Managing Financial Risk: Lecture 1
Terry Leitch
Course Objectives
This course will introduce and familiarize you with:
- Dynamic risk measures such as VAR
- One dimensional
- Historical data approach
- Multi-dimensional
- The nature of financial data and its impact on measurement techniques, especially kurtosis and the autoregressive behavior of market volatility
- The parameter estimation for risk models
- Robustness tests for risk measure
- Derivative tools that a risk manager can use to reduce the risks
Why Manage Risk?
Business Reasons
- If performance is stable, it is easier to plan
- It is difficult to get good terms in the middle of a crisis
- Stable performance enhances reputation and credibility
Two Firms:
| Q1Y1 |
$100,000 |
$900,000 |
| Q2Y1 |
$100,000 |
-$800,000 |
| Q3Y1 |
$100,000 |
$500,000 |
| Q4Y1 |
$100,000 |
-$700,000 |
| Q1Y2 |
$110,000 |
$600,000 |
| Q2Y2 |
$110,000 |
-$400,000 |
| Q3Y2 |
$110,000 |
-$200,000 |
| Q4Y2 |
$110,000 |
$940,000 |
| Total |
$840,000 |
$840,000 |
Why Manage Risk?
Financial Reasons
- Higher volatility of performance, higher rates of interest demanded by lenders
- Higher volatility, higher return on capital (shares are discounted) by investors
- Higher volatility, greater likelihood management will have to negotiate at a disadvantage
- Higher volatility, greater likelihood firm will be insolvent
The biggest motivator is leverage
Example - Market moves down 5%
- Firm with no leverage value drops to 95% of original value
- Firm with 3 to 1 leverage drops to 80% of original value
- Firm with 30 to 1 leverage has negative value
It is no conincidence that financial firms invest heavily in risk management
Types of Risk - Financial
- Market risk
- Equity risk
- Interest rate risk
- Currency risk
- Commodity risk
- Credit Risk
- Transaction risk
- Portfolio concentration
- Derivative counterparty
- Liquidity Risk
- Funding risk
- Trading risk
Types of Risk - Non-financial
- Operational risk
- Legal and regulatory risk
- Poltical risk
- Business Strategy Risk
- Natural phenomena
- Business continuity
- Accident
- Strikes
Types of Risk - Details
- Liquidity Risk - risk of insufficient funds to meet short term obligations
- Market Risk - risk that interest rate, equity, commodity and foreign exchange markets move in a way injurious to the firm
- Operational Risk - risk that process or human failure leads to significant financial loss
- Credit Risk - risk that someone who owes the firm money cannot or will not pay it
- Business Risk - risk that market changes for the business and affects the business plan
Types of Risk - Liquidity
- Liquidity risk is increased the greater the proportion of illiquid the securities you hold
- Examples of illiquid securities:
- Real Estate
- Structured Securities
- Leases
- Equipment
- Examples of liquid securities:
- Government Bonds
- Corporate Bonds
- Cash
Types of Risk - Market
- Broker dealer equity holdings exposed to equity markets
- Export company exposed to foreign exchange markets
- Mortgage lender exposed to interest rates
- Power company exposed to commodity markets
Types of Risk - Credit
- Over-lend to low-grade buyers on real estate
- Over-lend to highly-leveraged companies with poor cash flow
- Recent failures include GM, Lehman, RBS, etc.
Types of Risk - FX Market Risk Example
Microsoft sells products in almost every country in the world (see Hull p.112).
How should Microsoft bill customers outside the US? - in the local currency => this clearly exposes Microsoft to exchange rate risk - in dollars (this would apparently be the safest alternative…)
However, even when Microsoft bills customers in dollars it is still exposed to fluctuations of the exchange rate. If for example the dollar strengthens against the local currency, then customers will need more money to buy Microsoft’s products, which are now more expensive.
Microsoft might have to, at this point, lower the price of its product to remain competitive. => the exchange rate exposure remains even when billing in dollars!
