Financial Risk Measurement/Management

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1 Financial Risk Measurement/Management Week of September 16, 2013 Introduction: Instruments and Risk on the Trading Desk 2.1 Assignment For September 16 th (This Week) Read: Hull Chapters 5 & 7 (Trading, the Markets & Managing Trading Risk) Problems (Due September 23 rd ) Chapter 5: 3, 8, 10, 13, 21 Chapter 7: 1, 3, 6, 14; 16 Problems (Due Today) Chapter 1: 1, 2, 11, 12; 18 Chapter 2: 3, 8; 16, 18 Chapter 3: 5, 9, 15 Chapter 4: 7, 14, Assignment Schedule For September 23 rd (Next Week) Read: Hull Chapters 8-9 (Interest Rate Risk & VAR) Problems (Due September 30 th ) Chapter 8: 1, 5, 9, 12, 13, 16, 17; 19 Chapter 9: 1, 3, 4, 5; 14, 16 Problems (Due September 23 rd Chapter 5: 3, 8, 10, 13, 21 Chapter 7: 1, 3, 6, 14; 16 Lecture Encounters Monday & Wednesday, Noon -1:15pm, Shaffer 202 Section Section: Friday 12:00-12:50pm, Shaffer 202 Midterm: October 30, 2013 Final Exam Wednesday, December 18 th ; 9am - Noon Shaffer

2 Where we are Last week: Introduction to Risk and Capital, looked at the business model of Financial Institutions and the risks they take on Looked closely at banks This week: Introduction continues Business model and risks taken on by Insurance and Investment Companies Financial Instruments and Risk on the Trading Desk Next week: Interest Rate Risk and an Introduction to Value at Risk (VaR) 2.5 Banks Commercial Banking Taking Deposits & Lending Retail: Individuals & Small Businesses Wholesale: Larger Corporations & Funds Money Center Banks: Fund Operations in the Mkts. Investment Banking Corporate Capital Debt & Equity M&A; Corporate Finance 2.6 Insurance Companies Insurance Companies a Mortality Table Life vs. Property & Casualty vs. Health Life Insurance Whole Life vs. Term An investment vehicle for premiums Annuity Contracts from Life Insurers Fixed Annuity lump sum into payments Mortality Tables Longevity Risk & Mortality Risk

3 Insurance Companies a Mortality Table Insurance Companies Insurance Premiums Some entries can be calculated from others - consistency Probability of death = Conditional on reaching 90 Death in next year is / = which is consistent with the entry in column 2 Death in 2 nd year (91-92) ( ) x = Like for a Credit Default Swap Need to find breakeven value of premium so the PV of expected pay-outs equals PV of expected premiums Suppose a male 90-year old wants to buy a 2-year term life policy; term structure is 4% and flat; pay-outs midway in any year and premiums are paid at the beginning of each year PV of expected payoff: PV[.174x100K] + PV[.158x100K] = 17, ,894 = 31,954 PV of Premiums: 1 x X + PV[(1-.174) x X] = X X/(1.02) 2 = 1.79 x X Breakeven: 1.79X = 31,954 => X = 17, Insurance Companies Insurance Companies Property & Casualty Loss to property (fire, theft, home & auto, etc.) Legal Liability singular events and legacy risk Ratios (of payouts to premiums) 2.11 Health Insurance Attributes of Life and P&C Premiums can go up, but only with prevailing costs Don t increase as a function of the health of individual Moral Hazard & Adverse Selection Moral Hazard P&C Insurance can promote imprudent behavior Adverse Selection Insurance attracts bad risks Reinsurance against large losses

4 Insurance Companies The Balance Sheet (Life vs. P&C) Risks? Inadequate Reserves Liquidity Duration Hedges: Longevity Derivative & CAT bonds Insurance Companies Regulation of Insurance Companies States and the NAIC Pension Plans Defined Contribution Defined Benefit Are defined benefit plans viable? Investment Companies Mutual Funds & Hedge Funds Mutual Funds Small Investor Diversification Big Business - $10 trillion in US Open End vs. Closed End Index Funds Investment Companies Mutual Funds Cost Structure Load: Front End vs. Back End Annual Expense Fee Once a day NAV Purchase and Redemption

