Past Exam Questions Flashcards
Arguments for using semi-variance as a risk measure
Most investors do not dislike uncertainty of returns as such; rather they dislike the possibility of low returns.
One measure that seeks to quantify this view is downside semi-variance.
Arguments against using semi-variance as a risk measure
- not easy to handle mathematically
- takes no account of variability above the mean
- if returns are symmetrically distributed, semi-variance is proportional to variance - so it gives no extra information
- Semi-variance measures downside relative to the mean rather than another benchmark that might be more relevant to the investor
4 Assumptions of mean-variance portfolio theory
- Investors select their portfolios on the basis of the expected return and variance of that return over a single time horizon
- The expected returns, variance of returns and covariance of returns are known for all assets and pairs of assets.
- Investors are never satiated. At a given level of return, they will always prefer a portfolio with lower variance to one with higher variance.
8 Key arguments against modelling market returns using a Gaussian Random walk
- Market crashes appear more often than one would expect from a normal distribution
- While the random walk produces continuous price paths, jumps or discontinuities seem to be an important feature of real markets
- Days with no change, or very small change, also happen more often than the normal distribution suggests
- Assumption of independent increments is contradicted by empirical evidence of mean reversion and momentum effects
- Assumption of a constant volatility is contradicted by empirical evidence.
- Can be argued that expected returns on shares are likely to vary with bond yields, which contradicts the assumption of a constant mean
- Random walks have a fractal dimension of 1.5, whereas investigations of market returns often reveal a fractal dimension around 1.4
- Markets are often (negatively) skewed.
7 Properties of the one-factor Vasicek model
- incorporates mean reversion
- time homogenous, ie future dynamics of r(t) only depend upon the current value of r(t) rather than what the present time t actually is
- Arbitrage free
- Allows negative interest rates
- Easy to implement since the characteristic functions of all related quantities are available
- Constant volatility
7 Properties of the Cog-Ingersoll-Ross model
- Incorporates mean reversion
- Arbitrage free
- Time homogenous
- Volatility depends on the level of the rates (high when rates are high)
- does not allow negative interest rates
- More involving to implement than the Vasicek model as its linke to the chi-squared distribution
- one-factor model
State 12 limitations of CAPM
- Unrealistic assumptions
- Empirical evidence don’t support it
- Investors don’t always use the same “currency”
- Markets are not always perfect
- Investors don’t always have the same expectations
- Cannot borrow/lend unlimited amounts at the same risk-free rate
- Difficult to check as need to think about investment markets as well as capital markets
- Unrealistic to invest in the market portfolio in practice as so many stocks
Does not consider:
- multiple time periods
- or optimisation of consumption over time
Does not account for:
- taxes
- inflation
- situations in which no riskless asset exists
3 Forms of the Efficient Market Hypothesis
- Strong
- Semi-strong
- Weak
Strong form EMH
Market prices reflect all current information relevant to the stock, including information which is not public
Semi-strong form EMH
Market prices reflect all current, publicly available information relevant to the stockq
Weak form EMH
Market prices reflect all information available in the past history of the stock price
7 reasons why its hard to test any of the 3 EMH forms in practice
- need to make assumptions such as normality of returns / stationarity
- Transaction costs may prevent the exploitation of anomalies, so EMH might hold net of transaction costs
- Allowance for risk - EMH does not preclude higher returns as a reward for risk; however the EMH does not tell us how to price such risks
- Testing strong form EMH is problematic & requires access to info that’s not public
- It can be difficult to define “public information” or to determine exactly when information becomes public
- Impossible to test all of the possible trading rules that might be used by technical analysts
- Assumptions made about how security prices should react to new information may be invalid
4 Axioms of Expected Utility Theorem
- Comparability
- Transitivity
- Independence
- Certainty equivalence
Comparability
An investor can state a preference between all available certain outcomes
Transitivity
If a is preferred to B and B is preferred to C, then A is preferred to C.
Independence
If an investor is indifferent between two certain outcomes, A and B, then he is also indifferent between the following two gambles:
- A with probability p and C with probability (1-p)
- B with probability p and C with probability (1-p)
Certainty equivalence
Suppose A is preferred to B and B is preferred to C.
Then there is a unique probability, p, such that the investor is indifferent between B and a gamble giving a with probability p and C with probability (1-p).
B is known as the certainty equivalent of the gamble.
Non Satiation in terms of U(w)
U’(w) > 0
Risk-neutrality in terms of U(w)
U’‘(w) = 0
2 limitations of using Value at Risk to measure the downside risk in an investment portfolio
- VaR does not illustrate the size of the loss in the tail of the distribution, only the likelihood
- Usefulness of VaR may be limited by a lack of data to determine the tail of the distribution.
3 Main Assumptions of Mean-Variance portfolios
- Investors select their portfolios on the basis of the expected return and the variance of that return over a single time horizon
- The expected returns, variance of returns and covariance of returns are known for all assets and pairs of assets.
- Investors are never satiated. At a given level of risk, they will always prefer a portfolio with a higher returnn to one with a lower return.
3 types of credit risk model
- Structural model
- Reduced-form models
- Intensity-based models
Structural models
Explicit models of a corporate entity issuing both debt and equity.
aim to link default events explicitly to the fortunes of the issuer.
Reduced-form models
Statistical models which use market statistics (credit ratings) rather than specific data relating to the issuer, and give statistical models for their movement.
Intensity-based models
Model the factors influencing the credit events which lead to default and typically do not consider what triggers these events.
Describe how the Merton model can be used to estimate the risk-neutral probability of default.
The company is modelled as having fixed debt L and variable assets Ft.
This means equity holders can be regarded as holding a European call on the assets with a strike of L.