Types of Risk - Credit Example
- Where: US
- When: October 2008
- How: House prices kept rising since late 80s. In the 2000s we witnessed a huge development of ABS, specifically mortgage-backed-securities. These instruments should allow trading in potentially very illiquid primary assets and also provide implicit diversification because the cash ows of ABS come from a pool of different (geographically for example) assets. The real estate market was thought as been fairly independent form state to state. House price could fall for example in Florida, but raise in California. In 2008 all real estate markets started experiencing sharp declines. MBS lost most of their values which generated also a large liquidity problem!
Types of Risk - Liquidity Example
Funding risk: the risk of not been able to meet cash needs as they arise (very different from solvency)
- Who: Northern Rock (see Hull, p. 455)
- Where: United Kingdom
- When: 2007
- How: NR relied on selling SHORT-TERM debt to obtain its funding. In 2007 it became very difficult for NR to sell debt; investors were very reluctant to buy bank’s debt (NR was involved in the mortgage market). NR was well capitalized and had enough assets to meet all its obligations. Nonetheless NB could not fund itself. It asked for emergency funds, up to 25 billion pounds. It was eventually nationalized. ```
Types of Risk - Liquidity Example
Funding risk: the risk of not been able to meet cash needs as they arise (very different from solvency)
- Who: Northern Rock (see Hull, p. 455)
- Where: United Kingdom
- When: 2007
Types of Risk - Business Risk Example
- 100 years ago Eastern Buggy Whip was one of the biggest companies in the S&P 500
- There are no Buggy Whip Companies in the S&P 500 today
- To address this use Porter’s 5 Forces analysis
Types of Risk - Operational Risk Examples
- Insurance company miscalculates insurance policy payments
- Securities dealer loses a customer’s trade
- Real estate lender poorly documents a loan
Types of Risk - One Risk Leads to Another
- Equity market drops and a margin call causes an unexpected liquidity risk in a broker dealer
- Market for insurance products changes causing an insurance company to seek a buyer for its shares, pushing the shares of all insurers down
- A flood forces an insurer to liquidate many holdings at a loss causing the markets for bonds to drop
- An earthquake causes a manufacturer to shut down and is forced to default on their bonds causing a bank to default causing anothe manufacturer to fail etc., etc.
Network Connection of Different Risks
Types of Risk
One Risk Leads to Another Example
- In 1984, Continental Bank, the largest bank in the US at the time, had lent $600MM to Penn Square Bank while its equity capital was around $1B
- Penn Square lent to oil prospecting companies that flourished as the oil price had increased to very high levels
- The oil price dropped
- The oil companies began to fail and so did Penn Square
- When word got out Continental’s depositors withdrew $10B, roughly 25% of the bank’s deposits thereby causing a liquidity crisis
- To prevent Continental’s failure from causing other companies and banks to fail, the FDIC took it over and injected $4.25B
- Thus, “Too big to fail” entered our financial lexicon
Risk Measures
Severity and Frequency
- All approaches to monitor risk have two things in common: a loss amount at a confidence level
- There is a third thing that can change: time
- As time grows, so does uncertainty
- So, to understand the loss at a confidence level, you also need to know the time frame
Risk Measures
Severity and Frequency
| Extreme Earthquake |
0.00005 |
500,000,000,000,000,000 |
| Severe Earthquake |
0.16666667 |
500,000,000,000 |
| Severe Hurricane |
0.25 |
10,000,000,000 |
| Hurricane |
1 |
2,000,000,000 |
| Market Crash |
0.0333333 |
2,000,000,000,000 |
| War |
0.025 |
500,000,000,000,000 |
| Train Accident |
2 |
50,000,000 |
Rule of thumb:
- High loss, low frequency events -> insure
