Pre-Lecture: Stats Stuff
Interpolation
Takes weighted average of values between 2 data values
-
Add 1 to the number of data values first (e.g. 10 + 1)
Then apply percentile to the new number (e.g. 0.1 * 11 = 1.1)
The 1.1th position is where we need to look at (i.e. between data value #1 and #2)
“Interpolate” between #1 and #2, to get “1.1th position”
Extrapolation
Uses the previous difference in data value to “estimate”
-
After adding 1 to the number of data values  at certain percentiles, we may be required to
extrapolate instead of interpolate
Assumes that 𝑥th data point and (𝑥 − 1)th data point continues into (𝑥 + 1)th data point
E.g. 5, 6, 7, 9, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 39, 52, 60
o Calculate the extrapolated 97.8% of the distribution
o 97.8% * 20 = 19.56th position  60 + (60-52)*0.56 = 64.48
Normal Mixture
Normal mixture distribution takes a weighted combination of two independent normal distributions
𝑔(𝑥) = 𝜋𝜙1 (𝑥) + (1 − 𝜋)𝜙2 (𝑥)
-
Examples-wise, seem to be applied to mutually exclusive situations (e.g. good vs. bad states,
recession vs bull years, etc.)
And then, there’s conditional probabilities being applied to each situation
Simple vs Log Returns
𝑆𝑡
Log returns = ln (𝑆
𝑡−1
-
)
Time additive
o
𝑆𝑡
i.e. ln (𝑆
𝑡−1
-
𝑡−2
𝑆𝑡
𝑡−2
)
o e.g. 2-period log returns yield same result as adding both single-period log returns
But not portfolio additive
o If you apply portfolio weights to constituent assets’ log returns, the weighted average
will not equate the portfolio’s log returns itself
𝑆𝑡
Simple returns = 𝑆
𝑡−1
-
𝑆
) + ln (𝑆𝑡−1 ) = ln (𝑆
−1
Portfolio additive but not time additive
Lecture 1: Financial Risk Mgmt. Fundamentals
Definition of Financial Risk
Potential of gaining or losing value
Possibility that a stakeholder does not get the investment promised/needed
Types of Financial Risk
Market (price) risk
-
Risk that financial assets fluctuate in price due to market-wide news (systematic risk)
Interest rate risk
𝐶𝐹
-
𝑡
𝑃 = ∑𝑡 (1+𝑅)
𝑡
-
Risk that asset value changes due to interest rate (R) changes
Managing interest rate risk requires matching duration/maturity/cash flows, but for banks, this
may reduce profitability
Credit risk
-
Risk that promised CF may not be repaid in full
Includes possibility of partial/late payments
Most credit risks are idiosyncratic (not systematic)  diversifiable
Liquidity risk
-
Risk of being forced to borrow/sell an asset in a very short period of time
Measures the cost incurred by having to sell the asset immediately  how far is it sold from fair
market value?