It follows from the Black-Scholes model that we can deduce the (risk-neutral) default probability from the share price.
Utility
The satisfaction that a consumer obtains from a particular course of action.
Marginal rate of substitution
The amount of one good that a consumer is prepared to swap for one extra unit of another good.
Indifference curves
Joins all the consumption bundles of equal utility.
The slope of a consumer’s indifference curves will depend on his/her individual preferences and is equal to the marginal rate of substitution.
Consumption bundle
A given combination of goods.
3 Assumptions of consumer choice theory
- A consumer can rank any two bundles
- Consumers prefer more of a good to less of it
- Consumer preferences exhibit diminishing marginal rates of substitution
Assumptions of the budget constraint
- Prices of goods are fixed
- Consumer’s income is fixed
These two assumptions determine which consumption bundles are affordable. The budget line joins all points that a consumer can afford, assuming that all income is spent.
Utility Maximisation
Economists assume that consumers’ choices exhibit rational behaviour.
A rational consumer will choose the consumption bundle that maximises his own utility. This is the concept of utility maximisation.
Efficient portfolio
A portfolio is efficient if the investor cannot find a better one in the sense that is has a :
- higher expected return with the same variance
- lower variance with the same expected return.
Assumptions required for the existence of efficient portfolios
- Investors are never satiated
- Dislike risk
- Select assets based on mean and variance of return only
- Mean return, variance (or standard deviation) and co-variances are known for all assets.
Arbitrage pricing theory
Equilibrium market model that does not rely on the strong assumptions of the capital asset pricing model (CAPM).
Requires that the returns on any stock be linearly related to a set of factor indices as shown:
3 Main elements to the traditional theory of consumer choice
- Consumer’s preferences
- Budget constraint
- How the consumer decides which consumption bundle to choose
How consumers choose
Implications of utility maximisation:
- rational consumer will choose a consumption bundle such that the marginal rate of substitution is equal to the slope of the budget line - that is, where the ratios of marginal utilities equal the ratios of prices.
8 Key findings in behavioural finance
- anchoring and adjustment
- prospect theory
- framing
- myopic loss aversion
- estimating probabilities
- overconfidence
- mental accounting
- effect of options
Anchoring
Term used to explain how people will produce estimates.
They then adjust away from this initial anchor to arrive at their final judgement.
Prospect theory
Theory of how people make decisions when faced with risk and uncertainty. Replaces the conventional risk averse / risk seeking decreasing marginal utility theory.
Framing
The way a choice is presented (framed) and particularly, the wording of a question in terms of gains and losses, can have an enormous impact on the answer given or the decision made.
Myopic loss aversion
Similar to prospect theory, but considers repeated choices rather than a single gamble.
3 issues that might affect probability estimates
- Dislike of “negative” events
- Representative Heuristics
- Availability
Dislike of “negative events
The valence of an outcome has an influence on the probability estimates of its likely occurrence.
Valence
The degree to which it is considered as negative or positive
Representative Heuristics
People find more probable that which they find easier to imagine. As the amount of detail increases, its apparent likelihood may increase.
Availability
People are influenced by the ease with which something can be brought to mind.
Overconfidence
People tend to overestimate their own abilities, knowledge and skills.
This may be a result of
- hindsight bias
- confirmation bias
Hindsight bias
Events that happen will be thought of as having been predictable prior to the event.
Events that do not happen will be thought of as having been unlikely prior to the event.
Confirmation bias
People will tend to look for evidence that confirms their point of view (and dismiss evidence that does not justify it)
Mental accounting
People show a tendency to separate related events and decisions and find it difficult to aggregate events.
Effect of options
- Primary effect
- Recency effect
- People might be more likely to choose an intermediate option
- A range of options tend to discourage decision-making.
- Status Quo bias
- Regret aversion
- Ambiguity aversion
Primary effect
People are more likely to choose the first option presented
Recency effect
In some instances, the final option that is discussed may be preferred (The gap in time between the presentation of options & decisions may influence this)
Status Quo bias
People have a marked preference for keeping things as they are
Regret aversion
By retaining the existing arrangements, people minimise the possibility of regret.
Ambiguity aversion
People are prepared to pay a premium for rules.
Reasons why standard Brownian motion might not be suitable for the short-rate
- Interest rates may not be positive
- Interest rates do not display mean reversion
- Model won’t give a realistic range of yield curves
- Model whon’t fit historical data well
- It cannot be calibrated to current market data
- Not very flexible (single factor model_
- Model is Arbitrage-free
Outline evidence against normality assumptions in models of market returns
- Market crashes appear more often than expected from a normal distribution (jumps/discontinuities are an important feature of real markets)
- Days with now change / very small change happen more often than the normal distribution suggests.
- Q-Q plots of the observed changes in FTSE All Share against those that would be expected if returns were lognormally distributed show substantial differences. Actual returns have many more extreme events.
Expected utility theorem
A function, U(w) can be constructed representing an investor’s utility of wealth, w, at some future date.
Decisions are made on the basis of maximising the expected value of utility under the investor’s particular beliefs about the probability of different outcomes.
4 Axioms for Expected Utility Theorem
- Comparability
- Transitivity
- Independence
- Certainty equivalence
Name 3 types of multifactor models
- Macroeconomic factor models
- Fundamental factor models
- Statistical factor models
Macroeconomic factor models
Use observable economic time series as the factors.
Fundamental factor models
Use company specific variables as the factors
Statistical factor models
Principle components analysis is used to determine a set of indices which explain as much as possible of the observed variance.
These indices are unlikely to have any meaningful economic interpretation and may vary considerably between different data sets.
Cross-sectional properties of statistical distributions
Cross-sectional property fixes a time horizon and looks at the distribution over all the simulations.
