Discuss the assessment of different types of market risks
All risk (credit, op, insurance, etc) should be analysed and model together and in a consistent fashion
In addition to analyzing risk at a particular time horizon, their evolution over time should also be examined (e.g. dynamic solvency testing)
Additional notes on Task list:
Sweeting 14.2.5 Black Scholes is excluded but can read for review
Might be assumed to be pre-requisite knowledge
Good to just know the underlying assumptions
Time series can be used to model equity and other returns
Key characteristics of a financial time series analysis of historic equity market return
For return on individual equities:
Returns are rarely iid
Volatility vary over timer
Extreme returns appear in clusters (volatility clustering)
Return series are leptokurtic (heavy tail, non symmetric)
For return on protfolios of equities:
Correlations exist between returns of different series (e.g. different equities) at the same point in time (true for other asset classes as well)
Correlations between different series vary over time
Multivariate returns data show little evidence of cross-correlation
(i.e. between time periods \(t\) and \(t+1\))
Multivariate series of absolute or squared returns do show strong evidence of cross-correlation
Extreme returns in one series often coincide with extreme returns in several other series
(i.e. level of dependence are higher during periods of high volatility)
Model assumptions vs key characteristics
Models we typically do use (incorrectly) assume that log-returns are iid
e.g. Random walk for the development of log-prices in discrete time
e.g. Geometric Brownian motion model for the development of prices in continuous time
Trends and correlation
There are little serial correlation
Returns do follow trends over short periods
Correction do happen over the longer terms but any such correlation is insufficient to support significant profit taking
Overall it is very difficult to predict the return in the next period given only the history of the process
\(\therefore\) best estimate of tomorrow’s return given the daily return up to today is 0
Time horizon
As the time period over which we calculate the return increases, the features noted above tend to become less pronounced
Volatility clustering is less marked and returns appear to be more iid and less heavy tailed (as expected from CLT)
Volatility
Volatility:
Conditional standard deviation of returns given historical data
Volatility clustering:
When extreme values tend to be followed by other extreme values (not necessarily of the same sign)
Serial correlation between absolute or squared returns is consistent with the phenomenon of volatility clustering
Volatility clustering supports the idea that conditional s.d. are changing in a way that is to some extent predictable
This is the justification for the \(ARCH\) and \(GARCH\) previously
XS kurtosis
Kurtosis: “Peakedness” of a distribution
Kurtosis for daily financial return is higher than that of a normal distribution
\(\therefore\) has XS kurtosis
Higher kurtosis \(\Rightarrow\) more of the variance is due to infrequent extreme deviations (vs frequent modestly-sized deviations)
Distribution of daily financial returns data is leptokurtic
i.e. more acute peak around the mean (higher probability than a normally distributed variable of value near the mean) and fatter tails (higher probability than a normally distributed variable of extreme values)
Observed data suggest that the distribution of daily financial return has tails that decay more slowly than those of a normal distribution
\(\hookrightarrow\) We tend to get more extreme values from returns data than we would for normally distributed data
The kurtosis \(\downarrow\) as the period over which we are measuring returns \(\uparrow\) (e.g. to monthly)
Which is consistent with volatility clustering
Can be modeled with the full range techniques from Module 15
historical simulation, forward looking factor (or data) based approachExample:
corporate bond yields might describe the complex links between variables such as:Risk-free yield
Coupon rates
Credit spread
Caveat:
Be aware not to project history blindly into the future by recognizing an unsustainable trend
Should also use the past to model the uncertainty of yields (rather than their absolute level)
…Using the multivariate normal distribution
Forward looking data based approaches may assume that changes in the log returns are linked by multivariate normal distribution
Steps for modeling:
Pick the frequency of calculation (e.g. daily, weekly, etc)
Pick the time-frame of historical data to be used
(Trade off between volume and relevance)
For each asset class, choose the total return index \(S_t\) to use
For each asset class, calculate the log-return:
\(X_t = \ln \left(\dfrac{S_t}{S_{t-1}}\right)\)
Calculate the average returns and variance of each asset class and the co-variance between each class (and sub-set of classes)
Simulate a series of returns with the same characteristics based on a multivariate normal distribution
Limitations
As noted previously (Module 16), multivariate normal is not a good description of reality in RM applications
Can involve extensive calculations particularly as the number of asset classes increases
…Using PCA
PCA (from Module 19):
Can reduce the computational overhead when compared to application of the multivariate normal distribution
Goal: Determine only the main factors that contribute to deviations from the average returns
Ignore the factors with less influence
\(\hookrightarrow\) Reducing the complexity (dimensionality) of the analysis
Particularly useful when projecting returns on bonds (with a variety of duration)
\(\because\) Changes in bond yields can be explained largely by shifts in just a few factors
(i.e. level and shape of yield curve)
Steps for modeling:
Derive the matrix of deviations:
From the average returns by deducting the average return in each period for each asset class
Derive the PC that explain a sufficiently high proportion of the deviations from average past returns
Project this number of iid normal r.v.
