Quiz 2

Question 1

When we add a risk free asset to our portfolio, why does the variance equation not change?

Answer 1

No change = The variance equation remains the same as the level of risk is determined by the risky assets. In other words the risk free asset does not add risk.

Question 2

What is the one fund theorem?

What does the one fund theorem suggest?

Answer 2

One fund theorem = States that when a risk-free asset exists, that the entire set of efficient portfolios can be created by combining together one fund of risky assets, T, with the risk-free asset. The fund T is the tangency portfolio

Suggests = that an investment management companys only needs to create on fund of risky assets for all of their clients

Question 3

What is Tobins seperation theorem?

Answer 3

Tobins seperation theorem = states that an individuals investment decision can be separated into two parts

1: Determine tangency portfolio
2: Combine T with the risk-free asset in such a way that utility is maximised

Question 4

What do we mean by solve the markowitz problem?

Answer 4

Solve markowitz = find the weights of the assets which would achieve the specified portfolio Mu

Question 5

What do we mean when we say no borrowing exists at the risk free rate?

What does this imply for the one fund theorem?

Answer 5

No borrowing = we are referring to the fact that a portfolio consisting of risky assets and a risk free asset can achieve a higher return than the tangency portfolio if we place a negative weight into the risk free asset (and as such are borrowing) and thus continue in a straight line along the efficient frontier. Now when no borrowing exists we are unable to do this and as such if we want a return greater than the tangency portfolio we will have to invest 0 in the risk free asset and in fact follow along the efficient frontier of the bullet.

Implications of the one fund theorem = the one fund theorem no longer holds above Mu T

Question 6

What do we mean when we say risk free borrowing exists at a higher rate than the risk free lending rate?

Answer 6

Risk free borrowing at higher rate = It is possible to lend (invest positively) at the lower rate as we normally would until we hit the tangency portfolio. here it is possible to invest fully in the tangency portfolio or achieve a slightly higher return (same variance) portfolio by borrowing (investing negatively) at the higher borrowing rate. Obviously if we could simply continue borrowing at the lower rate then this would achieve the highest return portfolio

Question 7

In our course we use excel to solve when No short selling is applied as a restraint. As such we only need to know a few smaller questions concerning this part of the topic.

1. What does this mean for our weights?
2. What does the MVS where no short selling is allowed look like?

Answer 7

Weights = all weights must be greater than or equal to 1

MVS = a curved bullet that cannot exceed the returns of the highest performing asset nor underexceed the returns from the lowest performing asset. This bullet also stays inside the MVS (risky asset only) bullet and can touch it, but never go out of it).

Question 8

In the equilibrium model what two conditions hold?

Answer 8

1. All investors preferences are satisfied and they hold an optimal portfolio based on their risk preferences
2. Markets clear i.e. for every buyer of an asset there is a seller

Question 9

What does the capital market line determine?
Which equation defines the CML?

What does the security market line determine?
Which equation defines the SML?

Answer 9

CML = determines the expected return of efficient assets (with respect to the total risk)
Equation = same equation as we use for the MVS with a risk free asset, however we swap Mu T and Sigma T for Mu M and Sigma M

SML = determines the expected return of all assets (with respect to the systematic risk)
Equation = the CAPM equation (where we replace the gradient of a straight line with Beta)

Question 10

Under the CML what happens to the market portfolio when:
Investors are more risk-averse
Investors are less risk-averse

Answer 10

More = they will not place much money into the market portfolio (preferring the Rf), and it will have a high expected return

Less = they will put more money into the market portfolio and it will have a low expected return

Question 11

Under the CAPM Assumption 1 states that investors choose between portfolios on the basis of only two things? What are they?
Which 2 conditions justify this assumption?

Answer 11

They are = expected return and standard deviation of returns

1: Asset returns are jointly normally distributed
2: utility functions are quadratic

Question 12

What is an Isobeta line? What does it help us to understand?