(e.g. what will inflation be next year?)
The estimates are implicitly conditional on past information.
Can be deduced from prices of options and other derivatives.
Longitudinal properties of statistical distributions
Longitudinal property looks at the distribution over a long period of time.
E.g. what will the distribution of inflation be over the next 1000 years?
Unlike cross-sectional properties does not reflect market conditions at a particular time.
Alternative risk measures
- Value at Risk
- Tail Value at Risk
- Expected Shortfall
6 Assumptions underlying the Black-Scholes model
- Price of the underlying share follows a geometric Brownian motion
- There are no risk-free arbitrage opportunities
- Risk-free rate of interest is constant, the same for all maturities and the same for borrowing or lending.
- Unlimited short selling (negative holdings) is allowed
- There are no taxes or transaction costs
- Underlying asset can be traded continuously and in infinitesimally small numbers of units
Arbitrage opportunity
A situation where we can make a sure profit with no risk.
More precisely it means that
- We can start at time 0 with a portfolio which has a net value of zero.
At some future time T
- the probability of a loss is 0
- the probability that we make a strictly positive profit is greater than 0.
Outline relevant empirical evidence & theoretical arguments for:
- the volatility of returns over time
Direct statistical evidence shows volatility varies over time.
Volatility implied from option prices also shows volatility / volatility expectations vary over time.
Outline relevant empirical evidence & theoretical arguments for:
- the expected value of returns over time
Good theoretical reasons to expect the expected value of returns to vary over time.
Equities should give a risk premium over bonds and bond yields vary over time.
Empirically difficult to test.
Outline relevant empirical evidence & theoretical arguments for:
- Whether stock prices are mean reverting.
Empirically unsettled.
Some evidence for mean reversion but rests heavily on aftermath of a few dramatic crashes also conversely some evidence of momentum effects.
Outline relevant empirical evidence & theoretical arguments for:
- statistical distribution of returns
Strong empirical evidence that prices are non-normal.
Crashes happen more than would be expected. In addition more days with small/no changes than one would expect.
Discuss the extent to which a continuous time lognormal model of security prices can capture the statistical properties empirically observed or expected in the stock market.
- Random walk assumes constant volatility.
- Random walk assumes drift is constant
- No allowance for mean reversion
- Assumes normality.
State how investors are assumed to make decisions in modern portfolio theory.
Select on the basis of expected return and variance of return over a single time horizon.
Define an efficient portfolio in the context of MPT.
A portfolio is efficient if an investor cannot find a better one in the sense that it has both a higher expected return and a lower variance.
Define the market price of risk in CAPM
(Em - r)/σm
Name 3 approaches to modelling credit risk
- Structural models
- Reduced form models
- Intensity-based models
Structural models
explicit models of a corporate entity issuing both equity and debt.
Aim to link default events explicitly to the fortunes of the issuing corporate entity.
Reduced form models
Statistical models which use observed market statistics rather than specific data relating to the issuing corporate entity.
Market statistics most commonly used are the credit ratings issued by credit rating agencies such as Standard and Poor’s and Moody’s.
Reduced form models use market statistics along with data on the default-free market to model the movement of the credit rating of the bonds issued by a corporate entity over time. The output of such models is a distribution of the time to default.
Intensity-based models
model the factors influencing the credit events which lead to default and typically (but not always do not consider what actually triggers the credit event.
Explain “excessive volatility”
When the change in the market value of stocks (observed volatility), cannot be justified by the news arriving.
Claimed to be evidence of market over-reaction which was not compatible with efficiency.
3 Examples of empirical evidence of “under-reaction” of share prices to events
1) Stock prices continue to respond to earnings announcements up to a year after their announcement. An example of under-reaction to information which is slowly corrected.
2) Abnormal excess returns for both the parent and subsidiary firms following a de-merger. Another example of the market being slow to recognise the benefits of an event.
3) Abnormal negative returns following mergers. The market appears to over-estimate the benefits from mergers, the stock price slowly reacts as its optimistic view is proved to be wrong.
Discuss whether one-factor models are good models for the short-rate of interest.
All are arbitrage-free
Vasicek: Easy to implement but problem of possible negative interest rates.
CIR: more tricky to implement but positive rates.
HW: more flexible as time-inhomogenous, so better fit to market data, but negative rates are possible.
Credit event
An event which will trigger the default of a bond.
4 examples of credit events
- failure to pay either capital or a coupon
- loss event
- bankruptcy
- rating downgrade of the bond by a ratings agency.
4 Outcomes of a default
The contracted payment stream may be:
- rescheduled
- cancelled by the payment of an amount which is less than the default-free value of the original contract
- continues but at a reduced rate
- totally wiped out.
Recovery rate
In the event of a default, the fraction of the defaulted amount that can be recovered through bankruptcy proceedings or some other form of settlement is known as the recovery rate.
Explain how GAMMA can be used in the risk management of a portfolio that is delta-hedged.
If the portfolio is delta-hedged and has a high value of gamma, then it will require more frequent rebalancing or larger trades than one with a low value of gamma. The need for rebalancing can, therefore, be minimised by keeping gamma close to 0.
Explain how VEGA can be used in the risk management of a portfolio that is delta-hedged.
The value of a portfolio with a low value of vega will be relatively insensitive to changes in volatility. Since σ is not directly observable, a low value of vega is important as a risk-management tool.
Furthermore, it is recognised that σ can vary over time. Since many derivative pricing models assume that σ is constant through time, the resulting approximation will be better if vega is small.
6 Assumptions underlying Black-Scholes
- Price of the underlying share follows a geometric Brownian motion.
- No risk-free arbitrage opportunities.
- Risk-free rate of interest is constant, the same for all maturities and the same for borrowing/lending.
