Portfolio Analysis

Question 1

OUTCOMES OF MARKET

Answer 1

• Price and quantity
• Return - ex-post return, ex-ante return, measured in %, return on single asset/ return on portfolio of
assets
• Risk - measure of dispersion of returns, many measures (eg. Standard deviation, variance, range), individual asset risk/ portfolio risk - diversification effects
• Outcomes change according to asset type

Consider inflation effect

Question 2

Return on a portfolio

Answer 2

The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights.
rp = W1r1 + W2r2
W1 = Proportion of funds in
Security 1
W2 = Proportion of funds in
Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2

Question 3

Portfolio risk

Answer 3

• Portfolio Risk: a measure that estimates the extent to which the actual outcome is likely to diverge from the expected outcome
• Portfolio risk looks at the benefits of diversification - portfolio risk less than weighted average of each assets risk

formula = sum each pair of combination of covariances multiplied with corresponding weights

• COVARIANCE
– DEFINITION: a measure of the relationship between two random variables
– possible values:
» positive: variables move together
» zero: no relationship
» negative: variables move in opposite directions

• CORRELATION COEFFICIENT
– rescales covariance to a range of +1 to -1

covariance of i and j divided by stdev of i and j

Question 4

Other method for capturing expected return and risk

Answer 4

• Expected Risk and Return variables can also be measured using subjective probabilities
• All outcomes are determined according to likelihood (probability) of occurrence

Question 5

Sharpe Ratio for Portfolios

Answer 5

it is The Reward-to-Volatility

= risk premium / Stdev of excess return

Question 6

Use of Normal distribution

Answer 6

• Investment management is easier when returns are normal.
– Standard deviation is a good measure of risk when returns are symmetric.
– If security returns are symmetric, portfolio returns will be, too.
– Future scenarios can be estimated using only the mean and the standard deviation.

Normality and Risk Measures
• What if excess returns are not normally distributed?
– Standard deviation is no longer a complete measure of risk
– Sharpe ratio is not a complete measure of portfolio performance
– Need to consider skew and kurtosis

=average( R - Rbar)cube / sigma cube ( Skew)
=average( R - Rbar)power of 4/ sigma power of 4
(Kurtosis)

Question 7

Value at Risk ( another measurement of risk to capture potential large downside)

Answer 7

• A measure of loss most frequently
associated with extreme negative returns
• VaR is the quantile of a distribution below which lies q % of the possible values of that distribution
– The 5% VaR , commonly estimated in practice, is the return at the 5th percentile when returns
are sorted from high to low.

Question 8

Expected Shortfall

Answer 8

• Also called conditional tail expectation (CTE)
• More conservative measure of downside risk than VaR
– VaR takes the highest return from the worst cases
– ES takes an average return of the worst cases

Question 9

Historic Returns on Risky Portfolios

Answer 9

• Returns appear normally distributed
• Returns are lower over the most recent half of the period (1986-2009)
• SD for small stocks became smaller; SD for longterm bonds got bigger
• Better diversified portfolios have higher

Sharpe Ratios
• Negative skew

Question 10

Efficient Portfolio

Answer 10

Combinations of Two Risky Assets Revisited: Short Sales Not Allowed
• Case 1 – Perfect Positive Correlation (p = +1)
• Case 2 – Perfect Negative Correlation (p = -1.0)
• Case 3 – No Relationship between Returns on the Assets (p = 0)
• Case 4 – Intermediate Risk (p = 0.5)

imagine this graph, the more negatively correlated the more graph bends to vertical horizon

Can you extract and graph the efficient frontier ( no short selling)
(With short selling)

Remember it is only the part that is above min variance point

Question 11

The Efficient Frontier with Riskless Lending and Borrowing

Answer 11

• The introduction of a riskless asset into our portfolio possibility set considerable simplifies the analysis.
• We can consider lending at a riskless rate as investing in an asset with a certain outcome.
• Borrowing can be considered as selling such a security short; thus borrowing can take place at the riskless rate.

Question 12

Graph expected return and risk when the risk-free

rate is mixed with portfolio A.

Answer 12

portfolio A joined to risk free asset point, strait line. in between points reflect lending, the part further from A portfolio up reflect borrowing ( at risk free rate not realistic but for now)

Question 13

Graph The efficient frontier with lending but not borrowing at the riskless rate

Answer 13

jus efficient frontier and tangent from risk free point

Question 14

Graph The efficient frontier with riskless lending

and borrowing at different rates.

Answer 14

ledning and borroing rates would differ