Primary measure of illiquidity: bid-ask spread
Weakness of this measure: bid-ask assumes quotes are unaffected by “my” order, quotes are
valid only for a limited number of assets, market depth
Foreign exchange risk
Operational risk
-
Risk of loss due to physical catastrophe, technical failure and human error in firm’s operation
(incl. fraud, failure of mgmt., process errors)
Anything else that affects balance sheet
Exposure offers little return  makes no sense and ideally, should be entirely
mitigated/eliminated
Other forms of traditional risks
-
Regulation risk
Competition risk
Reputational risk
New forms of non-traditional risk
-
Climate risk
ESG risk
Lecture 2: Dynamics of Asset Prices
Different Types of Analysis
Technical Analysis
Fundamental Analysis
Statistical Analysis
Statistical Analysis
Why it is useful
-
Any risky investment’s returns can generally not be (fully) predicted  assume as random
Use probabilistic models that incorporate random variables (outcome of uncertain events) with
probability distributions (probability of event)
Theoretical models tend to fail empirically (e.g. CAPM)
Fundamental analysis could help, but firm analysis takes much effort and time, easy to identify
mispricing, but hard to determine magnitude
Price vs. Price returns
-
Prices known to be non-stationary
Generally have an increasing trend
We don’t care about prices, only the %change
Discrete and Continuous Time
-
Discrete time analysis: data in form of discrete observations
Continuous time analysis: approximation required, to fill gaps between observable datapoints 
use of ugly equations/notations but easier to work with
Returns (simple vs. logarithmic)
-
Simple returns
o Used to calculate return of portfolio
o
-
-
𝑅𝑖,𝑡+1 =
𝑆𝑖,𝑡+1 −𝑆𝑖,𝑡
𝑆𝑖,𝑡
Log returns
o Used to evaluate time-series properties
𝑆𝑖,𝑡+1
o
𝑅(𝑖,𝑡+1) = log (
o
Log refers to natural log (i.e. ln )
𝑆𝑖,𝑡
)
Both simple and log returns should yield similar results (𝑑 log 𝑥 =
o
o
o
𝑑𝑥
)
𝑥
But log returns takes into account continuous compounding
Simple returns assumes no compounding
Difference should matter only if investment horizon is long or if there is significant shock
in sample
Stylized Facts of Asset Returns
#1: Means and Standard Deviation
-
Typically, not possible to statistically reject a zero-mean return  hence our assumption
#2: Higher moments
-
Unconditional distribution of daily returns typically has fatter tail than normal distribution (i.e.
skew, negative specifically)
#3: Time-varying correlations
-
Correlations among assets change across time  dynamic hedging is required (costly)
In highly volatile down-markets, correlation between almost all asset pairs increase 
diversification benefit decreases when diversification is most needed
#4: International investments (currency element)
-
Currency returns generally positively related to stock and bond returns
#5: Serial correlations (a.k.a autocorrelation)
-
Correlation of one variable at time 𝑡, and the variable at time 𝑡 − 1
E.g. positive serial correlation of volatility  high volatility (i.e. risk) at time 𝑡 implies high
volatility in time 𝑡 + 1
Can be applied to longer periods (e.g. 5-day daily returns, lagged by 1 period i.e. 1 day)
Daily returns found to have very little serial correlation
Volatility is persistent and predictable, whereas serial correlation in squared returns (magnitude
of returns) often is haphazard (hence not predictable)
#6: Leverage effect
-
-
Equity index displays negative correlation between variance and returns
Explanation 1: Drop in stock price  increase leverage of firm (balance sheet equity down,
liabilities and debt increase as a proportion of firm)  equity value moves more in response to