- Unlimited short selling (negative holdings) is allowed
- no taxes or transaction costs
- underlying asset can be traded continuously and in infinitesimally small numbers of units
4 Defining characteristics of Brownian motion with Z0 = 0
- Independent increments
- Stationary increments
- Gaussian increments
- Continuous sample paths
4 Main assumptions of modern portfolio theory
- Assumes that investors select their portfolios on the basis of the expected return and the variance of that return over a single time horizon.
- Assumed that the expected returns, variance of returns and covariance of returns are known for all assets and pairs of assets.
- Investors are never satiated. At a given level of risk, they will always prefer a portfolio with a higher return to one with a lower return.
- Investors dislike risk. For a given level of return, they will always prefer a portfolio with lower variance to one with higher variance.
5 factors that affect the price of a European put option on a non-dividend paying share.
premium would INCREASE
- as the strike price increased.
- time to expiry increased.
- the volatility of the underlying share increased
premium would DECREASE
- as the underlying share price increased
- interest rates increased.
Efficient portfolio
If the investor cannot find a better one in the sense that is has:
- the SAME Expected return and a lower variance
- the SAME variance and a higher expected return.
3 categories of models
- Macroeconomic factor models
- Fundamental factor models
- Statistical factor models
Macroeconomic factor models
Use observable economic time series as the factors.
Macroeconomic factors
- annual rates of inflation
- economic growth
- short-term interest rates
- yield on long-term government bonds
- yield margin on corporate bonds over government bonds
Fundamental factor models
Use company specific variables.
Fundamental factors
- level of gearing
- price earnings ratio
- level of R&D spending
- industry group to which the company belongs
Statistical factor models
Do not rely on specifying the factors independently of the historical returns data.
Instead, use principal components analysis to determine a set of indices which explain as much as possible of the observed variance.
8 desirable characteristics of a model for the term structure of interest rates
- Arbitrage free
- Positive interest rates
- Mean reversion of rates
- Ease fo calculation of bonds and certain derivative contracts
- Realistic dynamics
- Goodness of fit to historical data
- Ease of calibration to current market data
- Flexible enough to cope with a range of derivative contracts
Define Value at risk (p= α)
VaR(X) = -t
where
P(X
Describe the 2-state model for credit ratings
In the 2-state model, the company defaults at time-dependent rate λ(t) if it has not previously defaulted.
Once it defaults, it remains permanently in the default state.
It is assumed that after default, all bond payments will be reduced by known factor (1 − δ), where δ is the recovery rate.
Now we need to change this risk neutral measure, which will change the default rate to λ’(t).
This rate is implied by market prices.
Jarrow-Lando-Turnbull model
generalises the 2-state model to n-1 credit ratings plus the default state with transitions possible between any pair of states except for the default state which is absorbing.
2 Major weaknesses of Arbitrage Pricing Theory
- In order to apply APT, we need to define a suitable multi-index model.
- We also need to come up with the correct factor forecasts.
The simplest approach is to calculate a history of factor returns and take their average. this implicitly assumes an element of stationarity in the market.
5 Measures of investment risk
- Variance of return
- Downside semi-variance of return
- Shortfall probability
- Value at Risk
- Tail value at Risk
Advantages of Variance of return
- Mathematically tractable
- fits neatly with a mean-variance portfolio construction framework
Disadvantages of “Variance of Return”
- Symmetric measure of risk. The problem of investors is really the downside part of the distribution.
- Credit risky bonds have an asymmetric return distribution and as defaults are often co-dependent on economic downturns, portfolios can have fat tails.
- Neither skewness or kurtosis of returns is captured by a variance measure.
Advantages to Downside Semi-variance of return
- Takes into account the risk of lower returns
- can be decomposed into systematic and non-systematic risk contributions
Disadvantages of downside semi-variance of return
- Semi-variance is not easy to handle mathematically and takes no account of variability above the mean
- if returns on assets are symmetrically distributed, semi-variance is proportional to variance.
- does not capture skewness or kurtosis
Advantages to using “shortfall probability”
- Gives an indication of the possibility of loss below a certain
- It allows a manager to manage risk where returns are not normally distributed
Disadvantages of using “shortfall probability”
- choice of benchmark level is arbitrary
For a portfolio of bonds, the shortfall probability will not give any information on:
- upside returns above the benchmark level
- nor the potential downside of returns when the benchmark level is exceeded.
Advantages to using Value at Risk (VaR)
- VaR generalises the likelihood of underperformance by providing a statistical measure of downside risk
Disadvantages to using Value at Risk
- Portfolios exposed to credit risk, systematic bias or derivatives may exhibit non-normal distributions.
- Usefulness of VaR in these situations depends on modelling skewed or fat-tailed distributions of returns.
- The further one gets out into the “tails” of the distributions, the more lacking the data and, hence, the more arbitrary the choice of the underlying probability distribution becomes.
Advantages of Tail Value at Risk
Relative to VaR, TailVaR provides much more information on how bad returns can be when the benchmark level is exceeded.
Disadvantages of Tail value at Risk
Same modelling issues as VaR in terms of Sparse data, but captures more information on the tail of the non-normal distribution.
5 Key features of Brownian motion
- independent increments
- stationary increments
- gaussian increments
- continuous sample paths
- B0 = 0
3 Limitations of 1-factor term structure models
- Imperfect correlation across maturities
- Different volatility phases
- Pricing complex derivatives
1-factor term structure models:
Imperfect correlation across maturities
If we look at historical interest rate data, we can see that changes in the prices of bonds with different terms to maturity are not perfectly correlated as one would expect to see if a one-factor model were correct.
Resent research has suggested that around 3 factors, rather than 1, are required to capture most of the randomness in bonds at different durations.
