Risk Goverance
Two Types:
Decentralized  each unit is responsible
Centralized (aka ERM)  one central unit is responsible
Allows overview
Economies of scale
Enterprise Risk Management (ERM) Evaluation
Goal: Identify and take profitable risks
 Aggregates risks
 Considers correlation
 Serious commitment and expense
Market and Financial Risk Factors
Market Risks (manage by derivatives)
 Interest rates
 Exchange rates
 Equity prices
 Commodity prices
Financial Risk
 Credit risk
 Liquidity risk
Nonfinancial Risk Factors
NonFinancial Risks (manage by insurance)
 Operational  computer, human, or weather events
 Settlement (Hersatt) other party fails to pay
 Model
 Sovereign
 Regulatory
 Tax, accounting and legal
Value at Risk (VaR)
Analytical Method
Also called variancecovariance method
Formula: [R_{p}  (z)(std)] * V_{p}
Know the following for z:
5% = 1.65 2.5% = 1.96
1% = 2.33 0.5% = 2.58
Example: Calculate 5% annual VaR for 150M
R = 9.55, std =14.87
9.55  1.65(14.87) = 14.99%
.1499 * 150M = 22.49M loss
Value at Risk (VaR)
Analytical Method  Monthly/Weekly
Formula: [R_{p}  (z)(std)] * V_{p}
To compute weekly:
R_{p} = R_{p }/ 52
std = std / √52
To compute monthly:
R_{p} = R_{p }/ 12
std = std / √12
Example: Calculate 5% weekly VaR for 150M
R = 9.55, std =14.87
(9.55 / 52)  1.65(14.87 / √52) = 3.22%
.0322 * 150M = 4.83M weekly loss
Value at Risk (VaR)
Analytical Method  Disadvantages
 Some returns, like options, are skewed (assumes normal)
 Market distributions have fat tails (leptokurtosis)
 Std hard to estimate for large portfolios
Value at Risk (VaR)
Historical Method
Rank all returns from lowest to highest and identify the % you need
Advantages: reflects past distributions
nonparametric
Disadvantage: assumes historical returns will will repeat
Example: 100 daily returns, the 5 lowest are:
.0019, .0025, .0034, .0096, .0101
Calculate daily VaR at 5%
5th lowest .0019. Means a 5% chance of daily loss exceeding 0.19%.
Value at Risk (VaR)
Monte Carlo
Similar to Historical, ranks outcomes.
Advantages
 Can customize (normal distribution for some assets, skewed for others, etc.)
VaR Extentions
 Incremental VaR
 Tail Value at Risk (TVaR)  average value in the 5% tail
 Looks at whole tail (even past VaR)
 Cash Flow at Risk (CFAR)
 Earnings at Risk (EAR)
 Looks at whole tail (even past VaR)
Credit Risk and Credit Risk VaR (CVaR)
Credit Risk Loss depends on:
Probability of default
Amount that can be recovered
CVaR estimates loss due to credit events
Types of Stress Testing
Complement to VaR
 Factor Push: puts factors at worst combination
 Maximum loss optimization: models the worst combination of factors
 Worstcase scenario
Forward Contract
Value of Credit Risk (Currency)
value to long = S_{t} / (1 + R_{foreign})^{t}  F_{0} / (1 + R_{domestic})^{t}
Positive = long credit risk Negative = seller credit risk
F = foreign (must be BASE)
Example: US enters 2yr forward, purchase = 10,000 EUR at USD 0.90
6 months later: Spot = .862/EUR. 1.5 yr rates: US 6%, EUR 5%
Calculate potential credit risk
.862 / (1.05)^{1.5}  0.90 / (1.06)^{1.5} = 0.0235
Long is losing. NO credit risk
Seller is winning, has credit risk of 10,000 EUR * 0.235 = $235
Forward/Swaps/Options/Futures Credit Risk
Forward: only change hands at end. Party winning has risk. Highest risk near the end
Swaps: credit risk at each swap date. Highest risk in the middle
Options: only long positions faces credit risk
Futures: No credit risk
Option Credit Risk
Currrent credit risk: only at exercise
Potential credit risk: positive market value of the option
Example:
Dealer sold a call option, X = $35, value = $46
Current credit risk: none
Potential credit risk: None for dealer, $46 per share for buyer
Managing Market Risk
VaR manager example
VaR is not additive because it considers correlation
Example A B
Capital $100,000,000 $500,00,000
VaR $5,000,000 $10,000,000
Profit $1,000,000 $3,000,000
Return on Capital 1% 0.6%
Return on VaR 20% 30%
RoC has A winning, but RoVaR has B winning
Risk Budgeting
Determining where and how much risk to take through ERM
Types: (not important)
 VaR limits
 Liquidity limits
 Performance stop loss
 Risk factor limits
 Scenario analysis limits
 Leverage limits
How to Manage Credit Risk

