Fixed Income Zvi Wiener Fixed Income 6

FI - 6 slide 2 IR derivatives Forward and Futures Options Caps, Floors Swaps Structured notes Hedging Mathematical models

FI - 6 slide 3 Interest rate futures Legal agreement settlement, delivery date quantity and quality of deliverable asset futures price (it is NOT a price!) long and short positions margin requirements

FI - 6 slide 4 Margin requirements Initial margin Maintenance margin - margin call trigger Variation margin - after a margin call Mark to market procedure reduces counterparty risk!

FI - 6 slide 5 Marking to Market Your balance time Initial margin Maint. margin margin call

FI - 6 slide 6 Futures Contract T-Bills T-Notes T-Bonds notional is typically $100,000 deliverable bond is unknown, CTD option timing option (during the delivery month) wild card option

FI - 6 slide 7 Options This type of contract is an obligation of one side only, but it requires a payment to purchase the right to choose.

FI - 6 slide 8 Call Value before Expiration E. Call XUnderlying

FI - 6 slide 9 Put Value before Expiration E. Put XUnderlying premium X

FI - 6 slide 10 IR Options Call Put European American Bermudian Exotic: Asian, Digital, Knock-In, Knock-Out, path dependent and multiple asset options.

FI - 6 slide 11 Various IR Options Futures options Caps Floors Exchange options Swaptions

FI - 6 slide 12 Cap Is priced as a sequence of caplets time cap

FI - 6 slide 13 Floor time floor

FI - 6 slide 14 Collar time floor cap

FI - 6 slide 15 Option pricing Time value, intrinsic value underlying time to maturity interest rates strike coupons volatility

FI - 6 slide 16 Swaps Currency swap Interest rate swap Amortizing swap Swaption

FI - 6 slide 17 Currency swap

FI - 6 slide 18 Currency Swap 3Y3Y3Y3Y3Y3Y 100Y $5$5$5$5$5$5 $130

FI - 6 slide 19 IR swap

FI - 6 slide 20 IR Swap L+1L+1L+1L+1L+1L+1 100

FI - 6 slide 21 IR Swap L+1L+1L+1L+1L+1L a regular LIBOR loan for one year!

FI - 6 slide 22 Term Structure Models Binomial trees Short-term based analytical models LIBOR based analytical models Multi-factor models Simulations

FI - 6 slide 23 Binomial Trees 6% 6.5% 5.5% 7% 6% 5% 7.5% 6.5% 5.5% 4.5%

FI - 6 slide 24 6% 6.5% 5.5% 7% 6% 5% 7.5% 6.5% 5.5% 4.5% Interest rates Bond prices

FI - 6 slide 25 Typical yield curves time to maturity yield increasing decreasing humped

FI - 6 slide 26 Analytic Term Structure Models

FI - 6 slide 27 Analytic Term Structure Models Hull, White Black-Karasinsky Black-Derman-Toy Heath-Jarrow-Morton Affine TS modles Gaussian models

FI - 6 slide 28 Arithmetic BM dX =  dt +  dW   time X

FI - 6 slide 29 Geometric BM dX =  Xdt +  XdW time X

FI - 6 slide 30 Mean Reverting Process dX =  (  -X)dt +  X  dW time X 

FI - 6 slide 31 Ho and Lee Model Rates are normally distributed. All rates have the same variability. The model has an analytic solution.

FI - 6 slide 32 Bond Prices under Ho and Lee Where

FI - 6 slide 33 Option Prices under Ho and Lee A discount bond matures at s, a call option matures at T

FI - 6 slide 34 Monte Carlo

FI - 6 slide 35 Monte Carlo Simulation

FI - 6 slide 36 Callable Bond Payoff Straight Debt Debt Callable Bond Value of the firm’s call option

FI - 6 slide 37 Convertible Bond Payoff Stock Straight Bond Convertible Bond

FI - 6 slide 38 Protective Put Payoff XUnderlying X Put Protective Put Stock

FI - 6 slide 39 Covered Call Payoff X X Written Call Covered Call Stock

FI - 6 slide 40 Straddle Payoff X X Straddle Call Put

Fixed Income Zvi Wiener DAC

FI - 6 slide 42 Life Insurance yearly contribution 10,000 NIS yearly risk premium 2,000 NIS first year agent’s commission 3,000 NIS promised accumulation rate 8,000 NIS/yr After the first payment there is a problem of insufficient funds. 8,000 NIS are promised (with all profits) and only 5,000 NIS arrived.