1-factor term structure models:
Different volatility phases
Looking at the long run of historical data, we find that there have been sustained periods of both high and low interest rates with periods of both high and low volatility.
These features, which are difficult to capture without introducing more random features into a model.
The issue is especially important for 2 types of problems in insurance: pricing & hedging of long-dated insurance contracts with interest-rate guarantees;
and asset-liability modelling and long-term risk-management.
1-factor term structure models:
Pricing complex derivatives
Need more complex models to deal effectively with derivative contracts which are more complex .
3 Desirable features of the Vasicek model
- Arbitrage-free
- Instantaneous and other rates mean reverting
- Ease of computation/pricing of derivatives and bonds
5 Undesirable features of the Vasicek model
- Short rate not necessarily positive
- Does not generate realistic dynamics/yield curves
- Does not provide good historical fit (even with suitable parameter values)
- Not easy to calibrate
- Not sufficiently flexible, eg cannot price derivatives whose value depends on more than one interest rate.
2 Key properties of Value at Risk
- statistical measure of downside risk
- assesses the potential minimum loss over given time with given degree fo confidence.
Advantage of VaR
- normal distribution is easy to manipulate to calculate VaRs based only on 2 parameters
Disadvantage of VaR
results may be misleading with skewed or “fat-tailed” distribution
Outline Shiller’s approach to test for “excessive volatility”
- Shiller used a discounted cashflow model of equities going back to 1870
- A perfect foresight price was determined using actual dividends paid and a terminal value for the stock
- If markets are rational, there would be no systematic forecast errors
- If markets are efficient, the perfect foresight price matches with share price
Strong evidence was found that contradicted the EMH.
4 Criticisms of Shiller’s Work
- Criticisms of terminal stock price
- Choice of constant discount rate
- Bias in estimates of variance because of autocorrelation
- Non-stationarity of the series
Explain why a lognormal model may be used for share prices
- Log-normal distribution makes the maths for option pricing simple
- Returns in non-overlapping periods are independent, which is consistent with EMH
- It does not allow negative share prices
- Mean and variance are proportional to time period
Weaknesses of using the lognormal model for share prices
- variance may not be stable over time
- the mean (drift) may not be constant over time
- share prices may be considered to be mean-reverting
- share prices exhibit jumps
Define DELTA for an individual derivative
The rate of change in derivative price with respect to change in the price of the underlying asset.
Define GAMMA for an individual derivative
The rate of change of delta with respect to change in the price of the undelrying asset.
Define THETA for an individual derivative
the rate of change in the value of the derivative with respect to change in time to expiration.
Define LAMBDA for an individual derivative
The rate of change in the value of the derivative with respect to change in the assumed continuous dividend yield on the underlying asset.
Define RHO for an individual derivative
The rate of change in the value of the derivative with respect to change in the risk-free rate of interest.
Define VEGA for an individual derivative
The rate of change in the value of the derivative with respect to the (assumed) volatility of the underlying asset.
Arbitrage Pricing Theory (APT)
An equilibrium market model that does not rely on the strong assumptions of the capital asset pricing model (CAPM)
APT requires that the returns on any stock be linearly related to a set of factor indices.
Explain the difference between the single index model and CAPM
the single-index model is purely empirical and is not based on any theoretical relationships between Beta and the other variables, which are assumed in CAPM.
Cross-sectional property
fixes a time horizon and looks at the distribution over all the simulations.
Longitudinal property
picks one simulation and looks at a statistic sampled repeatedly from that simulation over a long period of time.
State the conditions under which a market is arbitrage free (ito stochastic processes)
The market is arbitrage-free iff there exists a probability measure under which discounted asset prices are martingales.
Expected utility theorem (2004 version)
A rational agent with a preference ordering >= on certain outcomes maximises their expected utility, where their utility function u (over certain outcomes) models their preference in the sense that
u(A) >= U(B) <=> A >= B
“Risk Averse” agent
An agent is risk-averse if, for any amount of wealth w, they prefer a certain outcome of w to any gamble with expected outcome w.
“Non-satiated” agent
An agent with wealth w is non-satiated if they strictly prefer wealth v to w for any v > w.
CAPM formula for expected returns
E[Ri] = rf + Bi ( E[Rm] - rf )
where E[Ri] is the expected return on security i,
E[Rm] is the expected return on the market index
Bi = Cov(Ri, Rm) / Var(Rm)
APT formula
ri = r0 + Bi1 I1 + … + Bik Ik
Where Ik is the expected return on index k.
Bij is the sensitivity of security i to index j.
4 CAPM assumptions
- All investors have the same assesment of risk and return (means, variances, and covariances)
- All investors have same 1-period investment horizon
- Market is perfect
Requires the expected return on the market index to be identical to the expected return on the actual market.
2 APT assumption
- More is preferred to less
- Market is arbitrage-free
Implications of EMH on Technical analysis
If technical analysis produces excess returns, then the market is not weak form efficient.
Implications of EMH on fundamental analysis
If fundamental analysis produces excess returns then the market is not semi-strong form efficient.
Implications of EMH on insider trading
If insider trading produces excess returns then the market is not strong form efficient.
Insider trading
Involves trading on the basis of information that has not been published or known to the public.
the form of the utility function for an investor uses DOWNSIDE SEMI-VARIANCE OF RETURN
Utility function will be quadratic below the level of expected return and linear above it.
CAPM: 4 Reasons why actual returns on the portfolio will vary over successive years
- randomness around expectation E[Rp]
- changes in expected returns (betas) of assets (Bp)
- errors in estimating expected returns (betas)
- changes in expected market returns E[Rm]
Define “efficient frontier”
The representation in expected return - variance (or std) space of the set of efficient portfolios.