macroeconomic shocks, increase variance
Explanation 2: Investors dislike high variance (i.e. more risk)  higher risk premium required
when variance is high  also called Volatility Feedback
Lecture 3: Market Risk
Purpose of Risk Measures
-
Managerial decisions, to protect against substantial losses
Resource allocation  e.g. setting limits for traders
Performance evaluation
Regulatory reporting
Market Risk
Most generic measure of financial risk that
-
can be measured at firm, portfolio and asset level
applicable to majority types of risks
can be measured over any horizon
can be measured in terms of dollar exposure or % exposure
Two measures of market risk, three ways of measuring
-
Value-at-Risk and Expected Shortfall
Parametric (variance-covariance) approach, historical simulation, monte-carlo simulation
Measure 1: Value-at-Risk (VaR)
Defined as potential loss, in 𝑥% worst cases, given time horizon 𝐾
-
Represented in positive terms, be it % or $ amt
E.g. 95% VaR refers to 5% worst cases (i.e. 100% - 95%)
“There is 𝑥% chance the portfolio value decreases by more than the VaR, after 𝐾 days”
In discrete VaR, we will be conservative and choose the higher number (e.g. Lecture 3 Slide 33)
Supplementary improvements made to VaR
-
-
Relative VaR: calculate relative performance, then mean and standard deviation of relative
performance
o Compares VaR relative to a benchmark’s VaR
Marginal/Incremental VaR: calculates VaR with/without given asset, computes difference
o Since VaR does not tell about marginal contribution each asset has to the portfolio
o Look at an asset’s stand-alone VaR
o Look at its contribution to a portfolio’s VaR (as compared to another asset’s
contribution)
Drawbacks of VaR
-
Extreme losses are ignored  VaR only tells us the bottom 𝑥% threshold, nothing about what
happens below the threshold
Not entirely clear how 𝐾 (time horizon) and 𝑝 (VaR threshold) should be chosen
Measure 2: Expected Shortfall (ES) (Tail-VaR or Conditional VaR)
-
Overcomes the fact that VaR does not tell how bad things get below the 𝑥% threshold
Accounts for magnitude of large losses, and their probability of occurring
Expected value of loss, conditional on it being worse than VaR  calculated by summing losses,
1%
weighted on probability of occurrence over threshold (e.g. 5%)
Approach 1: Parametric (Variance-Covariance) Model
JPMorgan’s RiskMetrics Model
-
-
Assumes particular normal distribution of returns
Key is to measure standard deviation of portfolio
Daily Earnings at Risk (DEAR) ≈ Market value × Daily volatility (%)
Establish confidence intervals (e.g. 90%, 95%, 99%) and adjust for time horizon
Zero average returns are assumed if time horizon < 1 month
Important figures:
o 95%: 1.645𝜎
o 99%: 2.326𝜎
Longer time horizon  nonzero mean  VaR need to be adjusted by the expected return
𝑝
o 𝑉𝑎𝑅𝑡 = 𝜙𝑝−1 𝜎𝑃𝐹,𝑡 − 𝜇
o If you just NORM.INV(1%, 𝜇, 𝜎)  the answer already adjusts for expected return
o If you NORM.S.INV(1%)  need to adjust for mean and standard deviation
Conversion between %VaR and $VaR
-
-
DEAR = $ S.D. and… DEAR ≈ S.D. (%) * Market Value  %VaR × Portfolio value = $VaR
Time horizon adjustment (based on trading days)
o S.D. of daily returns = 𝜎𝑟1𝑑𝑎𝑦
o
S.D. of 2-days returns = √2𝜎𝑟1𝑑𝑎𝑦
o
S.D. of 1-month returns = √22𝜎𝑟1𝑑𝑎𝑦
o
S.D. of 1-year returns = √252𝜎𝑟1𝑑𝑎𝑦
Portfolio adjustment
o 𝜎 2 (𝑅𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 ) = 𝑤 2 𝜎𝑅2𝑎 + (1 − 𝑤)2 𝜎𝑅2𝑏 + 2𝑤(1 − 𝑤)𝜌𝑅𝑎,𝑅𝑏 𝜎𝑅𝑎 𝜎𝑅𝑏
2
o %𝑉𝑎𝑅𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
= 𝑤 2 %𝑉𝑎𝑅𝑅2𝑎 + (1 − 𝑤)2 %𝑉𝑎𝑅𝑅2𝑏 + 2𝑤(1 −
𝑤)𝜌𝑅𝑎,𝑅𝑏 %𝑉𝑎𝑅𝑅𝑎 %𝑉𝑎𝑅𝑅𝑏
2
o $𝑉𝑎𝑅𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
= $𝑉𝑎𝑅𝑅2𝑎 + $𝑉𝑎𝑅𝑅2𝑏 + 2$𝑉𝑎𝑅𝑅𝑎 $𝑉𝑎𝑅𝑅𝑏 𝜌𝑅𝑎,𝑅𝑏
Approach 2: Monte Carlo simulation
-
