 2

Assumes only 1 macroeconomic factor causing systematic risk affecting all stock returns and this factor can be represented by the rate of return on a market index, such as the S&P 500.

Question 3

How do you calculate the Return of an asset?

Answer 3

Ri = αi + βiRm + εi

Where:
Rm = Excess Return on mkt index (Calc by Return on M - Rf)
αi = Security excess return index when E(Rm)=0 -> Lower the alpha the lower return of security i
εi = Random firm specific component covering factors on notes

Question 4

How do you calculate the expected return of an asset?

Answer 4

E(Ri) = αi + βiRm

εi = 0 here

Question 5

What does the return of an asset equation yield?

Answer 5

The Systematic and Firm Specific Component of overall risk of each security:

Question 6

How do you calculate total risk?

Answer 6

Total Risk = Systematic Risk + Firm Specific Risk

δi squared = βi squared δim squared + δε Squared

Question 7

How do you calculate the correlation between 2 securities?

Answer 7

βiδm squared x βjδm squared / δiδm x δjδm

= Corr (Ri,Rm) x Corr (Rj,Rm)

Question 8

What are the benefits of the single index model?

Answer 8

1. Reduces dimensionality of the estimation problem
   -> Reduces amount you need to calculate
   (In total (3n +2) (SI Model) estimates compared to (2n+ n(n-i)/2) in mean variance

i.e If N= 1000 -> Becomes 3002 compared to 501,100

2. Correlation between 2 securities equation allows for specialisation of effort in security analysis
   - > Can have someone in tech (αi,βi) and someone in another field (αj,βj) since we can now understand the correlation between them

In mean variance required jack of all trades

Question 9

What are the costs of the Single Index Model?

Answer 9

1. Assumes correlation across assets depend only on one factor -> State of economy
2. Ignores other sources of uncertainty -> Like industry stuff

Question 10

How do we estimate α AND β

Answer 10

βi = COV (Ri,Rm) / VAR (Rm)

αi = Rbar (Excess return for i) - βi X Rmkt

Question 11

How do we calculate the security characteristic line?

Answer 11

Ri(t) = αi + βi x Rmkt (S&P500 for instance)

Question 12

What is the Security Characteristic Line?

Answer 12

Plotting performance of a particular security or portfolio against that of the market portfolio at every point in time.

Question 13

How do you calculate ST DEV of the systematic risk?

Answer 13

σ systematic squared = βi x σ squared MKT Index

Question 14

How do you calculate ST DEV of Firm Specific Risk?

Answer 14

σε mkt index squared

Question 15

How do you calculate the return on a portfolio?

Answer 15

Rp = αp + βp + εp

Where:
ap = Sum of wi ai
bp = Sum of wi bi
εp = Sum of wi εi

Question 16

How do we calculate VAR of a portfolio?

Answer 16

Since we assume COV (Rm,εp) = 0, VAR equals:

σp squared = βp squared x σm squared x σεi squared

Question 17

How do we calculate total risk of asset i?

Answer 17

σi squared = βi squared x σm squared + σεi squared

Question 18

What is the Non-Systematic Risk of Portfolio?

Answer 18

σεp squared = Sum wi squared σεi squared

Question 19

What is the Non-Systematic Variance?

Answer 19

Sum Wi squared x σεi squared =

Sum of 1/n squared (n=0 since infinite) x σεi squared

Question 20

What happens under diversification and more securities added?

Answer 20

Sum of 1/n squared (n=0 since infinite) x σεi squared = 0
Thus, non-sys risk removed via diversification

As diversification increases, total VAR of portfolio approaches VAR of mkt portfolio