case study 1

Introduction to Finance Case Study 1: Basics of Modern Portfolio Theory

Team members of Group 18:

Felix Starke 371747 [[email protected]]

Tom Eggers 416931 [[email protected]]

Jonas Bungert 416749 [[email protected]]

Tim Echterling 424032 [[email protected]]

Group 18: Starke, Felix; Eggers, Tom; Bungert, Jonas; Echterling, Tim 1

Index of abbreviations:

CAL Capital allocation line

SD Standard deviation

SR Sharpe ratio

T-Bills Treasury-Bills

Index of symbols:

X,Y Random variables (e.g. rate of return)

Standard deviation of the random variables X and Y

Standard deviation of the complete portfolio

Standard deviation of the risky portfolio

Standard deviation of the risk free asset

Risk free interest rate

Return of the risky portfolio P

Correlation of the random variables X and Y

, Weight of the random variables X and Y

A Risk aversion factor

List of figures:

Figure 1 Illustration of the efficient frontier, CAL, investor's indifference curve and optimal risky or complete portfolio


Exercise 1: a. Shorting/short sales: is the selling of borrowed financial assets which will be subsequently purchased in the future enables the investor to make a profit from declining asset prices

b. Covariance: shows the linear relation of two random variables, however it does not describe the magnitude of the relation

Correlation:

;

describes the magnitude of the linear relation of two random variables

c. Set of risky assets: sum of all possible risk-return combinations for portfolios of risky assets Efficient frontier: dominant portfolios of risky assets that provide the best risk-return combinations, given or . Left boundary of the opportunity set, upwards from the minimum variance portfolio

d. Global minimum variance portfolio: The efficient portfolio of risky assets that has the lowest

variance :

e. Risk-free asset: security without risk ( , e.g. T-Bill Capital allocation line (CAL): represents all possible risk-return combinations of the optimal risky

portfolio and the risk-free asset:

Sharpe ratio =

=

= slope of the CAL measures the risk-return

combination of a portfolio the higher the SR the better the portfolio performance

f. Optimal portfolio and the related indifference curve/utility function: The combination of risky and risk-free assets that generates the highest feasible utility for an individual investor Optimization condition: slope of indifference curve = slope of the CAL


Exercise 2: Figure 1: Illustration of the efficient frontier, CAL, investor's indifference curve

and optimal risky or complete portfolio

Optimal risky portfolio

Optimal complete portfolio

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

0% 5% 10% 15% 20% 25% 30%

E (r

)

σ

Inefficient frontier Efficient frontier

Capital allocation line Investor's indifference curve

Minimum variance portfolio

Source: own illustration.

Optimal risky

portfolio


a. The efficient frontier:

Starts upwards from the Minimum-Variance-Portfolio (MVP) with

b. The capital allocation line:

The capital allocation line starts at with the slope 0.4479

c. The investor´s indifference curve:

Can be determined by converting the utility function

to

with Ū=0.0657 from the optimal complete portfolio

d. The optimal risky portfolio: The point of intersection of the investor´s indifference curve and the CAL represents the location of the optimal portfolio Can be determined by 1. Calculating the optimal composition of the risky portfolio by using either

i. Excel Solver

ii. Formula

iii. Estimation through a data table and maximizing the Sharpe ratio 2. Calculating the optimal weights of the risky portfolio P and the risk-free asset by using either

i. Excel Solver

ii. Formula

e. Weights of X, Y and of the risk-free asset in the optimal complete portfolio: The portfolio is composed of , thus is leveraged by 69%,

generating with


Exercise 3: Influence of the following ceteris paribus input variations on the portfolio optimization:

a. The risk-free rate :

Slope of the CAL = Sharpe ratio =

The higher the smaller the excess return The CAL has a higher intercept and

flatter slope, resulting in a riskier tangential portfolio The investor allocates a higher proportion to the risk-free security his indifference curve is tangent to the CAL further left, i.e. towards

- For a decreasing the contrary logic holds

b. The risk aversion A:

The higher A the lower the utility of a given portfolio

combination Investor is more risk averse The investor specific complete portfolio therefore allocates more in and less in

- For a declining the opposite is true

c. The :

; ;

1. For

Diversification not useful Optimal risky portfolio via specialization into the asset with the higher Sharpe ratio Investor specific portfolio is a mix of and that asset, with maximized utility

2. For

A perfect hedge is possible:

with If then , the individual invests all in , else in

3. For

Diversification is sufficient with weights depending on

Optimal complete portfolio splits between resulting combination and , maximizing utility

For : The higher the less risky is , so the larger is the investment in the

risky portfolio, due to hedging possibilities

For 0 : The higher the riskier is , so the smaller is the investment in the

risky portfolio

Exercise 4: Weights of X and Y if borrowing and lending at the risk-free rate is prohibited:

If borrowing and lending at the risk-free rate is prohibited, only the risky assets X and Y are available, thus representing the tangential portfolio with and ,

case study 1

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