Collateral

Credit default swap/forward

Mark to market  settle contract now to reprice

Minimum credit standards

Limit exposure (position, loss, factors, VaR, leverage, liquidity)
Sharpe Ratio vs Sortino
Sharpe
R_{p}  R_{f} / std_{p}
Assumes normal distribution (no skew)
Sortino (use if std is inflated)
R_{p}  MAR / std_{downside}
Only downside being considered
RiskAdjusted ROC
RAROC = R_{p} / capital at risk
capital at risk = VaR, etc.
Return over Maximum Drawdown (RoMAD)
RoMAD = R_{p} / maximum drawdown
maximum drawdown = largest historical % decline from high to low
Beta Formula
B_{i} = Cov(i,m) / std_{m}^{2}
Beta Contracts
(B_{T}  B_{P}) / B_{f} * V_{p} / P_{f} (multiplier)
Example: 5M portfolio w/ beta of 0.8.
Futures contract beta = 1.05 and price = 240,000
Calculate # of contracts to get beta of 1.1 and 0.0
# of contracts = (1.1  0.8 / 1.05) * (5M / 240,000) = 5.95
Means buy 6 contracts at 240K
# of contracts = (0  0.8 / 1.05) * (5M / 240,000) = 15.87
Means sell 16 contracts at 240K
Target Duration with Futures
# of contracts = ((D_{T}  D_{p}) / D_{F}) * [V_{p} / P_{F} (multiplier)] * Yield Beta
Use yield beta if not parallel shift
Example: bond portfolio 103,630, 1 year period. Futures = 102,510
duration p = 1.793, duration f = 1.62, yield beta = 1.2
Calculate # of contracts to get duration to 0 and 3
(0  1.793) / 1.62 * (103,630/102,510) * 1.2 = 1.34
(3  1.793) / 1.62 * *103,630/102,510) * 1.2 = 0.9
Ex Post Results (Effective Beta)
effective beta = % change in V_{p} / % change in the index
Example:5M portfolio increased to 5.255M and futures increased 240K to 252,240. Market return was 5.2%.
Bought 6 contracts
contracts went up 12,240 * 6 = 73,440
hedged portfolio value = 5,255,000 + 73,440 = 5,328,440
Hedged return = (5,328,440/5,000,000)  1 = 6.57%
effective beta = 6.57 / 5.2 = 1.26
What is Basis Risk?
Describe each cause/type
When hedging is not perfect
Type: Cross hedge (hedging an index with few stocks)
Type: Contract expiration differs from hedge time frame
Type: Portfolio and contracts perform differently
Type: Rounding contracts
Synthetics
More precisely replicates outright ownership. Must calculate FV
Synthetic Stock: buy contracts and hold enough cash earning R_{f} to pay for shares
Synthetic Cash: sell contracts and hold enough shares of stock w/ dividends reinvested to deliver shares at expiration
Synthetic Equity/Cash Position Formula
# of contracts
Remember its the same: (B_{T}  B_{P}) / B_{F} * V_{P} / P_{F }(multiplier)
but the FV for V_{P} = V_{P}(1 + Rf)^{t}
So: (B_{T}  B_{P}) / B_{F} * V_{P}(1 + R_{f})^{t} / P_{F }(multiplier)
Example: Convert 100M to cash for 3 months
Rf = 3.5%, Equity price = $325,000, Index dividend yield = 2%
No beta given so (1  1) / 1 * 100M(1.035)^{.25} / 325,000 = 310.35
Synthetic Equitized/Initial Cash Position
Amount required today:
(# of contracts)(multiplier)(price) / (1 + R_{f})^{t}
For EV do it again: (1 + R_{f})^{t}
Example: 300M synthetic in R2000, 3mo futures price = 498.30
Multiplier: 500, Rf: 2.35%, Dividend yield: 0.75%
# of contracts: (300M)(1.0235)^{3/12} = 1211.11
Amount equitized: (1211)(500)(498.30) / (1.0235)^{3/12} = 299,973,626
EV at contract expiration: 299,973,626(1.0235)^{3/12} = 301,720,650
Adjusting Asset Allocation with Futures (Example)
Current Portfolio: 80/20 stocks/bonds
Size: 300M, Beta: 1.1, Duration: 6.5
Desired Temporary Allocation: 50/50
Stock futures price: 200,000, Beta: 0.96
Bond futures price: 105,250, Duration: 7.2
Need to sell stock contracts, buy bond contracts for 30%
300M * .3 = 90,000,000
Sell stock contracts: ((0  1.1) / 0.96) * (90,000,00 / 200,000) = 515.63
Buy bond contracts: ((6.5  0) / 7.2) * (90,000,000 / 105,250) = 717.97
Sell 516 stock contracts at 200,000
Buy 772 bond contracts at 105,250
Types of Foreign Exchange Risk