Using the associated eigenvalues as the variances
Weight these projected series of deviations by the appropriate element of the relevant eigenvectors
Add these weighted projected deviations to the expected returns from each asset class
Other multivariate distribution
Combine non-normal marginal distributions using an appropriate copula
However, volume of extreme data may be insufficient to form a strong view as to the best choice of distributions
Recall from Module 5, Basel II requires banks to hold additional capital if they invest their capital in risky assets
For this purpose, market risk may be quantified by using an internal model of the assets to calculate a 10-day 99% VaR
This lead to a regulatory capital requirement under Pillar I (which is a multiple of this VaR loss)
Banks can also use standardized approach rather than internal VaR model
Historic data may not give a realistic view of future experience
\(\therefore\) Need to incorporate:
Changes in fundamentals or other subjective viewpoints
The effect of tax
Reasonable estimate of the expected return is the gross redemption yield on the government bond of a similar term as the projection period
Other relevant factors:
Term premium: part of the risk premium that is a function of term
Term premium will varies by market and investors
Purchasing power parity: So that for risk-free overseas government bonds, the domestic risk-free rate is suitable
In theory purchasing power parity will compensate for any difference in yield
Bootstrapping: Can be used to calculate the implied forward spot yield curves based on the gross redemption yields of the bonds in a portfolio
Such detailed analysis may be spurious in the context of a stochastic modeling exercise
For bonds with a risk of default the expected return will be higher due to the credit spread
Measure of the difference between the yield on a risky and a risk-free security
Typically a corporate bond and a government bond respectively
\(\Delta\) in value of an asset due to changes in the credit spread is considered a market risk by Sweeting (Others can include it in credit risk)
Credit spread reflects the following factors:
\(\mathrm{E}[\Pr(Default)]\) and \(\mathrm{E}[LGD]\)
Any risk premium attached to the risk of default
(i.e. uncertainty surrounding the expected probability of default and the expected loss given default)
liquidity premium
(i.e. more difficult to sell the corporate bond when required (at an acceptable price))
Residual value of bond after default has happened
Common measures of credit spread
Nominal spread:
Difference between the gross redemption yields of risky and risk free bonds
Static spread:
Addition to the risk free rate at which discounted cash flows from risky bond will equate to its price
Option-adjusted spread:
Further adjust the discount rate through the use of stochastic modeling to allow for any options embedded in the bond
Observed market credit spreads vs historical default
Observed market credit spreads are generally higher than based on historical default on bonds
Difference can be put down to risk premia in respect of:
Higher volatility of returns relative to the risk-free asset (credit beta)
Higher uncertainty of returns
Particularly the possibility of unprecedented extreme events
Greater skewness of the future returns (more significant downside) due to the possibility of default
Lower marketability of corporate debt and the associated higher costs of trade
Differences in taxation
Potential negative correlation between credit spreads and interest rates may offset the above to some extent for investors focusing on absolute returns (i.e. those exposed to both interest and credit risk)
When assessing credit risks, the risk premia above should be:
Modeled as expected volatility in returns
Reflected in liquidity planning (rather than affect the modeled expected return)
Note that not all government bonds are risk-free so it is important to clarify the reference asset
Methods to determine risk premium is different as other assets such as equity and property have uncertain income
Historical risk premiums:
Calculated by: Deducting the observed return on risk free asset from the observed return on the risky asset, averaging over the periods that data is available
Need to further adjust the average to reflect expected future changes
(could be subjective or based on fundamental structural changes in the asset classes)
For overseas investments:
Important to consider volatility in each asset’s domestic currency and allow seperately for exchange rate risk in the correlation calculation
CAPM
CAPM1 gives a structure for analyzing risk premiums and ensuring their consistency
Market risk should be measured relative to a suitable benchmark
(Typically based on market indices or the investor's liabilities)
Features of a good benchmark
Unambiguous
Investable
Tractable
Measurable on a reasonably frequent basis
Appropriate
(e.g. to the investor’s objectives)
Reflective of current investment opinion (positive, negative, neutral)
Specified in advance
Might be appropriate to measure against benchmark that:
Contains a high proportion of the assets held in the portfolio
(i.e. Has similar assets)
Has a similar investment style (growth or value, etc)