- High frequency losses, low loss events -> process change
In the first part of this course we will focus on market, or asset price, risk
- Our goal is to identify the storms and pull in the sails before getting capsized
- We will do this by using information
- We will identify a risk limiting paradigm to follow called VaR
- It requires an estimate of the future deviation of returns
- We could make this a static estimate, but it turns out there are benefits to a dynamic one
- There is “information” embedded in stock returns that we will use to design a better process
- At the end we will check if this yields a better result (ie better than a monkey throwing darts)
Calculating Risk
Methodology
To get a loss frequency and severity, we need to assume a distribution (unless we’re using a historical distribution, more on that later) and calculate parameters
- Usual process is log-normally distributed returns
- Nice math properties (Can’t go below zero, variance scales as “t”)
- Is widely used as the tendency is for i.i.d’s to converge to a normal (central limit theorem)
- But we will see it has issues
- Process is:
- Calculate returns
- Estimate standard deviation of returns
- Scale S.D for the time period
Calculating Returns
Simple (arithmetic) returns are calculated by: \[ r_{t+1} = (S_{t+1}-S_t)/S_t = S_{t+1}/S_t -1 \]
- Continuous compounded (geometric), log returns are calculated by: \[ R_{t+1} = ln(S_{t+1})-ln(S_t) \]
Which are similar, except for large moves, i.e. \(r_{t+1}<<1\)
\[ R_{t+1}= ln(S_{t+1})-ln(S_t)=ln(S_{t+1}/S_t)=ln(1+r_{t+1})\approx r_{t+1} \]
Calculating Returns
Portfolio of Assets
- The value of a portfolio with n assets at time t is the weighted average of the asset prices \(S_{i,t}\) using the current holdings \(N_i\) of each asset as weights: \[ V_{PF,t}= \sum_{i=1}^n N_iS_{i,t}\]
- The arithmetic portfolio returns are given as \[ r_{PF,t+1}=V_{PF,t+1}/V_{PF,t}-1\]
- And log returns are \[ R_{PF,t+1}=ln(V_{PF,t+1})-ln(V_{PF,t})\]
Calculating Returns
Portfolio returns under simple returns
Advantage of simple, arithmetic returns is they add, because if we set \[ V_{PF,t}= \sum_{i=1}^n N_iS_{i,t}\] where \(N_i\) is the number of units held in asset \(i\) and \(V_PF,t\) is the value of the portfolio on day \(t\). Then the portfolio return is: \[ r_{PF,t+1}=\frac{V_{PF,t+1}-V_{PF,t}}{V_{PF,t}}=\frac{\sum_{i=1}^n N_iS_{i,t+1}-\sum_{i=1}^n N_iS_{i,t}}{\sum_{i=1}^n N_iS_{i,t}}= \sum_{i=1}^n w_ir_{i,t+1}\] where \(w_i=N_iS_{i,t}/V{PF,t}\) is the portfolio weight in asset \(i\) The formula for a portfolio of log-normal asset returns is much more complicated.
Calculating Returns in R
library(quantmod)
sp500=getSymbols('^GSPC',from="1900-01-01",warnings=F,verbose = F)
rets=dailyReturn(GSPC,type='log',leading=TRUE) # type=log gives lognormal returns
plot(rets,type="l") # plots returns, function knows how to handle dates
Return Phenomena
Serial Autocorrelation of Returns
Returns show very low autocorrelation, i.e. \[ Corr (R_{t+1},R_{t+1-\tau})\approx 0, for \tau = 1,2,3,...,100 \]
Return Phenomena
Low correlation of price changes over time
- The bands indicate statistical significane
- No correlation => no useful information in time behavior of returns
Return Phenomena - Leptokurtosis
- Actual data shows higher frequency in center of distribution than a “normal” distribution
- This is offset by a higher frequency of large moves than predicted by a normal distribution
Return Data Phenomena
Standard Deviation Dominates
- The standard deviation of returns completely dominates the mean of returns at short horizons such as daily
- The S&P 500 data supplied with the course has a daily mean trend of \(0.0056%\) per days and a daily standard deviation of \(1.3771%\)
- Recall from stats, t-statistic is estimate/standard error
- Here, \(t=0.0056/1.3771 = 0.004 << 2\)
- T>2 target for rejecting null
- Can also do p-value test using normal
Would you bet on making \(.0056%\) per day (red line) given the noise (black dots)?