Investment Companies ETFs (Example: Spider & S&P500) Created by Institutions (Creation Units exchanged for Underlying Shares) Traded on Exchanges (continuous trading) Can be borrowed and sold short

17 Investment Companies Hedge Funds Unregulated Can use short positions and leverage Limited Disclosure (including the NAV) Restrictions on Deposits and Redemptions Fees Management Fee plus

5 Investment Companies ETFs (Example: Spider & S&P500) Created by Institutions (Creation Units exchanged for Underlying Shares) Traded on Exchanges (continuous trading) Can be borrowed and sold short Exchangeable: ETF and Underlying Assets Insures no arbitrage in pricing/valuation Mutual Funds and ETFs are regulated by the SEC 2.17 Investment Companies Hedge Funds Unregulated Can use short positions and leverage Limited Disclosure (including the NAV) Restrictions on Deposits and Redemptions Fees Management Fee plus Performance Fee Hurdle Rate, High Water mark, & Clawback (% of fees go to recovery account vs. future losses) Prime Brokers (de facto regulator) 2.18 Investment Companies Investment Companies Hedge Fund Strategies Long/Short Equity Dedicated Short Distressed Situations Merger Arbitrage / Event Driven Convertible Arbitrage Fixed Income Arbitrage Emerging Markets Global Macro Managed Futures

Introductory Ideas about Risk Gains or Losses mainly losses Consider the notation P : Profit or Loss over a specifed horizon in the risk currency $ Also, that this is the product P R P where P is the

Consider the historic characterization ( ) : the mean R P P 2 ( ) : the standard deviation (variance) VAR : Value at Risk; cutoff point there is a low probability of greater loss 2.22 VAR = 14.

6 Introductory Ideas about Risk Gains or Losses mainly losses Consider the notation P : Profit or Loss over a specifed horizon in the risk currency $ Also, that this is the product P R P where P is the initial investment and R P is the horizon (period) return Then R P is a random variable and should be described by its probability density (or distribution) function Introductory Ideas about Risk Consider the historic characterization ( ) : the mean R P P 2 ( ) : the standard deviation (variance) VAR : Value at Risk; cutoff point there is a low probability of greater loss 2.22 VAR = 14.4% : Monthly loss level that is exceeded only one time in 100 (99% confidence) 2.23 Introductory Ideas about Risk Absolute Risk Measured in terms of shortfall relative to the initial investment, P, and its change, P The risk measure is standard deviation ( P) ( P ) P ( RP ) P P Relative Risk Measured relative to a benchmark, B, and its return, R B The deviation e RP RB is known as the tracking error In dollar terms this is e P And the risk is () ep ( RP RB) P P where ω is called the tracking error volatility (TEV) 2.24 Introductory Ideas about Risk Risk Management Process To estimate future profit or LOSS Straightforward if we have distributions of future P/L That distribution is the crux of the problem Could extrapolate from history but There are difficulties with historical extrapolation Known knowns factor models Known Unknowns liquidity, model risk Unknown unknowns new laws, counterparties

7 Introductory Ideas about Risk Risk Management starts with the price of assets Then considers how those prices can change Generate P/L for the holder of those assets In considering how prices can change Use asset pricing models and factor relationships CAPM, APT, Efficient Frontier, Optimal (PF) Combinations Systematic Risk vs. Idiosyncratic Risk Known knowns vs. Known Unknows (model risk) 2.26 Introductory Ideas about Risk Aside from survival, another Key Objective is to guide the choices in return management Measurements for comparison ( RP ) RF Sharpe Ratio (SR): SR ( RP ) Could use VAR vs. SD ( RP ) ( RB ) Information Ratio (IR): IR ( RP RB) CAPM Treynor Ratio (TR): ( RP ) RF TR & if CAPM holds, same for all as M P CAPM Jensen s alpha: ( R ) R ( R ) R P P F P M F 2.27 Introductory Ideas about Risk The Risk Manager is Concerned with Total Risk We only Manage vs. SD of Returns (last slide) We need to Protect Against Conditional VAR VAR shows where the crossover occurs Financial Markets One Source of Hedging/Risk Management Exchange traded Traditionally exchanges have used the open-outcry system, but increasingly they are incorporating more electronic trading Contracts are standard; there is virtually no credit risk Over-the-counter (OTC) A computer- and telephone-linked network of dealers, financial institutions, corporations, and fund managers Contracts can be non-standard; there is some amount of credit risk

