1. Risk and Return

2. The objective to learn risk and return:
The appropriate discount rate used in capital budgeting is the firm’s average cost of capital, that is, the weighted average of costs of debt and equity.
While cost of debt is easy to estimate, the cost of equity is difficult to determine.
The realized cost of equity does not always apply.
We need to learn how equity holders determine their required rate of return when investing equity.
The force of capital market would make the equity holders’ required rate of return equal to the true cost of equity.

3. Risk and Return
Due to the return will be realized by the end of the period, so the investor will not know the return for sure, so the rate of return is only a random variable to the investor. The investor needs to estimate what will be the return.

4. The subjective way to describe random variable – Probability Distribution
Outcomes
Probability
T-bill
Corp. Bond
Common Stock
Serious Recession
0.05 8%
12%
-3%
Mild recession
0.20
10% 6%
stable
0.50 9%
11%
Mild prosperity
8.5%
14%
Extreme
prosperity
19%
Expected Return
9.2%
10.3%
Standard Deviation 0%
0.84%
4.39%

5. Expected Return
Possible outcome
probability
Standard deviation

6. The difficulty in using the subjective Probability Distribution
It is difficult to describe the probability distribution of an individual.
It is not certain that an individual’s probability distribution is in accordance with the probability distribution of investors of market as a whole, which is more relevant in determining cost of equity.

7. The objective way to describe random variable – historical data
We can measure risk and return employing historical returns. We basically believe that history will repeat itself.
Expected Return
Standard Deviation

8. The problems in using historical data in estimating risk and return
Some of firms, start-up or private firms, may lack of stock trades information in estimating realized risk and return.
This historical estimation sometimes may not be representative for future expectations.

9. Risk and return for two assetsＡ Ｂ
Expected return
Standard deviation
Investment ratio
Expected return for the portfolio
Standard deviation for the portfolio

10. The definition of Covariance
The covariance is to measure the co-movement of two random variable.
When covariance is positive, then the two random variables tend to move into same direction.
When covariance is negative, then the two random variables tend to more to opposite directions.

11. Risk and return for three assetsＡ Ｂ Ｃ
Expected return
Standard deviation
Investment ratio
Expected return for the portfolio
Standard deviation for the portfolio

12. Risk and return for N assets
Expected return
Standard deviation

13. Risk Diversification
When number of assets increases, the majority of risk for the portfolio comes from the co-movement among assets. The distinctive risk comes from each asset becomes less important.

14. How diversification worksＡ Ｂ
Expected return
10%
14%
Standard deviation 4% 6%
Investment ratio
50%

15. = = =

16. Combining Stocks with Different Returns and Risk
Assets may differ in expected rates of return and individual standard deviations
Negative or small positive correlations reduce portfolio risk
Combining two assets with -1.0 correlation is able reduces the portfolio standard deviation to zero.

17. Constant Correlation with Changing Weights
rij = 1.00

18. Portfolio Risk-Return Plots for Different Weights2
With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk-return along a line between either single asset
Rij = +1.00 1
Standard Deviation of Return

19. Constant Correlation with Changing Weights
rij = 0.00

20. Portfolio Risk-Return Plots for Different Weights2 g
With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset h i j
Rij = +1.00 k 1
Rij = 0.00
Standard Deviation of Return

21. Portfolio Risk-Return Plots for Different Weights2 g
With correlated assets it is possible to create a two asset portfolio between the first two curves h i j
Rij = +1.00 k
Rij = +0.50 1
Rij = 0.00
Standard Deviation of Return

22. Portfolio Risk-Return Plots for Different Weights
With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset
Rij = -0.50 f 2 g h i j
Rij = +1.00 k
Rij = +0.50 1
Rij = 0.00
Standard Deviation of Return

23. Constant Correlation with Changing Weights
rij = -1.00

24. Portfolio Risk-Return Plots for Different Weights
Rij = -0.50 f
Rij = -1.00 2 g h i j
Rij = +1.00 k
Rij = +0.50 1
Rij = 0.00
With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk
Standard Deviation of Return

25. A zero standard deviation for a two asset portfolio with correlation of -1
When and , the investor is able to form a zero standard deviation (that is, zero risk) portfolio.

26. The Efficient Frontier
The efficient frontier represents that set of portfolios, within all possibilities, with the highest rate of return for every given level of risk, or with the lowest risk for every given level of return
Efficient Frontier would be composed of portfolios rather than individual securities
Exceptions being the asset with the highest return and the asset with the lowest risk

27. Efficient Frontier for Alternative PortfoliosB
E(R) A C
Standard Deviation of Return

28. The Efficient Frontier and an Investor’s Utility
The optimal portfolio chosen has the highest utility for a given investor
The optimal portfolio lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

29. Selecting an Optimal Risky Portfolio
U3’
U2’
U1’ Y U3 X U2 U1

30. Summary:
The efficient frontier derived from risky assets will be a curve, rather than a linear equation.
Investors with different risk preferences will choose different points on efficient frontier to be their optimal choices.
So the equilibrium will be on the whole efficient frontier.
Problem: it is difficult to describe and apply a non-linear relationship of risk and return.
Solution: introduce a risk-free asset into the model…

31. Combining a Risk-Free Asset with a Risky Portfolio
Since both the expected return and the standard deviation of return for such a portfolio are linear combinations for that risky portfolio and the risk free asset, a graph of possible portfolio returns and risks looks like a straight line between the two assets.

32. Risk-Free Asset An asset with zero standard deviation
Zero correlation with all other risky assets
Provides the risk-free rate of return (RFR)
Will lie on the vertical axis of a portfolio graph

33. Risk-Free Asset Covariance between two sets of returns is
Because the returns for the risk free asset are certain,
Thus Ri = E(Ri), and Ri - E(Ri) = 0
Consequently, the covariance of the risk-free asset with any risky asset or portfolio will always equal zero. Similarly the correlation between any risky asset and the risk-free asset would be zero.

34. Combining a Risk-Free Asset with a Risky Portfolio
Expected return
the weighted average of the two returns
This is a linear relationship

35. Combining a Risk-Free Asset with a Risky Portfolio
Standard deviation
Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

36. Capital Market Line (CML)D M C B A
RFR

37. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier
CML
Borrowing
Lending M
RFR

38. The Security Market Line (SML)
The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m)
This is shown as the risk measure
The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance:

39. Graph of Security Market Line (SML)
RFR

40. The Security Market Line (SML)
The equation for the risk-return line is
We then define as beta

41. Graph of SML with Normalized Systematic Risk
Negative Beta
RFR

42. Factors that influences the shape of SML
Inflation risk premium (inflation expectation)
Investors’ attitude toward risk (risk aversion)

43. Inflation risk premium

44. Investors’ attitude toward risk (risk aversion)

45. Capital Market equilibrium
The required rate of return:
The expected rate of return:
When expected return > required return
Then P0 increases, expected return decreases
When expected return < required return
Then P0 decreases, expected return increases

46. .A .B