Answer 12

Isobeta line = a line extending horizontally from the CML. This line measures the total risk of an asset (systematic + unsystematic). As we start on the line we have only systematic risk. As we extend through to the right hand side of the line we add more and more unsystematic risk, thus increasing our total risk. However all assets along the line will in fact have the same return.

The line is a good means for understanding that there is no risk premium for unsystematic risk

Question 13

Why do we use factor models?
What are risk factors?
What exactly does the SFM assume?
What do factor models hope to accomplish?

Answer 13

Factor models = reduce the number of parameters to estimate in OMEGA and can increase the reliability of the estimate

Risk factors = sources of randomness = market index, GDP, employment rate, interest rate, etc.

The SFM = assumes that the randomness of every asset in the market is generated by a single common factor - typically the market portfolio

Accomplish = explain the covariances between the returns of any pair of assets by focusing on one or more factors which affect the assets

Question 14

What are the three assumptions of the SFM?

What are the implications of assumption 3?

Answer 14

Assumption 1: Expected value of error term is 0
Assumption 2: covariance between the error term and the market return is 0
Assumption 3: covariance between the error term of one asset and the error term of another asset is 0

Implications of assumption 3 = the two assets in question must hence be related only through one factor, the return on the market portfolio

Question 15

What is a characteristic line?

Answer 15

Characteristic line = straight line plotted in stock returns - market returns space. The line is in fact a regression which achieves a line of best fit between an individual stocks returns and the corresponding market returns.
Equation = ri = ai + Brm

Question 16

Under the SFM we use our asset weights to determine the portfolio return etc. However what is unique about the situation where we use asset weights to determine our sigma^2epsilon?

Answer 16

Answer = we square our weights.

Question 17

How do we calcuate R^2?
What does R^2 = 1 imply?
If we have a portfolio of N assets and we continue to add assets until we reach infinity, what will happen to our unsystematic risk?

Answer 17

Calculate R^2 = This is super easy… in fact R^2 simply measures the systematic risk over total risk… Hence we would calculate systematic risk using our equations and then place it over the total risk to produce our R^2 value.

R^2 = 1 implies that our portfolio is fully diversified and includes only systematic risk.

Add assets = Our systematic risk will reach zero as we add more assets and hence diversify our portfolio

Question 18

How does assumption 2 vary between a single factor model and a multi factor model?

Answer 18

Assumption 2 = Assumption 2 extends from simply being “the covariance between the error term and the single factor is 0” to “ the covariance between the error term and all factors is 0”

Note: The other two assumptions remain the same.

Question 19

Define arbitrage?

What are the two fundamental assumptions of the APT?

Answer 19

Arbitrage = when there exists a zero-cost portfolio (or security) that yields a non-negative payoff for sure, and with the possibility of yielding a strictly positive payoff

1: Arbitrage opportunities do not exist in the market
2: Security returns are generated by a factor model

Question 20

What are the two advantages of APT over the CAPM?

Answer 20

1: Less restrictive = APT does not assume that all investors have homogeneous beliefs on which they act optimally (instead APT assumes that investors will exploit arbitrage opportunities so that mispriced assets are driven towards their equilibrium price)
2: Proponents of APT argue that the model can be verified empirically

Question 21

What is the fourth assumption under the APT?

What is the fifth assumption under the APT?

Answer 21

Assumption 4 = No arbitrage condition = all portfolios that (1) require no money and (2) have no systematic risk, generate no expected profits
- This condition ensures that all riskless portfolios must generate a common (risk-free) return, which we denote by Lambda 0

Assumption 5 = Each asset i can be treated as a diversified portfolio i.e. error term of i is aprox equal to 0

Question 22

Under the APT what is Lambda 1?

Under the APT what is Lambda 0

Answer 22

Lambda 1 = is called the FACTOR PRICE = it is the rate at which the market rewards investors for holding risk associated with this undiversifiable risk factor

Lambda 0 = Risk-free rate

Question 23

Under the APT we normalize our factor first.

