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Unit 5 Notes: Risk and Return

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Corporate Finance lecture notes for the EMBA at UNSW.

Past returns

Investors achieve returns by 2 ways:

  1. Dividends received;
  2. Price gain/increase on shares;

Price gains linked to dividend growth

Price increase with expected future dividends;
Future profits and dividends increase because:

  1. Total economy is expanding (GDP);Prices are increasing (CPI);
  2. Industry’s profit share of economy is expanding;
  3. Firm’s profit share of industry is expanding;

Dividend Yield
Simple formula ignoring time value.
D = Total dividend in period    P0= Price at start
d${}_{t}=\frac{Dividends(\$)}{Stock\ price} = \frac{{\rm D}}{{{\rm P}}_0}

Which stock price to use?
Current yield = use current price; Period yield = use price at start of period

Capital gains
Percentage return from price increase
\Delta P\%=\frac{{{{P}}_{{1}}{-}{{P}}_0}}{{P}_0}
Holding period return rt or Past total returns on one share
r${}_{t}$=p${}_{r}$+d${}_{t}$
=\frac{{{\rm P}}_{{\rm 1}}{\rm +D}}{{{\rm P}}_0}{\rm -}{\rm 1}
= 0.104 = 10.4%

Indices
Index shows only the capital gains but not the dividend yield.

Future returns

How to account for uncertainty about the future

Method 1: Use probability

Define various scenarios or economic states and then subjective probability of each economic state of occurring.

Calculate expected value E[r${}_{i}$]
Scenario 1: 25% probability of 30% return;
Scenario 2: 50% probability of 10% return;
Scenario 3: 25% probability of -10% return;
E[r${}_{i}$] = 0.25 x 0.30 + 0.50 x .10 + 0.25 x – 0.10
= 0.10 or 10%

Method 2: Use CAPM

Expected future returns of two assets
E[rp] = α1 x E[r1] + α 2 x E[r2]
E[rp] = α 1 x E[r1] + (1- α 1)x E[r2]

Expected future return of a portfolio

Weighted average of expected stock returns:
E[rp] = α 1 x E[r1] + α 2 x E[r2] + α 3 x E[r3] + ..
α 1 = % of stock 1 in portfolio based on $ value
α on all stocks must add to 100% or 1.00

If % weight of each stock is spread equal on a portfolio then α is dropped off and can use the average() expected returns of the portfolio

Total Risk

3 types of risk

1.    Default risk
Company goes bankrupt and shareholder get little. Shareholder returns ≈ -100%
2.    Total risk σ
Standard deviation or average spread
Total Risk = Systematic risk + Unsystematic risk
3.    Systematic risk (β)
CAPM

Measuring past total risk
For an individual stock

  1. Calculate weekly returns based on prices;
  2. Calculate std dev of stock’s returns;

For portfolio of stocks (equal weights)

  1. Calculate weekly returns for each stock;
  2. Weekly portfolio returns = average of total stock;
  3. Calculate std dev of portfolio returns

Need to ‘annualise’ this standard deviation as it is based on weekly returns. Annualise by multiplying std dev of portfolio by √52.

Portfolio Total Risk

Covariance \sigma $${}_{1,2}= Measure of how two variables move together

Correlation coefficient = Measure the same thing as covariance but as a ratio
[{\rho }_{{\rm 1,2}}{\rm =}\frac{{\sigma }_{{\rm 1,2}}}{{\sigma }_{{\rm 1}}{\rm \times \ }{\sigma }_{{\rm 2}}}\]
-1$<$$\rho $$<$ 1
\rho ≈ 1 means strong relationship between the variables

The variance of 2-asset portfolio

  • Must include relationship correlation coefficient

{\sigma }^{{\rm 2}}_{{\rm p}}{\rm =}{\alpha }^{{\rm 2}}_{{\rm 1}}{\sigma }^{{\rm 2}}_{{\rm 1}}{\rm +}{{\rm (1-}{\alpha }_{{\rm 1}}{\rm )}}^{{\rm 2}}{\sigma }^{{\rm 2}}_{{\rm 2}}{\rm +2}{\alpha }_{{\rm 1}}{\rm (1-}{\alpha }_{{\rm 1}}{\rm )}{\sigma }_{{\rm 1,2}}
{\sigma }^{{\rm 2}}_{{\rm p}}{\rm =}{\alpha }^{{\rm 2}}_{{\rm 1}}{\sigma }^{{\rm 2}}_{{\rm 1}}{\rm +}{{\rm (1-}{\alpha }_{{\rm 1}}{\rm )}}^{{\rm 2}}{\sigma }^{{\rm 2}}_{{\rm 2}}{\rm +2}{\alpha }_{{\rm 1}}{\rm (1-}{\alpha }_{{\rm 1}}{\rm )}{\rho }_{{\rm 1,2}}{\sigma }_{{\rm 1}}{\sigma }_{{\rm 2}}
where {\alpha }_{{\rm 2}}{\rm =(1-}{\alpha }_{{\rm 1}}{\rm )}

The definition of the correlation coefficient
{\rho }_{{\rm X,Y}}{\rm =}\frac{{\sigma }_{{\rm X,Y}}}{{\sigma }_{{\rm X}}{\sigma }_{{\rm Y}}}

The standard deviation
{\sigma }_{{\rm p}}{\rm =}\sqrt{{\sigma }^{{\rm 2}}_{{\rm p}}}
The weight \alpha* on the first asset in the minimum variance portfolio

E(r${}_{mvp}$) = $\alpha $*E(r${}_{1}$) + (1-$\alpha $*)E(r${}_{2}$)

where {\alpha }^{{\rm *}}{\rm =}\frac{{\sigma }^{{\rm 2}}_{{\rm 2}}{\rm -}{\rho }_{{\rm 1,2}}{\sigma }_{{\rm 1}}{\sigma }_{{\rm 2}}}{{\sigma }^{{\rm 2}}_{{\rm 1}}{\rm +}{\sigma }^{{\rm 2}}_{{\rm 2}}{\rm -}{\rm 2}{\rho }_{{\rm 1,2}}{\sigma }_{{\rm 1}}{\sigma }_{{\rm 2}}}

corporate finance mininum variance portfolio

Diversification

Diversification allows portfolio to obtain:
The return is equal to weighted average of each component with risk less than the weighted average of each components

Low covariance is at the heart of diversification.
\sigma{}_{ }{}_{1,2} = \rho {}_{ }{}_{1,2} \times \sigma {}_{ }{}_{1 } \times \sigma {}_{ }{}_{2}
In 2 asset case, the degree of correlation (and hence covariance) was a key concept. The lower the correlation the greater the gains from diversification.

In N-asset case, the covariance is a key concept. The variance of a well diversified portfolio is the average covariance of the assets that make up the portfolio.
e.g. the correlation between assets is 0.5 and each asset has std deviation of 20% so the average covariance is (0.5)(0.2)(0.2) =0.02. The portfolio variance will decrease to 0.02 giving a standard deviation of 14.14%.

Disadvantage when the portfolio has very poor assets.
Correlation and covariance may be very high – sub prime mortgages. Individual component are high risk but in a portfolio the risk is reduced.

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