8 Financial Products - Hedges Long/short positions Forwards Futures Swaps Options Exotics Short Selling Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way Short Selling (continued) At some stage you must buy the securities back so they can be replaced in the account of the client You must pay dividends and any other benefits the owner of the securities would receive What are the shortcomings of Short Selling 2.32 Forward Contracts A forward contract is an agreement to buy or sell an asset at a certain price at a certain future time Forward contracts trade in the over-thecounter market They are particularly popular on currencies and interest rates (also mortgages, MBS)

9 Profit from a Long Forward Position Profit from a Short Forward Position Profit Profit K Price of Underlying at Maturity, S T K Price of Underlying at Maturity, S T Futures Contracts Agreement to buy or sell an asset for a certain price at a certain time Similar to forward contract Whereas a forward contract is traded OTC, a futures contract is traded on an exchange How are they unlike a forward contract? Futures Contract continued Futures Contracts are settled daily (e.g., if a contract is on 200 ounces of December gold 2 contracts and the December futures moves $2 in my favor, I receive $400; if it moves $2 against me I pay $400) Both sides to a futures contract are required to post margin (cash or marketable securities) with the exchange clearinghouse. This ensures that commitments under the contract will be honored

10 Comparison of Forward & Future Contracts Swaps A swap is an agreement to exchange cash flows at specified future times according to certain specified rules An Example of a Plain Vanilla Interest Rate Swap An agreement to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million Next slide illustrates cash flows Cash Flows for one set of LIBOR rates Millions of Dollars LIBOR FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar.5, % Sept. 5, % Mar.5, % Sept. 5, % Mar.5, % Sept. 5, % Mar.5, %

11 Typical Uses of an Interest Rate Swap Converting a liability from fixed rate to floating rate floating rate to fixed rate Converting an investment from fixed rate to floating rate floating rate to fixed rate Quotes By a Swap Market Maker Maturity Bid (%) Offer (%) Swap Rate (%) 2 years years years years years years Other Types of Swaps Floating-for-floating interest rate swaps, amortizing swaps, step up swaps, forward swaps, constant maturity swaps, compounding swaps, LIBOR-in-arrears swaps, accrual swaps, diff swaps, cross currency interest rate swaps, equity swaps, extendable swaps, puttable swaps, swaptions, commodity swaps, volatility swaps, credit default swaps.. & total return swaps 2.44 Options A call option is an option to buy a certain asset by a certain date for a certain price (the strike price) A put option is an option to sell a certain asset by a certain date for a certain price (the strike price) Options trade on both exchanges and in the OTC market

12 American vs European Options An American option can be exercised at any time during its life A European option can be exercised only at maturity Up to this point, the Hedges were FREE! Options are like insurance, you pay a premium for a benefit/protection against an uncertain event 2.46 Hedging Examples A US company will pay 10 million for imports from Britain in 3 months and Decides to hedge currency exposure using a long position in a forward contract An investor owns 1,000 Microsoft shares currently worth $28 per share A 2-month put with a $27.50 strike costs $1 The investor decides to hedge against a price decline by buying 10 put option contracts on the exchange (100 shares each) 2.47 Optimal Hedge Ratio Hedging using Futures Proportion of the exposure in a cash position that should optimally be hedged is S F where S is the standard deviation of S, the change in the spot price (for the cash instrument) during the hedging period, F is the standard deviation of F, the change in the futures price during the hedging period is the coefficient of correlation between S and F. Optimal Futures Hedge Ratio Regressing change in asset price to change in futures Optimal Hedge Ratio minimum variance hedge