Transaction  cash flow at later date, can be hedged
 most common

Economic  ▲ in currency affect competitiveness
 Example; dollar ^, less competitive in foreign markets
 Can be difficult to hedge

Translation  risk of reporting in another currency
 most common
 Example; dollar ^, less competitive in foreign markets
 Can be difficult to hedge
How to Hedge a Currency Position
Receiving Foreign
Paying Foreign
Position Action
Receiving Foreign Long Sell forward
Paying Foreign Short Buy forward
Call Options
Right to buy underlying
Make money when goes up
Put Options
Right to sell
Make money when goes down
Relation to Calls/Puts
R_{f, }Volatility, and Asset Price
Input Calls Puts
Asset Price ↑ Positive Negative
R_{f} ↑ Positive Negative
Volatility ↑ Positive Positive
Covered Call
Buy underlying and selling call option
 profit = (S_{T}  X) + C_{0 }+ S_{T}  S_{0}
 max profit = X + C_{0}  S_{0}
 max loss = S_{0}  C_{0}
 Breakeven = S_{0}  C_{0}
Example:
(35  45) + 35  43 + 2.10 = 5.90
Protective Put
AKA portfolio insurance/hedged portfolio
Definition: Hold underlying buy a put option
 profit = (X  S_{T})  P_{0 }+ S_{T}  S_{0}
 max profit = S_{T}  P_{0}  S_{0}
 max loss = S_{0} + P_{0}  X
 breakeven = S_{0} + P_{0}
Example:
(35  30) + 30  37.5  1.4 = 3.90
Bull Spread
Profit when underlying increases
*Built with all calls or puts
 Buy call at X_{L}, sell call at X_{H}
 Buy put at X_{L}, sell put at X_{H}
Bear Spread
Profit when underlying decreases
*Built with all calls or puts
 Sell call at X_{L}, buy call at X_{H}
 Sell put at X_{L}, buy put at X_{H}
Box Spread
Bull and Bear Spread combined
 Creates an aribtrage relationship
 Known initial and ending cash flows
Straddle and Reverse Straddle
Making money from volatility. THE BIG V
Straddle
Buy call and put at same strike
Profit from high volatility
Reverse Straddle
Sell a call and put at same strike
Profit from low volatility
Straddle
Profit
Max Profit
Max Loss
Breakeven
Profit = (S_{T}  X) + (X  S_{T})  C_{0}  P_{0}
Max Profit = Infinite as stock increases
Max loss = C_{0} + P_{0}
Breakeven = X  C_{0}  P_{0} AND X + C_{0} + P_{0}
Butterfly Spread
Requires four options (2 long/2 short) with 3 strikes
Option 1:
Buy call X_{L}, sell 2 calls X_{M}, buy call at X_{H}
Option 2:
Buy put X_{L}, sell 2 puts X_{M}, buy put at X_{H}
Option 3:
Buy put X_{L}, sell put and call X_{M}, buy call X_{H}
Collar
Same as bull spread but done with owning the underlying
buy put X_{L}, sell a call X_{H}, Own underlying
Commonly used for interest rate options.