What does this mean and what are the implications.

Answer 23

Normalization = Allows all factors with their varying magnitudes to be read on the same scale.
Implications = Our equations for expected return, variance, and covariance become much simpler
1. The expected value of the factor = 0
2. The variance of the factor becomes 1

Question 24

What 3 conditions must hold in order for the return of a bond to equal the yield of the bond

Answer 24

1) The issuer of the bond does not default
2) The bond is held to maturity
3) All payments from the bond are reinvested at the original yield

Question 25

What are fixed income securities?

What are the two types of bonds?

Answer 25

Fixed income securities = are traded instruments that give the holder claim to a series of future payments, fully specified in advance

Zero-coupon bond = a bond where only one payment is made at maturity
Coupon paying bond = a bond where interest payments (coupon payments) are made over the bonds life (usually semi-annually) and the face value is paid at maturity

Question 26

Why is their an inverse relationship between market price of a bond and yield of a bond?

Answer 26

This is because the interest on a bond is a fixed coupon. This coupon, say $10, will represent less and less interest as the price of the bond goes up i.e. the $10 fixed coupon (doesn’t change in value) represents a smaller and smaller amount as the price of the bond rises.

Question 27

Define Yield to Maturity?

What is Curent Yield?

Define Holding Period Yield?

Answer 27

Yield To Maturity = is the unique interest rate that converts a stream of future payments to their present value.

Current Yield = bond coupon / current market price x 100 = the coupon expressed as a percentage of the current market price for the bond = only an approximation of the true YTM

Holding period yield = The HPY is the rate of return the buyer receives over the period the bond is held = if bond is held to maturity then HPY equals to YTM

Question 28

What does the risk-free rate depend on?
Define term structure of interest rates?
Define Bond stripping?
Outline the process for constructing a zero-coupon yield curve when only given regular (coupon paying) bonds.

Answer 28

Risk free rate = depends on the holding period of our investment

Term structure of interest rates = the relationship between the time to maturity of a certain class of bonds and their yields

Bond stripping = the process of converting a coupon paying bond to a zero-coupon bond

Bond stripping and yield curve process:

1. We first take the bond which will mature first and calculate its price using the bond pricing formula.
2. next we equate this first bonds price to the (face value + coupon payment) / (1+ zero coupon rate/ 2)
3. Rearrange the above to find the value of the first period zero coupon yield
4. Calculate the price of the second maturing bond
5. equate the price of this bond to the (coupon payment divided by 1 + first period zero coupon yield / 2) + (face value + coupon payment divided by 1 + second period zero coupon yield / 2) ^ 2
6. … continue the pattern

Question 29

Outline the market expectations theory? What are the implications?

Answer 29

MET = the term structure of interest rates is determined solely by the markets expectation regarding future interest rates
Implications = investors are indifferent between holding:
- an n-period security through to maturity or,
- a series of one period securities over the same n periods.
= if the market expects future short-term rates to be higher then the yield curve will be increasing. If it expects them to be lower it will be decreasing (inverted)

Question 30

Outline the Liquidity Preference Theory? What are the implications?

Answer 30

LPT = Investors require a premium for holding bonds with maturities that are different from their investment horizon

Implications = the expected return on a succession of 2 one-period bonds is less than the expected return on an equivalent long-term bond = the yield curve has a positive slope

Question 31

Outline the Market Segmentation Theory?

Outline the Preferred Habitat Theory?

Answer 31

MST = markets are segmented according to different times to maturity, with yields determined entirely separately by the supply and demand for funds in each segment = for a normal yield curve the supply and demand of funds produces lower short-term yields and higher long-term yields

PHT = combines MST and LPT. It assumes that while investors have preferred maturities, they will be prepared to modify the maturity of their investments so long as they receive a premium for doing so.

Question 32

When will the return of the bond we equal to the yield of a bond?