It is THE set of efficient portfolios.
Explain how an efficient frontier changes its shape with the introduction of risk-free lending and borrowing.
The existence of a risk-free asset has the effect of making the frontier curve a straight line that is tangent to the original frontier for risky assets (in mean-std space) and passes through the point (0,r) where r is the risk free rate of return.
At the point of tangency, the porfolio is diversified without risk-free assets. To the left of the point of tangency, the portfolios will have a mix of diversified assets and risk-free assets.
to the right of the point of tangency, the portfolios will consist of more than 100% diversified assets, as the investor would have borrowed at risk-free rate and invested in diversified assets
Describe informational efficiency in the context of the EMH
The EMH states that seet prices reflect information. However it does not explicitly tell us how nes information affects prices.
It is also emperically difficult to establish precisely when information arrives (for example, many events are widely rumoured prior to official announcements).
Many studies show that the market over-reacts to certain events and under-reacts to other events.
The over/under-reaction is corrected over a long time period. If this is true then traders could take advantage of the slow correction of the market, and efficiency would not hold.
6 Examples of effects that have been claimed to exist in stock markets that might be considered examples of informational inefficiency
- past performance predictability (past winners subsequently underperform)
- Accounting ratios have predictive power (P/E predicts low future returns)
- Firms coming to market have poor subsequent performance
- Stock prices take some time to react to earnings announcements
- Abnormally poor performance following mergers
- Abnormally good performance following demergers
6 Reaons why it is difficult to assess empirically whether or not the market is efficient
- Type I errors in hypothesis test: testing for many anomalies will inevitably generate some fake positives by chance.
- Some “information” may actually be a proxy for risk, which should be associated with differential return.
- Terminology: Is the market efficient only if transaction costs are taken into account, because these can stop anomalies being exploited.
- Invalid statistical tests: assumptions of normality of the return distributions may lead to the rejection of EMH only because the returns are not normally distributed.
- Timescale: clearly arbitrage possibilities do arise, but nevertheless, the market may be efficient on timescales which ignore fleeting arbitrage opportunities.
- Rare events (shocks): the market may reflect small probabilities of large “shocks” which nevertheless do not occur during long periods covered by a given dataset.
List 7 Steps involved in the process of conventional asset liability modelling
- Specify the time horizon and objectives
- Choose the parameters to be optimised
- For a particular set of input parameters, run a number of simulations.
- Record the distribution of the outputs
- Vary the inputs and repeat steps 3 and 4
- Repeat steps 3, 4, and 5 until efficient opportunity set has been mapped
- Present results and decide on choice of input parameters and outcome.
Outline 5 inputs to a conventional asset-liability model
- Investment model parameters
- Liability data
- Starting value of assets
- Objective function
- Dynamic modelling rules
4 Practical constraints to the asset liability modelling process
- number of considerations
- complexity of model
- time involved
- significance of modelling
“Excessive volatility”
Means that the change in market value of stocks could not simply be justified by the flow of new information.
Work of shiller
- considered discounted cashflow model of equities over 100 year period
- determined “perfect foresight price (PFP)” (i.e. the correct market price if markets could correctly predict future dividends)
- difference between PFP and actual price was due to forecasting errors
- If markets are rational expect no systematic errors. Also if markets are efficient broad movements in PFP should be correlated with moves in actual price.
Shiller’s conclusion
Strong evidence that the level of observed volatility contradicted the Efficient Market Hypothesis.
Criticisms of Shiller
Later studies with different formulations found that violation of Efficient Market Hypothesis was only borderline, reasons being:
- difficult to choose terminal value for stock price
- Schiller used a constant discount rate
- bias in estimates of the variances because of autocorrelation
- possible non-stationarity of the series measured.
6 Assumptions underlying Mean Variance Portfolio theory
- Investors select their portfolios on the basis of expected return and variances of return
- investors are risk-averse
- investors are non-satiated
- investors make their investment decisions over a single time horizon
- no taxes or transaction costs
- assets are infinitely divisible
Indifference curve
Curves where each point represents a portfolio that gives the same expected utility.
Optimal portfolio.
Point where the indifference curve is tangential to the efficient frontier. Requires quadratic utility or distribution of returns to be normal. Or, the portfolio with the highest attainable expected utility.
Opportunity set
Characteristics of each and every portfolio available to investors.
Efficient frontier.
Collection of efficient portfolios. A portfolio is efficient if we cannot find one with lower variance for the same expected return or higher expected return for the same variance.
How to go about quantitatively testing whether the WEAK EMH holds
Historical data, test whether patterns repeat themselves, try to fit time series models, test whether trading rules based on fitted models would yield higher than average returns.
How to go about quantitatively testing whether the SEMI-STRONG EMH holds
Test whether all off-balance sheet items priced, test whether changes to accounting standards led to changes in prices, run regressions of price changes on lagged accounting and economic data items.
How to go about quantitatively testing whether the STRONG EMH holds
Assess whether holders of privileged information made higher than average return.
Measure trading of directors before key events.
Assess whether any abnormal return generated before key event (eg announcement of merger should provide a one-off charge in abnormal returns rather than any gradual build-up of abnormal events.)
“Excessively volatile” market
Excessively volatile markets are those where the volatility of prices is greater than can be justified by the arrival of new information and any other returns to investors.
How would you test if a market is “excessively volatile”
You would require
- a long history of prices and cashflows from a market (i.e. 100 years of equity returns and details of all dividend payments and other returns to investors.)
- This enables the calculations of the prefect foresight price (PFP) which is the present value of all future cashflows and the terminal value of the asset.
- The difference between the PFP and the actual price arises from forecast errors of future dividends. If markets are rational then there should be no systematic pattern to the errors.