Return Data Phenomena
Squared Returns show autocorrelation
- While returns are not correlated, squared returns do show a degree of autocorrelation over time, especially over days and weeks (compare with return graph above):
Return Data Phenomena
- Variance is the standard measure of dispersion of the squared returns and it shows autocorrelation just as squared return show autocorrelation
- This tends to be a phenomena we see over short time frames as in weeks or days
- We will use this feature to build return distributions that will take advantage of this phenomena to help improve our results
- This should not be surirsing since the mean ~ 0 so returns^2 ~ variance
Return Data Phenomena
Variance and Returns
- Variance and returns show negative correlation
- Especially after large negative returns, variance of returns goes up
- It is thought this might be due to increased leverage as stocks outstanding debt stays the same while the equity value has decreased
- Called the leverage effect, we will build a model that captures this and see if this is useful in our quest for better risk management
Return Data Phenomena
Time behavior of asset correlations
- Correlation between assets appears to be time varying (ie unstable!)
- Importantly, the correlation between assets appear to increase in highly volatile down-markets and extremely so during market crashes
- “When it hits the fan, all correlations go to +/-1.”
Return Data Phenomena
Long term return distribution
- As time increases, returns look more and more normal
- This is probably due to the central limit theorem kicking in
- This has implications for multi-period risk management
Model for Future Returns
The Linear Model
The model we will use to estimate risk in one dimension is: \[R_{t+1} = \mu_{t+1} + \sigma_{t+1}z_{t+1}, z_{t+1}\sim i.i.d. D(0,1)\]
- D(0,1) is a standard normal variate with mean 0 and standard deviation of 1
- It is consistent with zero correlation and non-predictability of returns
- Can add structure to \(\sigma\) to capture negative correlation with returns
- Can add time behaviour to \(\sigma\) in order to capture the autocorrelation of squared returns
Since is assumed to be zero from accepting null hypothesis \[E_t[(R_{t+1}-\mu_{t+1})^2] = E_t[R_{t+1}^2]= \sigma_{t+1}^2\]
Model for Future Returns
Benefits of the model
Since is assumed to be zero from accepting the null hypothesis \[E_t[(R_{t+1}-\mu_{t+1})^2] = E_t[R_{t+1}^2]= \sigma_{t+1}^2\] since \(E_t[Z_{t+1}^2]=dt=1\) & \(E_t[Z_{t+1}]=0\)
Since \(Z_{t+1}\) is D(0,1), we get the nice property that we can use standard normal distribution tables to go back and forth between a risk amount and a probability
Model for Future Returns
Benefits of the Model
Introducing Value at Risk:
\[VaR_{t+1}= -\sigma_{t+1}\Phi_{p}^{-1}\]
- Loss intensity at probability “p”
- The inverse probability, or normal scaled loss \(\Phi_{p}^{-1}\) is a number between \(-\infty\) and \(\infty\) (but usually negative because this is risk management)
- Units of \(\sigma_{t+1}\) are either currency (eg $) or %, the latter can be applied to a currency denominated portfolio to get currency
Value at Risk
Inverse Probability Function
- If \(\Phi\) is the cumulative normal probability function, the \(\Phi^{-1}\) gives us the return level for a given probability
- For example, if we let \(p=0.01\), then we get \(\Phi_{p}^{-1}=\Phi_{0.01}^{-1}=\approx -2.33\)
- This is for the standard normal, we have to scale by the standard deviation
Value at Risk
\[VaR_{t+1}= -\sigma_{t+1}\Phi_{p}^{-1}\]
Value at Risk
Inverse Probability Function Example
- If \(\Phi_{p}^{-1}=\Phi_{0.01}^{-1}=\approx-2.33\) and \(\sigma=2\%\) for \(dt=1\) period then potential \(VaR=-2.33*2\% = 4.6\%\) of the portfolio value