Answer 32

Return = yield:

1) The issuer of the bond does not default - default risk
2) The bond is held to maturity - interest rate riAsk
3) All payments from the bond are reinvested at the original yield - reinvestment risk

Question 33

The relationship between bond price and coupon rate:

Identify the following. cy.

Answer 33

cy = premium bond, P>F

Question 34

Outline the two ways changes in interest rates effect the return on a bond?
Discuss the reinvestment risk of a zero-coupon bond?

Answer 34

(1) Price risk = the risk of a drop in a bonds market value due to a rise in interest rates if the bond has to be liquidated prior to maturity
(2) Reinvestment risk = the risk associated with a change in the future value of the investment due to a change in the rate at which coupons can be reinvested

Zero-coupon bond reinvestment risk = has zero reinvestment risk as there are no coupon payments in the first place

Question 35

Which bonds are riskier:
Long term bonds vs short term bonds?
Low coupon bonds vs high coupon bonds?

Answer 35

Long term bonds are, other things equal, riskier than short term bonds

Low coupon bonds are, other things equal, riskier than high coupon bonds

Question 36

In regards to bonds, what do we mean by “Duration”

Compare the duration of a zero-coupon bond and a coupon paying bond?

How do we use Duration to measure the riskiness of a bond?

Answer 36

Duration = a measure of the interest-rate sensitivity of a fixed-income security = a concept that attempts to convert a multi-payment security into an equivalent zero-coupon security = it is the weighted average time at which you receive future cash flows = the larger the D the more risky the bond.

Duration of ZCB and CB = the duration of a zero-coupon bond is equal to its maturity (D = n). The duration of a coupon-paying bond is less than its maturity (D < n)

Duration as a measure of riskiness = Given a certain change in interest rates, duration can be used to approximate the percentage change in bond prices = the larger D is, the more sensitive the bond’s price is to a change in interest rates, so that duration is a measure of bond risk.

Question 37

Define immunization?
Define Cash matching in regards to immunization and the relevant problem?
Define Duration matching in regards to Immunization?

Define the concept of dynamic hedging?

Answer 37

Immunization = the action taken to make a bond portfolio’s market value independent of changes in interest rates. A portfolio is only fully immunized when it has no exposure to interest rate risk.

Cash matching = Immunization can be achieved by exactly matching the future cash flows of assets and liabilities.
- Relevant problem = cash matching can be difficult to achieve in practice

Duration matching = Immunization can be achieved by matching the duration of assets and liabilities.

Dynamic hedging = If after immunizing a change in interest rates actually occurs you will notice that D(liabilities) no longer equals to D(assets) = hence we would have to re-immunize if we wanted to hedge against a second change interest rates

Question 38

Outline the methodology for calculating D?

Once we have D(L) and D(1) and D(2) how do we calculate the required weighting of each bond?

How do we calculate the number of contracts required of each Bond in our portfolio?

Answer 38

Calculating D = in essence we separate each coupon payment as well as the final payment and calculate the PV for each of the payments. Next we add all of the PV together to form a PV total. Then we take each of the independent PV and multiply by their corresponding periods (ti) and then add them all together to form a weighted PV. Take this weighted PV total and divide by just the PV total.

• The same process is required for calculating the Duration of a liability.

Bond weighting = Simply equate D(L) to D(A) to x1D(1) + (1-x1)D(2)

Number of contracts = simply multiply the PV total (liability) by the weighting of each bond. Then take this figure and divide it by the PV total of each bond. resultantly you will produce n (the number of contracts required of each bond

Question 39

In terms of an efficient market, what do pasive investors believe vs what active investors believe?

Answer 39

Pasive investors = believe that the market is in fact efficient, such that the current market price is seen to be the best estimate of the asset’s fair value
Active investors = believe that the market is less efficient (not-inefficient, but less efficient)

Question 40

Discuss the superiority of passive vs active funds?

Additionally discuss the 3 general factors upon which portfolio return is most likely to depend?