- The PFP and the actual price should be correlated because in an efficient market actual prices should respond to anticipated future increased cashflows which are reflected in PFP.
Speculator
Speculators take bets on future market movement and aim to make profits
by taking a view on the direction in which and/or the extent to which they
anticipate the market will move.
Explain why the volatility σ of a share must be estimated and explain what
is meant by“implied volatility”.
The volatility σ of a share cannot be observed directly in the market and
must therefore be estimated.
The implied volatility is calculated from the current market price of a derivative, such as an option. It relies on the principle that it is possible to
obtain St, K, T − t and ct directly from the market, then using this information it is possible to determine the assumed value for the volatility σ
that will reproduce the price of the option, using a particular pricing model.
When using the Garman-Kohlhagen formulae, the vega of the option
is strictly positive (i.e. an increasing function of σ), so as the volatility
increases so does the price, this indicates that the solution is unique
distinction between two
forms of the Efficient Markets Hypothesis (EMH) and briefly summarising the consensus view
on each based on empirical research, for each of the forms pointing out at least one difficulty
faced by researchers in testing the EMH.
A distinction needs to be drawn between strong and semi-strong form efficiency:
The semi-strong form postulates that market prices incorporate all publicly available
information. Directors would still be able to profit from trading on inside information under
this form of the EMH .
The strong form postulates that market prices incorporate all information, whether publicly
available or not.
Although the strong form is difficult to test as it requires access to non-publicly available
information and many studies of directors’ share dealings conclude that they fail to outperform
other investors, there is broad consensus that these results are due to the existence of insider
trading regulations and that the strong form does not hold. This would justify the
recommendation to restrict directors’ dealings in the period before financial statements are
published, when they might reasonably be expected to have access to privileged inside
information.
Evidence on the semi-strong form is mixed, with some studies showing evidence of
inefficiency and others concluding that markets are indeed efficient. There are numerous
difficulties involved in testing the semi-strong form; notably, the definition of publicly
available information, the allowance for time lapses between the publication of new
information and the setting of new equilibrium prices and the joint hypothesis problem, viz.
that any test requires a joint hypothesis of market efficiency and a particular equilibrium asset
pricing model, meaning that efficiency can never be empirically falsified. The mixed evidence,
coupled with the higher fees charged on actively managed funds, makes it reasonable to adopt
a strategy with the bulk of equity assets in passive funds, with the balance actively managed
(the ‘core-satellite’ approach).
State and briefly explain the theorem which asserts that, given that the assumptions of the
CAPM hold, the portfolio of risky assets chosen by both investors should be the market
portfolio.
Separation theorem: all rational investors will invest on the Capital Market Line, allocating
their portfolios between the risk-free asset and the portfolio of risky assets represented by
the tangency point where the CML meets the efficient frontier for risky assets. Since all
investors hold the same basket of risky assets, this must be the market portfolio, i.e. the
universe of all investable assets in proportion to their market capitalisations.
Identify the property of risky asset returns that must hold if the CAPM is valid in a market, where the power and expo-power utility functions describe the preferences of most
investors.
Given that the utility functions are not quadratic, it must be the case that risky asset returns
have elliptically symmetrical distributions
State 4 reasons why it is hard to test whether any of the three forms hold in practice.
Tests need to make assumptions (which may be invalid) such as normality of
returns or stationarity.
Transaction costs may prevent the exploitation of anomalies, so that the EMH
might hold net of transaction costs.
Allowance for risk: the EMH does not preclude higher returns as a reward for risk;
however the EMH does not tell us how to price such risks.
Testing the strong form EMH is problematic as it requires access to information
that is not in the public domain.
It can be difficult to define “public information” or to determine exactly when
information becomes public.
It is impossible to test all of the possible trading rules that might be used by
technical analysts.
The assumptions made about how security prices should react to new information
may be invalid.
Explain why insurance companies make use of run-off triangles.
There is normally a delay between incidents leading to claim and the insurance
pay out
Insurance companies need to estimate future claims for their reserve
It makes sense to use historical data to infer future patterns of claims
State four factors that affect the size of Ψ(U,t), for a given t. (Given X~N(μ,σ2) and N~ Poison(λ))
higher λ increases Ψ(U, t) as the process is faster – claims and premiums
come in quicker
higher μ increases Ψ(U, t) as claims amounts are larger, relative to the surplus
held
higher θ reduces Ψ(U, t) as premiums increase at a quicker rate, so more of a
buffer
higher U reduces Ψ(U, t) as more of a buffer to withstand claims
higher σ2 will typically increase Ψ(U, t), assuming that expected premiums
are higher than expected claims, since the likelihood of more
extreme claims increase, [but may reduce Ψ(U, t) if expected claims
are higher than expected premiums.]
Describe the main advantage of a multifactor model over the standard mean-variance portfolio
theory model, and suggest reasons why an investor may nevertheless choose to use the
standard portfolio theory model rather than a two factor model.
The main advantage of a factor model over the portfolio theory model is the large
reduction in the number of parameters that need to be estimated. This is primarily
because covariances are not required for each pair of securities, as it is assumed that any
correlation between securities is as a result of them responding to changes in the factors
in the model. For N securities, the number of data items reduces from N (N+3 )/2 to
4N+4 .
An investor may choose to use the portfolio theory model if:
*The securities of interest or the market in which he operates are such that it is not
possible to find a small number of factors that sufficiently explain the correlation
between securities.
*He has the information systems required to easily produce the large number of
parameter estimates.
*He believes a more detailed model will give him a competitive advantage in the
market.
*The factors that are most relevant statistically do not have any meaningful economic
interpretation.