Has a low turnover constituents
Has investible position sizes
Behaves in a similar way to the portfolio
(i.e. shows a strong (+) correlation between portfolio return (\(r_X\)) and the benchmark return (\(r_B\)) in XS of the market return (\(r_U\)))
(i.e. \(\rho(r_X - r_U, r_B - r_U) \gg 0\))
Has low correlation between the difference of the {portfolio return and the benchmark return }and {benchmark return and the market return}
(i.e. \(\rho(r_X - r_B, r_B - r_U) \approx 0\))
If the liabilities benchmark is used:
Reference point is often a set of cashflows (or notional assets) that represent the actual (uncertain) liabilities
(i.e. liabilities cash flow)
Risks and return can both be defined and measured with reference to the chosen benchmark(s)
Strategic risk
Risk of poor performance of the benchmark against which the manager’s performance will be judged (the strategic benchmark) relative to the liability-based benchmark
Active risk
Risk of poor performance of the manager’s actual portfolio relative to the manager's (strategic) benchmark
Active return
Difference between the return on the actual (active) portfolio and the return on the manager's (strategic) benchmark
Basic 1-factor approaches are useful for simulating short-term, single interest rates
Need more complex models for modeling: multiple points on a yield curve or changes to the shape of a yield curve
Exam Note:
Need to be familiar with manipulating interest rates (e.g. forward rates and yield curves)
See Sweeting 14.3.1 and 14.3.2 for revision
Brennan-Schwartz model
Considers \(\Delta\) in spot rates at 2 maturities
e.g. short term (\(r_{1,t}\)) and long-term (\(r_{2,t}\)) at time \(t\)
\(\Delta r_{1,t} = [\alpha_a + \beta_1(r_{2,t-1} - r_{1,t-1})] \Delta t + r_{1,t-1}\epsilon_{1,t}\)
\(\Delta r_{2,t} = [(\alpha_2 + \beta_2r_{1,t-1} - \gamma_2r_{2,t-1}] \Delta t + r_{2,t-1}\epsilon_{2,t}\)
Model assumptions
\(\Delta\) in short-term rates vary in line with the steepness of the yield curve
(the differential between long and short term rates)
i.e. \(\alpha_1 + \beta_1(r_{2,t-1}-r_{1,t-1})\) for some \(\alpha_1\) and \(\beta_1\)
Volatility of short term rates varies \(\propto\) the most recent level of short term rates
\(\Delta\) in long term rates vary \(\propto\) the square of the level of long term rates
(i.e. \(\gamma_2 r^2_{2,t-1}\) for some \(\gamma_2\))
But are also influenced by the short-term rates through the product term \(\beta_2 r_{1,t-1}r_{2,t-1}\)
Volatility of long term rates varies \(\propto\) the level of long term rates
Limitations:
Constraint to 2 factors
Difficult to select the appropriate parameters
PCA can be applied to gross redemption yields (or log to avoid negatives), forward yields or bond prices to determine the main factors that contribute to changes
For GRYs the process is similar to the market risk:
Pick frequency
Pick time-frame of historical data to use
Take the GRYs for bonds of different durations and calculate the average interest rate for the series
Deduct the average interest rate (to derive the deviation)
A set of factors is chosen and weighted, then projected using independent r.v. (normal) to produce expected interest rates for each term
For bond prices (\(p_t\)):
First calculate log returns \(\ln (p_t \big/ p_{t-1})\)
Then the deviations from average log-returns are analyzed
FX risk shares similarities to interest rate risk
Can be modeled in terms of the returns on short-term interest-bearing deposits denominated in different currencies
FX rates reflect different interest rates and the expectation of appreciation or depreciation of the currency
FX risk can be hedged away using currency forward agreements (See Module 27), hence there is no additional currency return to be gained (or modeled) if working in a single denomination
No-arbitrage argument:
\(\dfrac{e_0}{e_t}(1+r_{Y,t}) = 1 + r_{X,t}\)
\(r_{X,t}\) and \(r_{Y,t}\) are the annual effective spot interest rates in 2 countries at time \(t\)
\(e_t\) is the expected FX rate at time \(t\)
*Contagion risks
Usually seen as an extension of market risk
Can also apply to other risk (e.g. credit): Where the default of one org can cause its creditors and supplier to experience financial difficulties
Can be modeled as the interaction between different financial series
Certain series may be linked for extreme negative values
The increase level of dependence suggest we should use copula
Approaches to modeling contagion risks
Single financial series
Contagion can be considered a feedback risk
(i.e. there is some serial correlation that can be modeled)
However, that would presents an arbitrage opportunity which in theory should be eliminated by arbitrageurs
\(\therefore\) such serial correlation effects are usually ignored when modeling
Between related financial series
Fitting a t-copula using correlation parameter \(\rho\) that is situation dependent
i.e. \(\rho = \rho_0 + D_1 \rho_1 + D_2 \rho_2\)
\(\rho_0\):
Normal level of dependency
\(\rho_1\) and \(\rho_2\):
Additional dependency in different states
\(D_i\):
States indicators
e.g. \(D_1 = 1\) during financial crisis and 0 otherwise; \(D_2=1\) in the aftermath of a financial crisis and 0 otherwise
\(r_x = r_f + \beta_X(r_U - r_f)\)
Links the expected return \(r_X\) on asset \(X\) to:
Where: