Risk Management and Derivatives Flashcards
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
Non-Financial 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 variance-covariance method
Formula: [Rp - (z)(std)] * Vp
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: [Rp - (z)(std)] * Vp
To compute weekly:
Rp = Rp / 52
std = std / √52
To compute monthly:
Rp = Rp / 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)
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
- Worst-case scenario
Forward Contract
Value of Credit Risk (Currency)
value to long = St / (1 + Rforeign)t - F0 / (1 + Rdomestic)t
Positive = long credit risk Negative = seller credit risk
F = foreign (must be BASE)
Example: US enters 2-yr 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
Rp - Rf / stdp
Assumes normal distribution (no skew)
Sortino (use if std is inflated)
Rp - MAR / stddownside
Only downside being considered
Risk-Adjusted ROC
RAROC = Rp / capital at risk
capital at risk = VaR, etc.
Return over Maximum Drawdown (RoMAD)
RoMAD = Rp / maximum drawdown
maximum drawdown = largest historical % decline from high to low
Beta Formula
Bi = Cov(i,m) / stdm2
Beta Contracts
(BT - BP) / Bf * Vp / Pf (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 = ((DT - Dp) / DF) * [Vp / PF (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 Vp / % 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 Rf 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: (BT - BP) / BF * VP / PF (multiplier)
but the FV for VP = VP(1 + Rf)t
So: (BT - BP) / BF * VP(1 + Rf)t / PF (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 + Rf)t
For EV do it again: (1 + Rf)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
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
Rf, Volatility, and Asset Price
Input Calls Puts
Asset Price ↑ Positive Negative
Rf ↑ Positive Negative
Volatility ↑ Positive Positive
Covered Call
Buy underlying and selling call option
- profit = (ST - X) + C0 + ST - S0
- max profit = X + C0 - S0
- max loss = S0 - C0
- Breakeven = S0 - C0
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 - ST) - P0 + ST - S0
- max profit = ST - P0 - S0
- max loss = S0 + P0 - X
- breakeven = S0 + P0
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 XL, sell call at XH
- Buy put at XL, sell put at XH
Bear Spread
Profit when underlying decreases
*Built with all calls or puts
- Sell call at XL, buy call at XH
- Sell put at XL, buy put at XH
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 = (ST - X) + (X - ST) - C0 - P0
Max Profit = Infinite as stock increases
Max loss = C0 + P0
Breakeven = X - C0 - P0 AND X + C0 + P0
Butterfly Spread
Requires four options (2 long/2 short) with 3 strikes
Option 1:
Buy call XL, sell 2 calls XM, buy call at XH
Option 2:
Buy put XL, sell 2 puts XM, buy put at XH
Option 3:
Buy put XL, sell put and call XM, buy call XH
Collar
Same as bull spread but done with owning the underlying
buy put XL, sell a call XH, 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 1-unit change in the price of the underlying stock (speed)
Call Range from 0-1
Out-of-the-money is closer to 0
In-the-money is closer to 1
Put Range is from -1 to 0
Out-of-the-money is closer to 0
In-the-money 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
Important to draw out the diagram
Interest Rate Swap Duration
1
Dpay floating = Dfixed - Dfloating = +D paying floating INCREASES duration
Dpay fixed = Dfloating - Dfixed = -D paying fixed DECREASES duration
2
Floating rate duration resets and assume duration is half. (1 year = 0.5)
Example:
5-year 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 = VP [(MDT - MDP) / MDswap]
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 [(4-5.2) / -3.1)] = 23,225,806
Swaption
Definition: option to enter a pre-negotiated 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