Problem Sets Flashcards
“A change in the price of a corporate bond is a reflection of market risk.” Discuss this statement
- The market price of a corporate bond can change due to market risk (i.e. change in the general level of interest rates as summarized in the yield curve), but also due to credit risk (i.e. a change in the creditworthiness of the issuer).
Define risk in its broadest sense.
- In general terms, risk can be defined as the likelihood of a harmful consequence. The likelihood captures the ‘uncertainty’ of risk, while the harmful consequence relates to the outcome.
- These two dimensions can nicely be summarized in a distribution of profit and losses, or returns. For a complete description of risk also the attitude towards risk should be accounted for. However, as no unique ranking exists, this dimension of risk is typically not modelled.
“Micro-prudential regulation is focussed on monitoring idiosyncratic risks.” True or false? Motivate briefly.
- This statement is true. Idiosyncratic risk arises within an individual institution and is under direct control of the management of the firm. Micro-prudential regulation targets this individual-specific risk-taking behaviour. This contrasts to macro-prudential regulation that aims at monitoring systemic risks that are shared across firms.
Assuming a normal distribution for returns has clear advantages over more complex distributions. Discuss. Indicate also where the normal distribution fails from an empirical as well as theoretical perspective.
- Assuming a normal distribution has the advantage of exploiting the stability property, which allows portfolio extension of normal returns (the sum of normally distributed returns is also normally distributed), and multiperiod horizons for normal logreturns (the sum of single – period normally distributed logreturns is also normally distributed).
- In addition, the normal distribution is convenient to work with as only the first two moments are needed for a full description of the distribution. However, it does not capture the skewness and kurtosis as observed in most asset returns (mainly kurtosis is problematic; skewness is less of a problem in general, but can be problematic for particular payoff functions (e.g. options)).
- Also, normal returns can be larger than -100%, which violates the limited liability property of returns. Assuming normal logreturns is then more realistic as it excludes negative prices, and implies lognormal returns.
- Finally, while this stability property might seem attractive from a theoretical point of view, it does not capture the stylized fact that longer horizon logreturns become more normally distributed.
While the normal distribution is often used to represent stock returns, it does not capture the property of fat-tailed returns. Which distributions do account for fat tails?
- A number of distributions capture fat tails. The most simple way to account for kurtosis ois to assume a t-distribution. Such distribution has modest fat tails for low degrees of freedom, but converges to the normal when the degrees of freedom increases. (use 4 to 6 degrees of freedom to impose kurtosis).
- Alternatively, one can assume extreme value theory with GPD or GEV distributions. Such distributions specifically address the issue of fat tails, and are more flexible in capturing a large mass in the tails.
- While the GPD only models the tails of the distribution, the GEV models the full distribution.
- Finally, non-normal stable distributions (Levy, Cauchy) also allow for fat tails, but have the unattractive feature of infinite moments. This makes the distributions hard to work with, and violates what we observe in reality. Also, the stability property for logreturns is not in line with varying distributional properties at different horizons
How can we govern the volatility of volatility?
- It has been shown empirically than volatility is time-varying, with clusters of high/low volatility. Such volatility dynamics can easily be captured in an EWMA.
- The responsiveness of the model to most recent changes is captured by the parameter in the following specification: 𝜎𝑛 2 = 𝜆𝜎𝑛−1 2 + (1 − 𝜆)𝜎𝑅𝑛−1 2 The higher will be, the more weight is given to the previous forecast of volatility, and thus less to the most recently observed change.
- This translates into volatility dynamics that responds slowly to new information, and thus leads to a slow-mowing weighted average. Put differently, the volatility of volatility will be rather low.
What is the difference. between EWMA and GARCH?
- From a theoretical perspective, the main difference between both models is the presence of a long-run variance component in the GARCH specification. This long-run variance component makes sure that the variance is always pulled back towards its long-run value.
- For the current example, the weight given to the previous volatility forecast is identical ( = ), but the weight given to the most recent change is lower in the GARCH model than in the EWMA model, since in the former a weight of 0.03 is also given to the long-run component (in both models, the weights have to add up to 1).
- Within the GARCH model, the forecast is therefore pulled upwards towards this higher long run forecast.
The Spearman correlation is only meaningful for any 2 variables that have elliptical distributions. True or false?
This is false. It is the Pearson correlation that is only meaningful for 2 variables that have elliptical distributions. The Spearman correlation is a much more general measure of dependence as it is a non-parametric measure of association based on data ranks.
“In a fixed exchange rate regime, there is no forex risk.” Comment on this statement.
Even in a fixed FX regime, exchange rate risk exists by means of a devaluation/revaluation (i.e. adjustment to parity value) or by a permanent break in the regime. Though the risk can be limited, there is always the chance that the regime changes
“The delta of an option is a measure of moneyness.” Comment on this statement
This is true. Moneyness refers to the intrinsic value of an option, and this is reflected in the delta of the option (slope of the option value function). As delta is the slope of tangent at the price curve, it ranges between -1 and 1. The higher the absolute value of delta, the more value the option has.
“If you buy an option, you are positive vega”. Comment on this statement.
This is true. Vega is a measure of volatility risk and measures the degree to which the option value changes as the implied volatility changes. When you buy an option, you are buying volatility – i.e. you value volatility.
The Cornish-Fisher expansion to quantile estimation is highly relevant in the context of options. True or false? Motivate.
This is true. The Cornish-Fisher percentile corrects the standard normal percentile for the presence of skewness and kurtosis. Since the distribution of an option position is skewed (to left or right depending on the position), this gives a more accurate description of what is at risk.
Credit events can take on numerous formats: bankruptcy, downgrading, calling back a bond, and default on payments. True or false? Motivate.
This is not true. Calling back a bond occurs when the borrower wants to refinance its debt at a lower cost. This is not a credit event
Which type of company is expected to have the highest recovery rate, all other things equal:
- A trading company active in volatile markets
- An internet merchant of trendy consumer goods
- An asset-intensive manufacturing company
- A highly leveraged fund
Motivate your choice.
- Recovery rates are higher when the assets of the firm in default consist of tangible assets that can be resold easily. Manufacturing firm will therefore yield the highest expected recovery rates (note also that the more volatile the assets, the greater the probability of a fall in the market value upon liquidation).
In the Merton model of default, corporate debt can be viewed as risk-free debt, minus a put option on the firm value. True or false? Motivate.
- True. The payoff to the debtholder equals min(AT,B):
- (a) In case of no default the is repaid at full B
- (b) In case of default, he is repaid by what is left over of assets AT < B
- Such payoff can be rewritten as a long position in a risk-free bond with face value B plus a short put option on the assets with a strike B: 𝐵 − 𝑀𝑎𝑥(𝐵 − 𝐴𝑇, 0) = 𝑀𝑖𝑛(𝐴𝑇, 𝐵)