Hedging with Interest Rate Options
Borrowing/Lending
Borrowing
Risk: increasing rates
Hedge: buy an interest rate call
Lending
Risk: decreasing rates
Hedge: buy a interest rate put
Caps and Floors
Cap: Series of interest rate calls
Protects a floating rate debt payer from increasing rates
Floor: series of interest rate puts
Protects owner of floating rate debt from decreasing rate
Delta
Definition: change in the price of an option for a 1unit change in the price of the underlying stock (speed)
Call Range from 01
Outofthemoney is closer to 0
Inthemoney is closer to 1
Put Range is from 1 to 0
Outofthemoney is closer to 0
Inthemoney is closer to 1
Swap Risks
Floating and Fixed Rate
 Floating rate debt has minimal duration but uncertain cash flow
 High cash flow risk
 Low duration
 Low market value risk
 Fixed rate debt has higher duration but certain cash flow
 Low cash flow risk
 High duration
 High market value risk
 High cash flow risk
 Low duration
 Low market value risk
 Low cash flow risk
 High duration
 High market value risk
Important to draw out the diagram
Interest Rate Swap Duration
1
D_{pay floating} = D_{fixed}  D_{floating} = +D paying floating INCREASES duration
D_{pay fixed} = D_{floating}  D_{fixed} = D paying fixed DECREASES duration
2
Floating rate duration resets and assume duration is half. (1 year = 0.5)
Example:
5year pay fixed swap with quarterly settlement. Comparable bond 4.1 duration
4.1 + 0.25/2 = 3.975
Swap Cash Flow and Market Risk Strategies
Rates will increase
Scenario Strategy Result
Assets: Fixed Receive Float ↓ MVR, ↑ CFR
Assets: Float Do nothing Accept low MVR, high CFR
Liability: Fixed Do nothing Accept high MVR, low CFR
Liability: Float Receive Fixed ↑ MVR, ↓ CFR
Modified Duration Swap
NP = V_{P} [(MD_{T}  MD_{P}) / MD_{swap}]
Formula calculates size of swap. Make sure to state what type:
Pay fixed (decrease) or pay float (increase). Put that in the denominator
Example: 60M portfolio, duration 5.2, target 4, swap duration 3.1
Lower duration so we want to pay fixed
60,000,000 [(45.2) / 3.1)] = 23,225,806
Swaption
Definition: option to enter a prenegotiated swap
Two Types
 Payer: allows swaption buyer to enter a pay fixed
 gains value if rates rise
 Receiver: allows swaption buyer to enter a received fixed
 gains value if rates fall
EAR
Step 1: Premium from call
Premium * (1 + r)^{days/360}
Step 2: Loan  premium
Step 3: Loan interest
Step 4: Call payoff
Step 5: [(Loan + interest  call payoff) / loan proceeds]^{annual rate}