Risk Management

understand risks; determine when it’s appropriate to take risk

- identify risks
- set risk tolerances
- report risk to stakeholders
- monitor

Risk Governance

policies and procedures establishing risk management

- structure - centralized (best), decentralized
- reporting
- methologies
- infrastructure needs

Enterprise Risk Management

centralized risk management

- identify risk factors
- quantify risk
- aggregate to measure firm wide risk
- report/ allocate risk
- monitor

Financial Risk

due to events external to firm, in financial markets

- market risk
- credit risk
- liquidity risk

mitigate with derivatives (options, swaps)

Market Risk

- interest rate risk
- exchange rate risk
- equity price risk
- commodity price risk

Nonfinancial Risk

- operational risk
- settlement/ perf netting risk - one party pays, other defaults
- model risk - GIGO
- sovereign risk
- regulatory risk
- tax, accounting, political risk

mitigate with insurance

VaR

probability of expected loss over a specified time; comparable across asset classes, not managers

- analytical: VaR = [R - z * σ] V
- historical
- monte carlo simulation

One Tail SD

- 5% = 1.65 SD
- 1% = 2.33 SD

VaR Complements

- incremental VaR: risk from additional factor
- cash flow/ earnings at risk: min CF loss for given prob over time
- tail value at risk (TVaR): avg outcomes in tail

Credit Risk

possibility counterparty defaults; current and potential credit risk

prob default * PV losses

PV rec - PV paid

Credit Risk of Currency Forwards

long base currency

S_{0 }/ ( 1 + b )^{t} - F_{t }/ ( 1 + p )^{t}

highest credit risk in middle of forward’s life

Credit Risk of Currency Swaps

highest credit risk btwn middle/end of swap’s life

PV rec - PV pay

Credit Risk of Options

long position = credit risk

current credit risk when option is exercised

Managing Credit Risk

- limiting exposure
- marking to market
- collateral
- netting payments
- closeout netting
- credit derivatives

Risk Budgeting Factors

must consider correlation of risk in diff units

- VaR limits
- position limits
- liquidity limits
- performance stopout
- risk factor limits

Sortino Ratio

ratio of excess return to risk; doesn’t penalize manager for good performance

( R_{p} - MAR ) / downside deviation

Forward vs Future

Forward: custom, high default risk, less liquidity; currency, int payments

Futures: standardized, trade on exchange, low default risk; bond, equity

Modifying Equity Beta

contracts = ( Δβ / β_{f} ) * ( V_{p }/ Vf )

β

cov_{i,m} / σ_{m}^{2}

Effective Beta

%Δ value of portfolio / %Δ value of index

Why Effective Beta Deviates

basis risk = imperfect hedge

- num/ demon based on diff items
- evaluating before expiration
- # contracts rounded
- F and S not priced correctly

Modifying Bond Duration

contracts = β_{yield} * ( ΔMD / MD_{f} ) * ( V_{p }/ Vf )

Synthetic Stock

beta = 1

repliate buying contracts; buy futures contract, long T-bills

- N
_{f }= FV_{Vp}/ V_{f} - terminal shares replicated = N
_{f}* mult (beg = discount by div yield) - initial eqty eq = PV( N
_{f}* mult * price ) [terminal = FV]

Synthetic Cash

beta = -1

replicate selling contracts; long equity, short futures contract

- N
_{f}= - FV_{Vp}/ V_{f} - initial cash eq = PV( N
_{f}* mult * price ) [terminal = FV] - terminal shares = N
_{f}* mult

Exchange Rate Risks

- transaction risk
- economic risk
- translation risk (translating financial statements)

Hedging Currency Positions

- receiving foreign curr = sell forward
- paying foreign curr = buy forward

Covered Call

long stock, short call; exp lower volatility

- payoff: S
_{t}- max( 0, S_{t}- X ) - S_{0}+ C - max gain: (ex opt) X - S
_{0}+ C - max loss: (don’t ex opt, S
_{t}= 0) S_{0}- C - breakeven: (initial cost) S
_{0 }- C

Protective Put

long stock, long put; exp higher volatility

- payoff: S
_{t}+ max( 0, X - S_{t}) - S_{0 }- P - max gain: unlimited
- max loss: (ex opt) S
_{0 }+ P - X - breakeven: (initial cost) S
_{0 }+ P