Answer 40

1) growing evidence confirms the superior long-term performance of passive index investing (especially after the higher transaction and management fees of active funds are taken into consideration)
2) There is little predictability in active performance, outperforming the market one year provides no indication of the ability of an active fund manager to do so again.

three general factors = the targeted risk level of the portfolio itself, the market performance, and the skill level of the portfolio manager.

Question 41

Discusss the appropriate measure of risk for 1) an investor holding only one asset (or a less diversified portfolio) and 2) an investor holding multiple assets (or a diversified portfolio)?

Answer 41

one asset / low diversification = we use variance ad the correct measure of risk
Multiple assets / diversified portfolio = we use systematic risk or beta as the appropriate measure.

Question 42

Outline the Jensen index?
Outline the implications of the efficient market hypothesis here?

What do we mean by Depth and Breadth?
Discuss the sensitivity of the Jensen index to Depth and Breadth?

Answer 42

The Jensen index = represents the differential return of the managed portfolio given the return to a portfolio laying on the SML and with the same portfolio Beta as the managed portfolio

• Jp > 0 indicate better performance than that available using the SML
• Jp < 0 indicate worse performance

efficient market hypothesis = impossible to outperform market = all portfolios will fall on the SML and will therefore have Jp = 0

Depth = relates to the magnitude of the excess return earned by the manager
Breadth = relates to the number of different assets for which the manager can earn an excess return.

Jensen index = sensitive to only Depth and not Breadth.

Question 43

Outline the Treynor index?

Discuss the sensitivity of the Trynor index to Depth and Breadth?

Answer 43

Treynor Index = uses portfolio systematic risk instead of total risk. It measures the risk premium earned per unit of risk taken. It assumes that due to diversification investors are only concerned with the systematic risk component of their portfolios.

Treynor index sensitivity = the Treynor index is also insensitive to the breadth of portfolio performance ( as is the Jensen).

Question 44

Discuss the Sharpe index?

Discuss the sensitivity of the Sharpe index to Depth and Breadth?

Answer 44

Sharpe index = uses the capital market line (CML) as a benchmark. It measures the portfolio risk premium per unit of the total risk of the portfolio. The Sharpe index of the market is the slope of the CML.

Interpretation = portfolios lieing above the CML have a higher Sharpe index and have thus outperformed the market. (Below CML = under-performed the market)

Sensitivity of Sharpe index = is sensitive to both Depth and Breadth and can thus measure relative management skills.

Question 45

Discuss the usefulness of the Treynor and Sharpe index?

Answer 45

The Treynor index = is a useful measure if unsystematic risk of the portfolio is irrelevant. It is an apporpriate measure for funds to be included in a diversified portfolio.

The Sharpe index = is an appropriate measure for sole investment, since it includes all risk. It penalises funds not sufficiently diversified.

Question 46

Problems of Performance Measures based on the CAPM:
What happens if their is a mispricing?
What happens if you cannot borrow at the risk-free rate?

Answer 46

mispricing = performance indicators are likely to be biased

No risk-free borrowing = the CML will be kinked and the estimated SML will be different to the true SML

Question 47

Discuss the problem of mis-specification of the market index?

Answer 47

The problem is you don’t know whether you have chosen the true market portfolio and consequently there is uncertainty whether the resulting market
performance indices are related to:
- the quality of the portfolio management, or
- the inadequacy of the selected market index.

Question 48

Market efficiency:

Outline the Random walk hypothesis?

Answer 48

Random walk hypothesis = a theory stating that security market prices evolve according to a random walk and therefore cannot be predicted = a price series is said to obey a random walk if successive changes in price are independently distributed = the implication of independence is that price movements up to time t cannot be used to accurately forecast price changes at t+1 and beyond.

Question 49

Market efficiency:
outline the Efficient Market Hypothesis?
Outline the three forms of the EMH?

