Portfolio Risk and Return: Part 1

Question 1

Year AUM BOY Net return
1 30 10%
2 33 -5%
3 35 15%

Calculate
holding period return
arithmetic mean
geometric mean

Answer 1

HPR
1+.10 * 1+(-0.05) * 1+.15 = 20.18%

avg
10-5+15 / 3 = 6.67%

geo
[1+.10 * 1+(-0.05) * 1+.15]^1/3 - 1 = 6.317%
**don’t just use the HPR answer of 20.18%. re do it and dont forget to -1 at end

Question 2

When are the following used?
HPR
mean return
geometric mean return
money-weighted return/IRR
time-weighted rate of return 
annualized return
real return

Answer 2

HPR:
- the return on an asset during the period it was held. calculated as the some of cap gain and income/P0, or by 1+r1 * 1+r2 *1+rn
MEAN RETURN
- when averaging returns for a period
GEOMETRIC MEAN RETURN
- measuring performance over time
- its the compound rate of return on an investment
MONEY-WEIGHTED RETURN / IRR
- calculates the return on actual investment using cash inflows and outflows
- useful performance measure when the manager is responsible for timing CF
TIME-WEIGHTED RETURN
- measures the compound growth rate of $1 initially invested over a stated measurement period
ANNUALIZED RETURN
- converts the returns for periods that are shorter or longer than a year, to annualized number for easy comparison
REAL RETURN
- the return after deducting taxes and inflation

Question 3

Time-weighted rate of return
Jim purchases a share for $50 on Jan 1, 2011, makes another purchase on Jan 1, 2012 for $60. Each share paid a dividend of $3 at EOY. On Jan 1, 2013, Jim sold the 2 shares and collected $150. find TWR

Answer 3

*break the overall evaluation period into sub-periods based on dates of CF in/outflow. Cal HPR for each sub-period

Jan 1 2011 Jan 1 2012 Jan 1 2013
n = 1 n = 1
pv = -50 pv = -60
pmt = 3 pmt = 3
fv = 60 fv = 75
*it doesnt matter that there are 2 stocks in this per. just find the hpr return
cpt i/y = 26% cpt i/y = 30%

take geometric mean of the annual returns to get the TWRR:
[1+.26 * 1+.30]^1/2 - 1 = 27.98%

Question 4

Annualized return

2% return in 20 days annualized is

Answer 4

annualized return = ( 1 + return of the per) ^c - 1
c = number of periods in year

= [ (1 + .02) ^ (365/20) ]- 1

Question 5

Real return

Answer 5

(1 + nominal rate) = ( 1 + real rate ) * ( 1 + inflation )

Question 6

Portfolio expected return

Answer 6

E(Rp) = w1 * ER1 + (w2 *ER2)

Question 7

Portfolio Variance

Answer 7

Sp^2 variance of the portfolio = w1^2 * STD1^2 + w2^2 * STD2^2 + (2 * w1 * w2 * STD1 * STD2 * CORR)

**dont forget to * correlation

Question 8

Utility formula

Answer 8

Utility = E(r) - ( 0.5 * A * S^2)

U = expected return - ( 0.5 * measure of risk aversion * variance of the investment)

**always use decimal values

Question 9

An investor with A=2 owns a RF asset returning 5$. What is his utility?
Then, he considers an asset with STD = 10%, at what level of return will he have the same utility?

Answer 9

1. U = .05 - ( 0.5 * 2 * 0) = .05
2. .05 = E(r) - (0.5 * 2 * 10^2)
   solve for E(r) = 6%

Question 10

The risk-return relationship for a risk-seeking investor is most likely to be:

Answer 10

negative

Question 11

Portfolio’s STD formula

Answer 11

Sp = STDV of [ w1^2 * S1^2 + w2^2 * S2^2 + (2 * w1 * w2 * STD1 * STD2 * CORR) ]

Question 12

As correlation decreases, the diversification benefit ______

Answer 12

increases

ie the risk of the portfolio decreases

Question 13

Covariance formula

Answer 13

COV = STD1 * STD2 * CORR

Question 14

Minimum-variance frontier

Answer 14

• the line combining portfolios with minimum variance

- looks like a sideways U

Question 15

Global minimum-variance portfolio

Answer 15

• the portfolio on the efficient frontier with the lowers risk (minimum variance) among the other portfolios of risky assets
• it is not possible to hold a portfolio of risky assets that has less risk than the global min-var portfolio

Question 16

Efficient frontier

Answer 16

• the part of the minimum variance frontier that is above the global minimum-variance portfolio
• represents the set of portfolios that will give the highest return at each risk level
• consists of only risk assets (no RF)
• still a curved line

Question 17

Capital allocation line and optimal risky portfolio

Answer 17

• a straight line: the combination of risky assets and a risk-free asset

optimal risky portfolio:

• the point at which the capital allocation line is tangential to the efficient frontier
• based on the market and not the investor’s risk preferences

Graph of:
efficient frontier and capital allocation line
y-axis: portfolio expected return: rf asset return starts the optimal risky portfolio line
x-axis: portfolio stdv