Assume specific price dynamics
Repeatedly sample values under a probability distribution with specified parameters
Approach 3: Historical simulation
-
Use historical data to evaluate how current portfolio holdings would have performed historically
 derive return distribution
Assumes similar economic conditions will persist
Lecture 5: Market Risk (Historical Simulations and Back-testing)
Historical Simulations
Assumptions and characteristics
-
Assumes history will continue in the near future
Looks at past 𝑇 days to derive a VaR
No distributional assumptions  no estimations needed
Extremely misleading following large shocks
Simple historical simulations (think about it as rolling VaR of 𝑇 number of days)
-
Equally weighted 𝑇-days of historical observations (e.g. {𝑅𝑡−1 , 𝑅𝑡−2 , … 𝑅𝑡−𝑇 })  i.e. each
historical observation takes up
-
100
%
𝑇
Sort and pick the bottom 𝑝% for (100 − 𝑝)% VaR
Use interpolation if necessary (e.g. 1.5 percentile with 100 observations)
Determining size of 𝑇
o Too large: most recent observations carry little weight  smaller change and longer
effects on VaR
o Too small: VaR depend on a very small number of tail events (sensitive to shocks)
Weighted historical simulations
-
Weighted 𝑇-days of historical observations, typically set with exponentially declining weights
o If 𝑤𝑇 is the weight for 𝑅𝑡−𝑇 for 𝑘 < 1
 𝑤𝑇 = 𝑘 ∗ 𝑤𝑇 − 1
 E.g. 𝑘 = 0.9  1, 0.9, 0.92 , …
 Then re-weigh each observation as a fraction of total
o

-
-
-
𝑎(1−𝑟 𝑛 )
1−𝑟
𝑤1 (1−𝑘 𝑇 )
Sum of geometric series =
Sum of 𝑤𝑇 =
1−𝑘
1−𝑘
= 1  𝑤1 = 1−𝑘 𝑇
Sort and accumulate weights  until 1% is reached, for 99% VaR
Pros
o Weighting function makes choice of 𝑇 less crucial
o WHS responds quickly to large shocks, and dies out quickly, consistent with
Autocorrelation Function plots we observed (i.e. serial correlation)
Cons
o No guidance on how to choose decay factor 𝑘
o Unidirectional: if we are short the market, a market crash has no impact on our VaR
since WHS does not respond to large gains
o WHS still requires a lot of historical observations
Persistence/decay factor 𝑘
o Smaller 𝑘 (e.g. 0.1 vs. 0.9)  shock will have shorter influence (because the recent
datapoints have more weight)
Portfolio simulations
-
-
-
When looking at investing in a fund, should we look at historical daily returns of the fund for
VaR?
o No, should look at today’s portfolio holdings, and historical prices of these holdings
o Combined with the portfolio weights to derive fund returns
Hypothetical value of portfolio with 𝑛 assets:
o 𝑉̃𝑃𝐹,𝑡−1 = ∑𝑖 𝑁𝑖,𝑡 𝑆𝑖,𝑡−1 (assets at time 𝑡, multiplied by their spot prices at time 𝑡 − 1)
o Pseudo log returns are computed from pseudo portfolio values
o 𝑅̃𝑃𝐹,𝑡 = ln(𝑉𝑃𝐹,𝑡 ) − ln(𝑉̃𝑃𝐹,𝑡−1 )
Sort the returns and evaluate the percentiles for VaR
Short positions
-
VaR does not move much following positive shock to returns
Problematic particularly since market tends to reverse during high-volatility times
RiskMetrics Model
99% 𝑉𝑎𝑅10−𝑑𝑎𝑦 𝑅𝑀 = √10 × 2.33𝜎𝑡
Add some variance dynamics to arrive at standard RiskMetrics EWMA model (Exponentially-Weighted
Moving Average):
2
𝜎𝑡2 = 0.94𝜎𝑡−1
+ 0.06𝑅𝑡2
-
Variance at time 𝑡 is predicted using variance at time 𝑡 − 1 and the return of time 𝑡
Recursion happens where the new variance reweighs the old variance by 𝜆 = 0.94 and adds
effect of latest return into calculation by 1 − 𝜆 = 0.06
VaR Limit
Trading limits
-
Often imposed in units of $VaR
Assume a trader is given 10-day 99% $VaR limit of $100,000 (assigned value by manager)
-
Trader hence allowed to invest $𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛 ≤ 99% %𝑉𝑎𝑅
$100,000
10−𝑑𝑎𝑦
 e.g. if your 99% %VaR is 5% loss
 you can have up to $2,000,000 of position
-
𝑆𝑡+1
)
𝑆𝑡
