Chapter 4 - Portfolio Theory

Question 1

What does MPT stand for?

Answer 1

Mean-variance Portfolio Theory

Question 2

What does the MPT assume about what investment decisions are based on?

Answer 2

MPT assumes that investment decisions are base solely up risk (variance) & return (mean).

It also assumes investors are will to accept more risk for higher expected return.

Question 3

What is MPT used for?

Answer 3

The MPT provides an investor a method to construct a portfolio that gives the maximum return for a specific risk, or the minimum risk for a specified return i.e. the investors optimal portfolio.

Question 4

What are the 7 assumptions of MPT?

Answer 4

1. All expected returns, variances and covariances of pairs of assets are known
2. Investors make their decisions purely on the basis of expected return and variance
3. Investors are non-satiated
4. Investors are risk-averse
5. There is a fixed single-step time period
6. There are no taxes or transaction costs
7. Assets may be held in any amounts, ie short-selling is possible, we can have infinitely divisible holdings, and there are no maximum investment limits

Question 5

Define an inefficient portfolio.

Answer 5

A portfolio is inefficient if the investor can find another portfolio with the same (or higher) expected return and lower variance, or the same (or lower) variance and higher expected return.

Question 6

True or False: An investor for is risk averse and non-satiated would consider an inefficient portfolio

Answer 6

False

Question 7

What are the two assumptions that allow for the definition efficient portfolios?

Answer 7

• Investors are never satiated. i.e. at a given level of risk they will always prefer a portfolio with higher return to a lower return.
• Investors dislike risk. For a given level of return they will always prefer a portfolio with lower variance to one with a higher variance

Question 8

What do we need to be able to rank efficient portfolio?

Answer 8

An investors utility function - this will then determine the investors portfolio.

Question 9

What is the formula to calculate the proportion of asset A & asset B which give you the minimum variance?

Answer 9

X_a = (V_b -C_ab)/(V_a - 2*C_ab +V_b)

Question 10

When is the efficient frontier a straight line for a portfolio with 2 risky assets?

Answer 10

When the correlation coefficient is either 1 or -1.

When -1 we need positive holdings of both
When +1 we need negative of 1 and positive of the other

Question 11

Define indifference curves.

Answer 11

Portfolios lying along a single curve which all give the same value of expected utility and so the investor would be indifferent between them. They slope upwards for a risk averse investor.

Question 12

Why is the optimal portfolio on the efficient frontier at the point where the frontier is at a tangent to an indifference curve?

Answer 12

The optimal portfolio occurs at the point where the indifference curve is tangential to the efficient frontier for the following two reasons:

1. The indifference curves that correspond to a higher level of expected utility are unattainable as they lie strictly above the efficient frontier.

2.Conversely, lower indifference curves that cut the efficient frontier are attainable, but correspond to a lower level of expected utility.
The highest attainable indifference curve, and corresponding highest level of expected utility, is therefore the one that is tangential to the efficient frontier.

The optimal portfolio occurs at the tangency point, which is in fact the only attainable point on this indifference curve, which is why it is optimal.

Question 13

What is Lagrangian function formula?

Answer 13

W = Variance of portfolio - lambda(Mean of portfolio - E_p) - mu(Xa+Xb+Xc + … -1)

Where,
- E_p & 1 are the constraining constants
- lambda & mu are known as the lagrangian multipliers

Question 14

What do we use lagrangians function for?

Answer 14

To find the proportions of assets which minimise the variance.
(This will be a function of E)

Question 15

If assets are independent, what is the formula for the portfolio variance?

Answer 15

V_p=sum[1,n] (x_i)^2V_i + sum[i=1,n]sumj=1,n Cij

If independent C_ij=0 so we get

V_p=sum[0,n] (x_i)^2*V_i

or alternatively,

V_p= X_1^2V_1 + (X_2)^2V_2 + … + (X_n)^2*V_n

Question 16

Prove that the specific risk (i.e. risk associated with each asset( of a portfolio approaches zero when the covariance between assets are 0.

Answer 16

V_p=sum[0,n] (x_i)^2*V_i

If we assume equal amounts in each N assets i.e. 1/N then we have
V_p=sum[0,n] (1/N)^2V_i
V_p=(1/N)sum[0,n] (1/N)V_i
V_p=(1/N)*Vbar

Where Vbar is the average variance.

As N tends to inf, then V_p tends to 0.

Question 17

What is the variance of a portfolio where assets are not independent.

Answer 17

V_p = Vbar/N + ((N-1)/N)*Cbar

C_bar average of convariances
Vbar average variance

Question 18

Define the Efficient frontier.

Answer 18

The efficient frontier is the line that joins the points in expected return-standard deviation space that represent efficient portfolios.
A portfolio is efficient if the investor cannot find a better one in the sense that it has either a higher expected return and the same (or lower) level of risk (measured in terms of standard deviation of returns) or a lower level of risk and the same (or higher) expected return.

Question 19

Define an Optimal portfolio.

Answer 19

The investor’s optimal portfolio is the portfolio on the efficient frontier that gives the highest possible level of expected utility, given the investor’s particular indifference curves.
It is represented by the point in expected return-standard deviation space where the efficient frontier is tangential to the highest attainable indifference curve

Question 20

What is an indication that an investor is risk averse if they invest in two assets with E_A = 10%, Simga_A=20% and E_B=15%, Sigma_B=30%?

Answer 20

If we compare asset A to asset B, asset B has 50% higher expected return for higher risk but the investor chooses to also invest in asset A which has lower risk but at a cost of lower expected returns.

Question 21

How many data points do we need for Portfolio Theory?

Answer 21

N means
N variance
N(N+1)/2 Covariances

N+N+N(N+1)/2 = N(N+3)/2

Question 22

What is the disadvantage to MPT?

Answer 22

The computational difficulty and the amount and type of data needed.

Question 23

What is the formula of the variance if the correlation coefficient of the portfolio is 1?

Answer 23

Vp= (XAVA + XBVB)^2