Discuss the limitations of using Value at Risk to measure the downside risk in
an investment portfolio.
VaR does not illustrate the size of the loss in the tail of the distribution, only
the likelihood.
The usefulness of VaR may be limited by a lack of data to determine the tail of
the distribution.
State in words the four axioms of the Expected Utility Theorem.
- Comparability
An investor can state a preference between all available certain outcomes. - Transitivity
If A is preferred to B and B is preferred to C, then A is preferred to C. - Independence
If an investor is indifferent between two certain outcomes, A and B, then
he is also indifferent between the following two gambles:
(i) A with probability p and C with probability (1 - p); and
(ii) B with probability p and C with probability (1 - p). - Certainty equivalence
Suppose that A is preferred to B and B is preferred to C. Then there is a
unique probability, p, such that the investor is indifferent between B and a
gamble giving A with probability p and C with probability (1 -p). B is
known as the certainty equivalent of the above gamble.
State the main assumptions of mean-variance portfolio theory.
Investors select their portfolios on the basis of the expected return and the
variance of that return over a single time horizon.
The expected returns, variance of returns and covariance of returns are known
for all assets and pairs of assets.
Investors are never satiated. At a given level of risk, they will always prefer a
portfolio with a higher return to one with a lower return
Outline the three types of credit risk model.
The three types of credit risk model are:
* structural models: these are explicit models of a corporate entity issuing
both debt and equity. They aim to link default events explicitly to the
fortunes of the issuer.
- reduced-form models: these are statistical models which use market
statistics (such as credit ratings) rather than specific data relating to the
issuer, and give statistical models for their movement. - intensity-based models: these model the factors influencing the credit
events which lead to default and typically do not consider what triggers
these events
Describe how the Merton model can be used to estimate the risk-neutral
probability of default.
In the Merton model, the company is modelled as having a fixed debt, L and
variable assets Ft
. This means the equity holders can be regarded as holding a
European call on the assets with a strike of L. It follows from the BlackScholes model that we can deduce the (risk-neutral) default probability from
the share price.
Explain why an insurance company might purchase reinsurance
To protect itself from the risk of large claims
Describe two types of reinsurance.
- Excess of loss reinsurance where the reinsurer pays any amount of
a claim above the retention. - Proportional reinsurance where the reinsurer pays a fixed
proportion of any claim.
Claims on a certain portfolio of insurance policies arise as a Poisson process with
annual rate lambda. Individual claim amounts are independent from claim to claim and
follow an exponential distribution with mean mu. The insurance company has
purchased excess of loss reinsurance with retention M from a reinsurer who calculates
premiums using a premium loading of theta. Denote by Xi
the amount paid by the
reinsurer on the ith claim (so that Xi
= 0 if the ith claim amount is below M).
A Poisson process is characterised by the probability of a single claim arising
in a small time interval dt being dt (with no probability of more than one
claim).
For the reinsurer, the probability of a claim arising in a small time interval dt
is given by
lambda x dt × P(Xi> M)
=….
=lambda x (exp(-m/mu)) x dt
=> Poisson proc with rate lambda x (exp(-m/mu))
If claims arise acc. to Poisson and claim amounts exponential, with excess of loss reinsurance.
the adjustment coeff of the REinsurer does not depend on M (retention level)
R does not depend on the retention M.
This is a surprising result at first glance, but arises because of the memoryless
feature of the exponential distribution
i.e. Xi -M|Xi > M is exponential with parameter 1/MU
so the reinsurers claim process is just a slower version of the insurers.
Explain why a tradeable asset has to be introduced in order to build an
arbitrage-free model.
We have started off with a process for r(t) which is not a tradable asset. An
arbitrage opportunity must relate to trading an asset, therefore arbitrage-free
models must allow for trading.
Explain what Market Price Of Risk is
dB(t,T1) = B(t,T1)[m(t,T1)dt + S(t,T1)dW(t)]
γ(t,T1)= (m(t,T1)-r(t))/S(t,T1)
γ(t,T1) represents the excess expected return over the risk-free rate per
unit of volatility in return for an investor taking on this volatility.
Explain why a deep out of the money call option in the Black-Scholes world
will experience a rate of return close to the risk-free rate of return
Deep out of the money, delta and gamma will be close to zero which implies
that theta will equal the risk free rate of return.
(looking BS PDE i.t.o delta and gamma)
Explain the difference between an efficient market and an arbitrage-free
market.
Attempts to explain this phenomenon gave rise to the efficient markets
hypothesis, which claims that market prices already incorporate the relevant
information. The market price mechanism is such that the trading pattern of a
small number of informed analysts can have a large impact on the market
price. Lazy (or cost conscious) investors can then take a free ride, in the
knowledge that the research of others is keeping the market efficient.
If we assume that there are no arbitrage opportunities in a market, then it
follows that any two securities or combinations of securities that give exactly
the same payments must have the same price. This is sometimes called the
“Law of One Price”.
Arbitrage-free markets can be inefficient
Explain how mean-reversion in the stock market can be consistent with an
efficient market.
Even mean reversion can be consistent with efficient markets. After a crash,
many investors may have lost a significant proportion of their total wealth; it
is not irrational for them to be more averse to the risk of losing what remains.
As a result, the prospective equity risk premium could be expected to rise
Explain the difference between the single-index model and the Capital Asset
Pricing Model
The single-index model is purely empirical and is not based on any theoretical
relationships between βi and the other variables, which are assumed in CAPM.
State the assumptions underlying the average
cost per claim method with grossing up factors.
The number of claims relating to each development year is a constant proportion of the total claim numbers from the relevant accident year.
Claim amounts for each development year are a constant proportion of the total claim amount for the relevant accident year.
Claims are fully run off after development year 2.