PnL computed as $𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑡 log (
HS VaR vs RiskMetrics VaR
-
HS VaR react too slowly to negative shocks  increases VaR too late  $Position will be too big
when market starts going down  huge losses
HS VaR react too slowly to upturn  decreases VaR too slowly  $Position too small when
market starts going up  small profits
Additional Analysis (Refer to Lecture Notes, not tested)
-
Volatility scaling
Bootstrapping
-
Backtesting HS VaR estimates
Lecture 6: Understanding Credit Risk
Credit Risk
Credit default risk  risk of credit event happening
-
Full default: Bankruptcy, default of reference entity
Partial default: Fail to pay in full (principal + interest) on time
Debt restructuring (decrease coupon or extend maturity)
Violation of any bond covenants
Credit spread risk
-
Risk of fluctuation of asset prices, due to changes in firm’s credit condition
Credit Ratings
Rating agencies
-
Baa3 and BBB- are lowest of IG
Ba1 and BB+ are highest of HY
-
Ratings are given either issue-specific or issuer-specific
Order of ratings: Secured > unsecured > subordinated, Sovereigns > corporates (usually)
Historical data on annual default rates based on ratings
Credit transition (or migration)  Credit spread risk
-
-
Changes to credit ratings  based on a probability distribution  corresponding change in asset
value/price
E.g. 3Y annually compounded pure discount bond with BBB-rating, priced at YTM = 5% currently.
What is the 99%VaR in returns for immediate credit changes if credit rating probability is:
o BB, E(YTM) = 6%, P(transition) = 3%
o B, E(YTM) = 7%, P(transition) = 1%
o CCC or lower, E(YTM) = 12%, P(transition) = 0.5%
Look at credit downgrade probability  (i) 0.5% CCC or lower, (ii) 0.5% B  B forms the 99%VaR
scenario
1000
o
B-rating bond price = (1.073 ) = 816.2979
o
Current price at YTM 5% =
o
∴ 99% $VaR = $863.8376 − $816.2979 = $47.5397  99% %VaR = $863.8376 =
1000
1.053
= 863.8376
$47.5397
0.055 = 5.5%
Issues with external ratings from agencies
-
Conflict of interest – issuers are clients, and governments can influence rating agencies
Correlated credit ratings – rating agencies share models
Difficult to model – quantifying black swan events like COVID-19
Internal Credit Ratings
Internal ratings-based approach
-
More tailored  at-the-cycle (predicts over short-term) or through-the-cycle (longer horizon)
Usually only large banks can fulfil requirements to engage in internal-ratings based approach
Determinants of credit ratings
-
Leverage ratios (D/E, D/A, etc.)
Coverage ratios (interest coverage, cash coverage, asset coverage, …)
Liquidity ratios (cash ratio, current ratio, etc.)
Profitability ratios (RoE, RoA, DPS, EPS, etc.)
Volatility of earnings
Debt covenants  restrictions imposed on borrowers by lenders
o Lower financing cost for borrower
o Lower credit risk for lenders
o E.g. financial ratio thresholds, dividend restrictions, subordination of future debt
issuances
o E.g. sinking funds  funds set aside periodically to gradually repay debt  similar to
amortization where principal gets paid off over tenor  more CF earlier for lenders 
reduces duration and risk
Expected Loss From Credit Event
Exposure at default (EAD)
-
Total amount outstanding including principal and interest
Probability of default (PD)
-
-
Measures probability of credit event happening
Real-world probability: used in everyday language
o Hazard rate (a.k.a default intensity 𝜆𝑡 ) = Pr(𝐷𝑒𝑓𝑎𝑢𝑙𝑡(𝑡)| 𝑁𝑜 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 (𝑇 < 𝑡))
 Conditional probability of default, given no earlier default
o E.g. Among firms that survive till Y4, 10% of firms default in Y5. What is probability of a
firm defaulting w/in 5Y assuming constant hazard rate?