Bull Call Spread

long CL, short CH; long P_{L}; short P_{H}; _{ }exp inc S

- profit: C
_{H}- C_{L }+ max( 0, S - X_{L}) - max( 0, S - X_{H}) - max profit: (both ex) X
_{H}- X_{L }+ C_{H}- C_{L } - max loss: (neither ex) C
_{L }- C_{H} - breakeven: (ex C
_{L}) X_{L }+ C_{L }- C_{H}

Bear Put Spread

long P_{H}, short P_{L}; long C_{H}, short C_{L}; exp dec S

- profit: max( 0, X
_{H}- S ) - max ( 0, X_{L}- S ) + P_{L }- P_{H} - max profit: (both ex) X
_{H}- X_{L }+ P_{L }- P_{H} - max loss: (no ex) P
_{H }- P_{L} - breakeven: (ex P
_{H}) X_{H}+ P_{L }- P_{H}

Butterfly Spread with Calls

long C_{L} and C_{H}; short 2 C_{M}; exp dec volatility

- profit: max( 0, S - X
_{L }) + max( 0, S - X_{H }) - 2max( 0, S - X_{M }) + 2C_{M}- C_{L}- C_{H} - max profit: (ex C
_{L}, C_{M}) X_{M}- X_{L}+ 2C_{M}- C_{L}- C_{H} - max loss: (no ex) C
_{L}+ C_{H }- 2C_{M} - breakeven: 2C
_{M}- C_{L}- C_{H }+ 2X_{M}- X- C
_{L}+ C_{H }- 2C_{M }+ X_{L}

- C

Butterfly Spread with Puts

long P_{L} and P_{H}; short 2 P_{M}; exp dec volatility

- profit: max( 0, X
_{H}- S ) + max( 0, X_{L }- S ) - 2max( 0, X_{M}- S ) + 2P_{M}- P_{L}- P_{H} - max profit: (ex P
_{H}, P_{M}) X_{H}- X_{M}+ 2P_{M}- P_{L}- P_{H} - max loss: (no ex) P
_{L}+ P_{H }- 2P_{M} - breakeven: P
_{L}+ P_{H }- 2P_{M}+- 2P
_{M}- P_{L}- P_{H}+

- 2P

Straddle

long call and put OR short call and put; exp inc price volatility

- profit: max ( 0, S - X ) + max ( 0, X - S ) - C - P
- max gain: unlimited
- max loss: (inv) C + P
- breakeven: X - C - P
*or*X + C + P

Collar

stock, long P_{L}, short C_{H}; exp low price volatility

- profit: max ( 0, X
_{L}- S ) + max ( 0, S - X_{H}) + S - S_{0} - max profit: X
_{H}- S_{0} - max loss: S
_{0}- X_{L} - breakeven: S
_{0}

Box Spread

combo of bull and bear spreads; arbitrage opp

- profit: X
_{H}- X_{L}+ P_{L}- P_{H}+ C_{H}- C_{L} - compare annualized HPR to r
_{f}

Interest Rate Call Payoffs

- net loan: loan - FV(premium)
- call payoff: NP [max[ 0, LIBOR - X] * D/ 360 ]
- eff $ int cost = loan int - call payoff
- EAR: [(loan + eff $ int cost) / net loan]
^{365/D}- 1

Interest Rate Put Payoffs

- net loan: loan + FV(premium)
- put payoff: NP [max[ 0, X - LIBOR] * D/ 360 ]
- eff $ int cost = loan int + put payoff
- EAR: [(loan + eff $ int cost) / net loan]
^{365/D}- 1

Interest Rate Cap

payment when rate > X

series of int rate calls; caplets

good for floating rate payer

Interest Rate Put

payment when X < rate

series of int rate puts; floorlets

good for floating rate receiver

Delta

change in option price/ change in underlying

Delta Hedging

hedge downside risk of short options (dealers); earn r_{f}

stock required = - delta * # options

Gamma

change in delta/ change in underlying

Vega

change in underlying/ change in volatility

Swap Duration

- fixed duration = 0.75 maturity
- floating duration = 0.5 reset period
- D
_{pay floating}= D_{fixed}- D_{floating}= +D - D
_{pay fixed}= D_{floating}- D_{fixed}= -D

Modifying Swap Duration

NP = [( MD - MD_{0} )/ MD_{swap}] V_{p}

rec floating = neg swap duration

Payer Swaption

buyer = right to be fixed-rate payer, receive floating

exp rates to go up

Receiver Swaption

buyer = right to be fixed-rate receiver, pay floating

exp rates to go down