Answer 49

Efficient market hypothesis = market efficiency measures how quickly the price of the securities adjust to reflect the new information = if prices respond to all relevant information in a rapid fashion, we say the market is relatively efficient = hence an informationally efficient market may be defined as one in which all available information is both fully and correctly reflected in security prices.

Weak form = stock prices reflect any information that may be contained in the past history of the stock price itself.
Semi-strong form = stock prices reflect all publicly available information
Strong form = stock prices reflect all information. This includes private or inside information as well as public information.

Weak = information in past stock prices
Semistrong = all public information
Strong = all available information

Question 50

Discuss market anomalies in relation to the EMH?
What are the three types of Share market anomalies?
What are Superstitious Market indicators?

Answer 50

Market anomalies = provide a challenge to the EMH = Anomalies refer to situations when a security or group of securities performs contrary to the notion of efficient markets i.e. patterns in prices that are (1) consistent and (2) persistent.

1) Calander anomalies = patterns of trading behaviour that occur at certain times of year (day of the week effect, january effect, end of the month effect)
2) Fundamental anomalies = regularity in a stock’s performance due to fundamental factors (size anomaly, neglected stocks, value anomaly)
3) Event anomalies = consistent market reactions to specific market events (reversal effects, stock split effects, post earnings announcement drift, merger arbitrage)

Superstitious market indicators = non-market signals that may indicate the direction of the market (the super bowl indicator, the hemline indicators, the aspirin indicator)

Question 51

Contrast Traditional finance and Behavioural finance

In regards to Behavioural finance, outline the following irrational behaviours: Information processing errors, and behavioural biases?

Answer 51

Traditional finance = assumes individuals are risk-averse, self-interested utility maximizers and that the markets in which they trade are efficient = what should actually happen

Behavioural finance = approaches decision making from an empirical perspective. It identifies patterns of individual behaviour without trying to justify or rationalise them = may be understood as the study of irrational behaviour = what actually happens

information processing errors = investors do not always process information correctly and therefore infer incorrect probability distributions for future returns. Such errors stem from basic statistical, information-processing, or memory errors. They are also the result of reasoning based on faulty thinking.

• Forecasting errors = people give too much weight to recent experience compared to prior beliefs when making forecasts
• Overconfidence = people overestimate their abilities and the precision of their forecasts
• Conservatism = investors are too slow in updating their beliefs, causing an initial under reaction to news.
• Representativeness bias = people do not properly account for small sample sizes, resulting in the (incorrect) inference of non-existent patterns.

Behavioural biases = even with a correct distribution of returns, investors often make inconsistent or systematically sub-optimal decisions. These often stem from impulse or intuition or are the result of reasoning based on feelings.
- Framing = decisions are affected by how choices are formed
- Mental accounting = decisions are often segregated into different portfolios with different objectives, rather than being treated from a total portfolio perspective
- Regret avoidance = people blame themselves for (or regret) more unconventional choices that turn out badly, so they avoid regret by making more conventional decisions.
Prospect theory = modifies the description of rational risk-averse investors by incorporating loss-aversion. Here, utility depends not on the level of wealth but on changes in wealth from current levels. The implication being that investors are risk-averse to gains but risk-seeking to losses.

Question 52

Discuss the limits of Arbitrage in relation to behavioural biases?

Answer 52

Behavioural biases shouldn’t matter for the market as a whole if there are at least some rational investors taking advantage of the resulting arbitrage opportunities.
However, there are Limits to Arbitrage!

• Fundamental risk: Incorrect pricing could get worse before it gets better with the arbitrageur falling insolvent before the mis-pricing is corrected.
  “Markets can remain irrational longer than you can remain solvent!”
• Implementation costs: Short selling is relatively costly and cannot be scaled infinitely. Furthermore, in many derivatives there are restrictions on the size of a position that can be held.
• Model risk: There is a risk that the arbitrage, if based on a model, might not be a true arbitrage. Consider the assumptions of the APT.