 P(default(Y5)|no default(T<Y5)) = 0.1  constant default rate means regardless
which year, the conditional probability of default remains at 0.1
 P(default w/in 5Y) = P(df Y1) + P(df Y2) + … + P(df Y5) = 0.1 + 0.1(0.9) + ⋯ +
0.1(0.94 ) = 40.95%
 = 1 – Pr(df > Y5) = 1 − 0.95 [sum of geometric series formula, but 𝑎 and 1 − 𝑟
cancel out because they’re equal]
Risk-neutral probability: probability implied by market prices
o Expected return for corporate bonds is not the YTM
 Assuming reinvestment at same rate, Treasury YTM = 𝐸[𝑅] (no default risk)
 YTM of corporate bond computed based on “no default” assumption  but
there is possibility of default  YTM ≠ expected return of loan
 ∴ 𝐸[𝑅𝑐 ] should account for expected %loss given default
 𝐸[𝑅𝑐 ] = (1 + 𝑌𝑇𝑀) ∗ (1 − 𝑃𝐷 ∗ 𝐿𝐺𝐷) − 1
o Estimating PD
 Possible to back-out PD from bond prices (but… when PD is estimated using
bond prices or YTM, the PD is not real probability  it is risk-neutral)

o
Risk-neutral explanation: e.g. an asset gives you $110 with p = 50%, and $90
with p = 50%  EV of payoff = $100  but one would only pay $100 if they
were risk-neutral (i.e. people are risk-averse, and would pay less than EV e.g.
$98)
 Extension: if the asset gives you $110 with p, and $90 with (1-p)  EV = 110p +
90 – 90p = market price (e.g. $98)  20p = $8  p = 0.4
 This is what we can infer about PD from market prices, assuming we take a riskneutral stance (i.e. no buying below value because of risk-aversion)  riskneutral distribution underweights bad states, overweighs good states compared
to risk-averse
Indifference between risky and risk-free instruments
 𝐸[𝑅] when adjusted for risk-neutral PD: 𝐸 ∗ [𝑅]  should exhibit 0 risk premium
over 𝑅𝑓  i.e. 𝐸 ∗ [𝑅] = 𝑅𝑓
 As such, 𝐸[𝑅] > 𝐸 ∗ [𝑅] always
 Risk premium will approximately reflect expected dollar loss of
investment (𝑃𝐷 ∗ × 𝐿𝐺𝐷)
 𝑌𝑇𝑀𝐺𝑜𝑣 = 𝐸 ∗ [𝑅𝐶𝑜𝑟𝑝 ] = (1 + 𝑌𝑇𝑀𝐶𝑜𝑟𝑝 ) × (1 − 𝑃𝐷 ∗ × 𝐿𝐺𝐷) − 1

𝑌𝑇𝑀𝐶𝑜𝑟𝑝 −𝑌𝑇𝑀𝐺𝑜𝑣

This roughly expresses that the difference between expected return of a
corporate bond, and the risk-free bond, will approximately equal the
expected dollar loss of the investment (in the risk-neutral sense)
1+𝑌𝑇𝑀𝐶𝑜𝑟𝑝
= 𝑃𝐷 ∗ × 𝐿𝐺𝐷
Loss given default (LGD)
-
-
Measures %loss given credit event
Recovery rate = 1 – LGD
o Recovery depends on seniority and collateralization
o Recovery rate usually defined as price of bond 30 days after default, as a % of face value
o Recovery rate difficult to measure, also depends on macroeconomic conditions
Moody’s estimates of average recovery rate = 59.33% − 3.06 ∗ 𝐷𝑓 𝑟𝑎𝑡𝑒(%)
o Can see that higher default rate is linked to lower recovery rate
Expected Loss
-
E[Loss] = EAD * PD * LGD
Forms foundation for credit default risk measurement
Lecture 7: Credit Risk Applications and Mgmt.
