Risk Management and Derivatives

Question 1

Risk Goverance

Answer 1

Two Types:

Decentralized - each unit is responsible
Centralized (aka ERM) - one central unit is responsible
Allows overview
Economies of scale

Question 2

Enterprise Risk Management (ERM) Evaluation

Answer 2

Goal: Identify and take profitable risks

• Aggregates risks
• Considers correlation
• Serious commitment and expense

Question 3

Market and Financial Risk Factors

Answer 3

Market Risks (manage by derivatives)

• Interest rates
• Exchange rates
• Equity prices
• Commodity prices

Financial Risk

• Credit risk
• Liquidity risk

Question 4

Nonfinancial Risk Factors

Answer 4

Non-Financial Risks (manage by insurance)

1. Operational - computer, human, or weather events
2. Settlement (Hersatt)- other party fails to pay
3. Model
4. Sovereign
5. Regulatory
6. Tax, accounting and legal

Question 5

Value at Risk (VaR)

Analytical Method

Answer 5

Also called variance-covariance 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

Question 6

Value at Risk (VaR)

Analytical Method - Monthly/Weekly

Answer 6

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

Question 7

Value at Risk (VaR)

Analytical Method - Disadvantages

Answer 7

• Some returns, like options, are skewed (assumes normal)
• Market distributions have fat tails (leptokurtosis)
• Std hard to estimate for large portfolios

Question 8

Value at Risk (VaR)

Historical Method

Answer 8

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%.

Question 9

Value at Risk (VaR)

Monte Carlo

Answer 9

Similar to Historical, ranks outcomes.

Advantages

• Can customize (normal distribution for some assets, skewed for others, etc.)

Question 10

VaR Extentions

Answer 10

• 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)

Question 11

Credit Risk and Credit Risk VaR (CVaR)

Answer 11

Credit Risk Loss depends on:
Probability of default
Amount that can be recovered

CVaR estimates loss due to credit events

Question 12

Types of Stress Testing

Answer 12

Complement to VaR

1. Factor Push: puts factors at worst combination
2. Maximum loss optimization: models the worst combination of factors
3. Worst-case scenario

Question 13

Forward Contract

Value of Credit Risk (Currency)

Answer 13

value to long = S_t / (1 + R_foreign)^t - F₀ / (1 + R_domestic)^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

Question 14

Forward/Swaps/Options/Futures Credit Risk

Answer 14

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

Question 15

Option Credit Risk

Answer 15

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

Question 16

Managing Market Risk

VaR manager example

Answer 16

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

Question 17

Risk Budgeting

Answer 17

Determining where and how much risk to take through ERM

Types: (not important)

1. VaR limits
2. Liquidity limits
3. Performance stop loss
4. Risk factor limits
5. Scenario analysis limits
6. Leverage limits

Question 18

How to Manage Credit Risk

Answer 18

1. Collateral
2. Credit default swap/forward
3. Mark to market - settle contract now to reprice
4. Minimum credit standards
5. Limit exposure (position, loss, factors, VaR, leverage, liquidity)

Question 19

Sharpe Ratio vs Sortino

Answer 19

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

Question 20

Risk-Adjusted ROC

Answer 20

RAROC = R_p / capital at risk

capital at risk = VaR, etc.

Question 21

Return over Maximum Drawdown (RoMAD)

Answer 21

RoMAD = R_p / maximum drawdown

maximum drawdown = largest historical % decline from high to low

Question 22

Beta Formula

Answer 22

B_i = Cov(i,m) / std_m²

Question 23

Beta Contracts

Answer 23

(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

Question 24

Target Duration with Futures

Answer 24

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

Question 25

Ex Post Results (Effective Beta)

Answer 25

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

Question 26

What is Basis Risk? Describe each cause/type

Answer 26

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

Question 27

Synthetics

Answer 27

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

Question 28

Synthetic Equity/Cash Position Formula of contracts

Answer 28

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

Question 29

Synthetic Equitized/Initial Cash Position

Answer 29

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

Question 30

Adjusting Asset Allocation with Futures (Example)

Answer 30

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

Question 31

Types of Foreign Exchange Risk

Answer 31

1. **Transaction** - cash flow at later date, can be hedged 1. most common 2. **Economic** - ▲ in currency affect competitiveness 1. Example; dollar ^, less competitive in foreign markets 2. Can be difficult to hedge 3. **Translation** - risk of reporting in another currency

Question 32

How to Hedge a Currency Position Receiving Foreign Paying Foreign

Answer 32

Position Action Receiving Foreign Long Sell forward Paying Foreign Short Buy forward

Question 33

Call Options

Answer 33

Right to buy underlying Make money when goes up

Question 34

Put Options

Answer 34

Right to sell Make money when goes down

Question 35

Relation to Calls/Puts R_f, Volatility, and Asset Price

Answer 35

**_Input Calls Puts_** Asset Price ↑ Positive Negative R_f ↑ Positive Negative Volatility ↑ Positive Positive

Question 36

Covered Call

Answer 36

Buy underlying and selling call option * **profit** = (S_T - X) + C₀ + S_T - S₀ * **max profit** = X + C₀ - S₀ * **max loss** = S₀ - C₀ * **Breakeven** = S₀ - C₀ Example: (35 - 45) + 35 - 43 + 2.10 = 5.90

Question 37

Protective Put

Answer 37

AKA portfolio insurance/hedged portfolio **Definition:** Hold underlying buy a put option * **profit** = (X - S_T) - P₀ + S_T - S₀ * **max profit** = S_T - P₀ - S₀ * **max loss** = S₀ + P₀ - X * **breakeven** = S₀ + P₀ Example: (35 - 30) + 30 - 37.5 - 1.4 = -3.90

Question 38

Bull Spread

Answer 38

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**

Question 39

Bear Spread

Answer 39

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**

Question 40

Box Spread

Answer 40

Bull and Bear Spread combined * Creates an aribtrage relationship * Known initial and ending cash flows

Question 41

Straddle and Reverse Straddle

Answer 41

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

Question 42

Straddle Profit Max Profit Max Loss Breakeven

Answer 42

Profit = (S_T - X) + (X - S_T) - C₀ - P₀ Max Profit = Infinite as stock increases Max loss = C₀ + P₀ Breakeven = X - C₀ - P₀ AND X + C₀ + P₀

Question 43

Butterfly Spread

Answer 43

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

Question 44

Collar

Answer 44

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.

Question 45

Hedging with Interest Rate Options Borrowing/Lending

Answer 45

**_Borrowing_** Risk: increasing rates Hedge: buy an interest rate call **_Lending_** Risk: decreasing rates Hedge: buy a interest rate put

Question 46

Caps and Floors

Answer 46

**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

Question 47

Delta

Answer 47

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

Question 48

Swap Risks Floating and Fixed Rate

Answer 48

* 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**

Question 49

Interest Rate Swap Duration

Answer 49

**_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: 5-year pay fixed swap with quarterly settlement. Comparable bond 4.1 duration -4.1 + 0.25/2 = -3.975

Question 50

Swap Cash Flow and Market Risk Strategies

Answer 50

**_Rates will increase_** ## Footnote **_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

Question 51

Modified Duration Swap

Answer 51

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 [(4-5.2) / -3.1)] = 23,225,806

Question 52

Swaption

Answer 52

Definition: option to enter a pre-negotiated swap _Two Types_ 1. Payer: allows swaption buyer to enter a pay fixed 1. gains value if rates rise 2. Receiver: allows swaption buyer to enter a received fixed 1. gains value if rates fall

Question 53

EAR

Answer 53

**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}