13 Optimal Hedge Ratio Minimum Variance Result The number of futures, NF, in ratio h (the hedge ratio, or NF = h x NA), required to hedge NA units of an asset between times t 1 and t 2 follows from P/L of the hedged position over that time: P/L = (NAS2 NFF2) (NAS1 NFF1) = NA ΔS NF ΔF = NA x (ΔS h ΔF) ( NF = h x NA ) where S 1, F 1 and S 2, F 2 ( ΔS & ΔF ) are the spot price of the asset, S, and the futures, F, at t 1 and t 2, respectively (or their respective changes, Δ) The minimum variance hedge ratio can be found by minimizing the variance of the P/L (w/r to the hedge ratio) 2.50 Optimal Hedge Ratio Minimum Variance Result Minimizing the variance of the P/L P/L = NA x (ΔS h ΔF) Is equivalent to min [Var(ΔS h ΔF)] where the variance of this linear combination of F & S, ν = Var [ΔS h ΔF] = S2 + h 2 F2 2 h S F We minimize ν when h is such that dν/dh = 0 if d 2 ν/dh 2 > 0 or 2 h F2 2 S F = 0 which implies S h F 2.51 What Hedging Achieves Hedging reduces risk. It does not increase expected profit. Hedging can result in an increase or a decrease in a company s profits relative to the situation it would be in with no hedging 2.52 Options vs Forwards Forward contracts lock in a price for a future transaction Options provide insurance. They limit the downside risk while not giving up the upside potential For this reason options are more attractive to many corporate treasurers than forward contracts

14 A Trader s Gold Portfolio How Should Trading Risks Be Hedged? Position Value ($) Spot Gold 180,000 Forward Contracts 60,000 Futures Contracts 2,000 Swaps 80,000 Options 110,000 Exotics 25,000 Total 117, Hedging on the Trading Desk using Delta Delta of a portfolio is the partial derivative of a portfolio with respect to the price of the underlying asset (gold in this case) Suppose that a $0.1 increase in the price of gold leads to the gold portfolio increasing in value by $100 The delta of the portfolio is 1000 The portfolio could be hedged against short-term changes in the price of gold by selling 1000 ounces of gold 10 futures. This is known as making the portfolio delta neutral 2.56 Linear vs Nonlinear Products Delta of the Option When the price of a product is linearly dependent on the price of an underlying asset a hedge and forget strategy can be used Non-linear products require the hedge to be rebalanced to preserve delta neutrality Or you use a non-linear hedge Options are an example of a non-linear hedge vehicle non-constant delta B Option price A Slope = Stock price

Example Delta Hedging A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S 0 =

to lock in a $60,000 profit? Initially the delta of the option is 0.

458, 6,400 shares must be sold to maintain delta neutrality The option is non-linear so the delta changes with price Tables 7.

15 Example Delta Hedging A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S 0 = 49, K = 50, r = 5%, = 20%, T = 20 weeks, = 13% The Black-Scholes value of the option is $240,000 How does the bank hedge its risk to lock in a $60,000 profit? Initially the delta of the option is This means that 52,200 shares are purchased to create a delta neutral position But, if a week later delta falls to 0.458, 6,400 shares must be sold to maintain delta neutrality The option is non-linear so the delta changes with price Tables 7.2 and 7.3 (pages 142 and 143) provide examples of how delta hedging might work for the option Example Example Offset by the $300K option premium 2.61 Offset by the $300K option premium

16 Gamma Gamma ( ) is the rate of change of delta ( ) with respect to the price of the underlying asset Gamma is greatest for options that are close to the money Gamma Measures the Delta Hedging Errors Caused By Curvature (Figure 7.4, page 145) C'' C' Call price C S S' Stock price Vega Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility Vega tends to be greatest for options that are close to the money Gamma and Vega Limits In practice a trader responsible for all trading involving a particular asset must keep gamma and vega within limits set by risk management

17 Theta Hedging in Practice Theta ( ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines Traders usually ensure that their portfolios are delta-neutral at least once a day Whenever the opportunity arises, they improve gamma and vega As portfolio becomes larger hedging becomes less expensive

Financial Risk Measurement/Management

550.446 Financial Risk Measurement/Management Week of September 23, 2013 Interest Rate Risk & Value at Risk (VaR) 3.1 Where we are Last week: Introduction continued; Insurance company and Investment company