Credit Risk Diversification
Credit portfolio
-
Corporate bonds have negatively skewed payoff (referring to default)  negative skew for
independent distributions can be easily and completely diversified away
Credit risk relatively uncorrelated across firms, as compared to other assets (e.g. stocks)
Thus, forming credit portfolio substantially reduces risk
Example
-
Single loan, $1 million, LGD = 1, PD = 0.01  E[L] = 0.01 * $1 million = $10,000
-
Therefore, SD = √0.01(0 − 990,000)2 + 0.99(1𝑀 − 990,000)2 = $994,987
Assuming independent defaults, two loans of $500,000 each  Same E[L] = 0.01 * $500,000 * 2
= $10,000
But… SD becomes lower  SD =
-
√0.012 (0 − 990,000)2 + 2(0.01)(0.99)(500,000 − 990,000)2 + 0.992 (1𝑀 − 990,000)2 =
$14,100 approximately
Credit VaR
Example 1
-
Single loan, $1,000, LGD uncertain, PD = 0.1  revealed that LGD = 0.3 w/ 60% chance, 0.7 w/
40% chance
95% VaR  of the 10% PD  bottom 4% is LGD = 0.7, next 6% is LGD = 0.3  so 95% VaR = 0.3
* $1,000 = $300
Example 2 (regarding EAD = principal + interest)
-
Single loan, one-year, $1,000, interest = 5%, EAD = $1,050, PD = 7%, LGD = 0.5
In the 7% PD  Recoverable amt. will be = 0.5 * $1,050 = $525
95% VaR for initial investment of $1,000 = $1,000 - $525 = $475
Example 3 (1 more bond on E.g. 2)
-
Two loans, one-year, $1,000, interest = 5%, EAD each = $1,050
PD Bond 1 = 10%, PD Bond 2 = 8%, PD Bond 2 only = 8% * 90%  LGD = 0.7  Loss $735, VaR =
$685
PD Bond 1 and 2 = 0.8%  Loss $1,050, VaR = $1,050
95% VaR
o PD Bond 1 only = 10% * 92% = 9.2%  VaR = $2000 – ($1,050 + $1,050 * (1 – 0.3)) =
$215
o PD Bond 2 only = 8% * 90% = 7.2%  VaR = $2,000 – ($1,050 + $1,050 * (1 – 0.7)) =
$635
o PD Bond 1 and 2 = 10% * 8% = 0.8%  VaR = $2,000 – ($1,050 * (1 – 0.3) + $1,050 * (1 –
0.7)) = $950
-
o Worst 0.8% VaR = $950
o Next worst 4.2% VaR  Bond 2 df only  VaR = $635
95% VaR = $635
95% ES = (0.8/5) * $950 + (4.2/5) * $635 = $152 + $533.4 = $685.40
Loan Concentration Risk
Concentration limits
-
Sets external limit on max amt. of loan that can be made to individual borrower/sector (as % of
capital)
Max loss as % of capital = Concentration limit * Loss rate
Goal is not to be concentrated to firms with similar characteristics (e.g. country, sector, size)
Example
-
Mgmt. caps losses at 15% of capital to a particular sector
Amt. lost per dollar of defaulted loans in the sector = 40cents  LGD = 0.4
If total capital of firm is $100M  what is maximum amount of loans to a single sector?