- If portfolio = \(\$100MM\) then the VaR says there is a \(1\%\) chance of a loss of \(\$4.6MM\)($VaR) or greater for the next period
- For 5% level, i.e. \(\Phi_{0.05}^{-1}=-.1.645\), we have a \(VaR=1.645*2\% =3.29\%\), or there is a \(5\%\) probability that there is a loss of \(3.29MM\) ($VaR) for our portfolio over the next period
Value at Risk
Inverse Probability Function Example
Model for Future Return Variability
Another example and inverse example
- If \(\Phi_{p}^{-1}=\Phi_{0.01}^{-1}=\approx-2.33\) and \(\sigma=2\%\) for \(dt=1\) period then potential \(VaR=-2.33*2\% = 4.6\%\) of the portfolio value
- If we want to go the other way, say, what is the chance we will have a loss of \(\$5MM\) or greater in the next period, \(\$5MM\) is \(5\%\) of the portfolio and \(-5\%/2\% = -2.5\)
- \(\Phi(-2.5)=.0062\), so there is a .62% chance that we will have a loss of \(\$5MM\) or greater in the next period
VaR
Period and Frequency
Notice I said “next period”. This is usually a few days, but it depends on the scale of \(\sigma\). If \(\sigma\) is a month or a year, then the VaR is for a month or a year. So this is a one period model. We will look at ways to do multiple periods later in the course.
- So we have two things to choose and one to estimate:
- What is the time period you are looking forward over?
- What confidence level are you interested in?
- Based on time period calculate \(\sigma\)
Note that time period plus confidence level defines frequency. For example, if we look at \(95\%\) level for a single days time frame, we are looking at a loss that happens once in twenty days
VaR
Steps
- Define time period
- Estimate \(\sigma\) for time period
- Define confidence level
- Calculate:
- Loss at confidence level by inverting probability
- Probability of loss exceeding a specified \(\$\) level by calculating \(Z\) value by \(\$\)level/\(\sigma\)
- \(\$\)VaR or monetary loss at given VaR level by taking VaR \(\%\) loss and turning it into \(\$'s\)
Caveat
- The models we will build are known to have the property that the information decays and so there is little impact beyond a month or so
- Typical focus is 3 to 10 days
VaR - A simple Time varying Model for \(\sigma\)
RiskMetrics \(\sigma\) Estimate
There are numerous ways to calculate \(\sigma\) such as a simple return calculation followed by a standard deviation calculation
One well-known model that incorporates the serial correlation of squared returns is RiskMetrics, where \(\sigma\) is given by: \(\sigma_{PF,t+1}^{2}=0.94\sigma_{PF,t}^{2} + 0.06R_{PF,t}^{2}\)
VaR
Sigma Calculation
To get \(\sigma\) under RiskMetrics, first need to bootstrap via regular standard deviation calculation and then move to weighted time correlated \(\sigma\):
VaR
Sigma Calculation
Much of this course reveolves around calculating \(\sigma_{t+1}\) using different methods
- RiskMetrics
- Historical Average
- GARCH
- Realized Variance
- Extreme Value
and comparing the methods via
- R^2
- Likelihood Ratio (when we have it)
- P&L impact
- Independence test
- Coverage test
VaR
Drawbacks
VaR gives good information but…
- The normal distribution tends to smooth things out and underestimates large moves
- The portfolio is assumed constant over the time period
- Choice of time frame and confidence levels is arbitrary
Next Time
- What is VaR for \(97.5\%\) confidence level if \(\sigma=2.3\%\)?
- What is \(\$\) VaR if portfolio value is \(\$12MM\)
- At what confidence level does the portfolio’s loss exceed \(\$1MM\)?
Also, look at the data that comes with text and get familiar with S&P data set
Read over chapter 2