Losses allowed for one sector = 15% * $100M = $15M
Losses = 0.4 of loan quantum = $15M  Loan quantum = $37.5M
Credit Default Swaps
Characteristics
-
-
Privately negotiated (OTC) bilateral contract
Specifications of contract:
o Reference entity (bond issuer)
o Notional amount (usually covering the bond principal)
o Premium
o Maturity
CDS terminates on credit event  no further premium to be paid after
Applicable cashflows
o Buyer: pays periodic premium (premium rate * notional)
o Seller: pays lump-sum on credit event (notional * (1-recovery rate))  only cover nonrecoverable portion
CDS pricing
-
-
𝐸 ∗ [PV of cash outflow] = 𝐸 ∗ [PV of cash inflow]
i.e. PV of expected CDS premium paid out has to equate PV of expected default payment
o Assumes zero protection premium
o Else the PV of premium paid > PV of expected default payment)
o CDS premium paid each year is also not guaranteed  depends on P(no default)
𝐸 ∗ derived using risk-neutral probability  if real-world probability were used, buyer pays more
because risk-averse  bad states are assigned higher probability  more premium required
Additional notes:
o Hazard rate ≈ CDS spread when LGD = 1 (difference stems from time-value of money)
o Insurance premium must be ≈ P(loss)
o
Actual spreads will be higher than those suggested by risk-neutral probabilities
CDS spread and default premium (preventing arbitrage)
-
Default premium often defined as corporate bond yield in excess of treasury w/ matching
maturity and tenor
CDS spread should approx. equal default premium  holding corp. bond + CDS contract (where
CDS notional = EAD) should equate holding risk-free bond with equal notional and maturity
Can back-out PD* × LGD from CDS spread
Swaps Trading
Swaps
-
-
Contract between 2 parties, agreement to exchange CF driven by 2 different assets
E.g.
o I/R swap: fixed vs. floating rate bonds
o FX swap: CFs of 2 fixed income securities from different CCYs
o CDS: CFs of fixed coupon treasury bond vs. CF of credit bond
Credit risk only arises when swap value goes positive (i.e. you gain, counterparty lose)
Credit value adjustment (CVA)
-
-
Expected loss due to counterparty’s credit risk  need to adjust value of swap for this exp. loss
o 𝐶𝑉𝐴𝑡 = Swap value (𝑉𝑡 ) * 𝑃𝐷𝑡 * 𝐿𝐺𝐷𝑡
To hedge credit risk for swap positions  CVA = CDS premium paid to hedge
Risk-adjusted value of swap (𝑉𝑅𝐴 )
𝑉 − 𝐶𝑉𝐴𝑡 , 𝑉𝑡 > 0
o 𝑉𝑅𝐴,𝑡 = { 𝑡
 i.e. min(𝑉𝑡 (1 − 𝑃𝐷𝑡 ∗ 𝐿𝐺𝐷𝑡 ), 𝑉𝑡 )
𝑉𝑡 , 𝑉𝑡 < 0
o In other words:
 Swap value positive  risk-adjusted swap value is swap value minus credit value
adjustment (i.e. take out credit risk)
 Swap value negative  risk-adjusted value of swap is the negative swap value
(i.e. whatever you’re bound to pay)
What if you long CDS to protect against a default, but both sides default?  note that short
positions of CDS are largely concentrated to the few IBs and insurance companies
o What happened in practice is also that parties started buying CDS to protect against CDS
defaults
o E.g. MS could be buying CDS from CS against Lehman default, but not trust CS to fulfil
the obligation if Lehman defaults  and goes to JPM to buy a CDS to cover the CDS with
CS
Key Question Bank
Week 2
7. Suppose you invest in a stock called "ABC". You expect ABC to perform well in the next
month with N(0.04, 0.072), but not as well in the following month with N(-0.02, 0.062).
Assuming you expect zero serial correlation between the two periods, what distribution do you
expect for the returns over the next two months?
 Average return over next 2 months is 2%
Variance of return over next 2 months is 0.07^2 + 0.06^2 = 0.0085
SD of return is the square root of it which is 0.0922
Two events in this example will happen together (there is no case where you live in period 1 but do not
live in period 2 or vice versa). Hence, the sum of the two normal random variable follows a normal
distribution.
Week 7
Question 7 of Wk 7 Quiz  Use power to adjust for compounding for fractional years
1)
2)
3)
4)
1000/1.03=970.87, 970.9~970.9 okay.
=0.005*1.645+0.03= 3.8225%, 3.82~3.83 okay
1000/((1+.038225)^(11/12)) = 966.2
970.9-966.2=4.7 [4.6~4.7 okay]