FI - 6 slide 43 10,000 NIS Risk 2,000 NIS Client’s 8,000 NIS Agent 3,000 NIS insufficient funds if the client leaves insufficient profits

FI - 6 slide 44 Risk measurement The reason to enter this transaction is because of the expected future profits. Assume that the program is for 15 years and the probability of leaving such a program is . Fees are – 0.6% of the portfolio value each year – 15% real profit participation

FI - 6 slide 45 Obligations The most important question is what are the obligations? The Ministry of Finance should decide Transparent to a client Accounted as a loan

FI - 6 slide 46 One year example Assume that the program is for one year only and there is no possibility to stop payments before the end. Initial payment P 0, fees lost L 0, fixed fee a% of the final value P 1, participation fee b% of real profits (we ignore real). Investment policy TA-25 (MAOF).

FI - 6 slide 47 Liabilities (no actual loan) Assets (no actual loan)

FI - 6 slide 48 Total=Assets-Liabilities Fair value

FI - 6 slide 49 Liabilities (actual loan) Assets (actual loan)

FI - 6 slide 50 Total=Assets-Liabilities (loan)

FI - 6 slide 51 2 years liabilities (no actual loan) 2 years assets (no actual loan) In reality the situation is even better for the insurer, since profit participation fees once taken are never returned (path dependence).

FI - 6 slide 52 2 years fair value, no loan

FI - 6 slide 53 2 years liabilities (with a loan) 2 years assets (with a loan)

FI - 6 slide 54 Stock index Profit No loan With a loan 10 years, L 0 =7%

FI - 6 slide 55 Partial loan - portion q Theoretically q can be negative.

FI - 6 slide 56 Mixed portfolio When the investment portfolio is a mix one should analyze it in a similar manner. Important: an option on a portfolio is less valuable than a portfolio of options. Another risk factor - leaving rate should be accounted for by taking actuarial tables as leaving rate.

FI - 6 slide 57 Conclusions It is a reasonable risk management policy not to take a loan against DAC. Up to some optimal point it creates a useful hedge to other assets (call options and shares) of the firm. Intuitively DAC is good when the stock market performs badly and profit participation is valueless. DAC performs bad when the market performs well.

Fixed Income Zvi Wiener Risk Management

FI - 6 slide 59 Qualitative Requirements An independent risk management unit Board of directors involvement Internal model as an integral part Internal controller and risk model Backtesting Stress test

FI - 6 slide 60 Quantitative Requirements 99% confidence interval 10 business days horizon At least one year of historic data Data base revised at least every quarter All types of risk exposure Derivatives

FI - 6 slide 61 Types of Assets and Risks Real projects - cashflow versus financing Fixed Income Optionality Credit exposure Legal, operational, authorities

FI - 6 slide 62 Risk Factors There are many bonds, stocks and currencies. The idea is to choose a small set of relevant economic factors and to map everything on these factors. Exchange rates Interest rates (for each maturity and indexation) Spreads Stock indices

FI - 6 slide 63 How to measure VaR Historical Simulations Variance-Covariance Monte Carlo Analytical Methods

FI - 6 slide 64 Historical Simulations Fix current portfolio. Pretend that market changes are similar to those observed in the past. Calculate P&L (profit-loss). Find the lowest quantile.

FI - 6 slide 65 Returns year 1% of worst cases

FI - 6 slide 66 Profit/Loss VaR 1% VaR 1%

FI - 6 slide 67

FI - 6 slide 68

FI - 6 slide 69 Variance Covariance Means and covariances of market factors Mean and standard deviation of the portfolio Delta or Delta-Gamma approximation VaR 1% =  P – 2.33  P Based on the normality assumption!

FI - 6 slide 70  Variance-Covariance 2.33    1%

FI - 6 slide 71 Weights Since old observations can be less relevant, there is a technique that assigns decreasing weights to older observations. Typically the decrease is exponential. See RiskMetrics Technical Document for details.

FI - 6 slide 72 Monte Carlo Distribution of market factors Simulation of a large number of events P&L for each scenario Order the results VaR = lowest quantile

FI - 6 slide 73 Example Your portfolio consists of two positions. The first one is a zero coupon bond maturing in 1 year with current market value of $10M. The second one is a zero coupon bond maturing in 10 years with market value of $1M. Which position contributes more to the risk of the portfolio?

FI - 6 slide 74 Real Projects Most daily returns are invisible. Proper financing should be based on risk exposure of each specific project. Note that accounting standards not always reflect financial risk properly.

FI - 6 slide 75 Example You are going to invest in Japan. Take a loan in Yen. Financial statements will reflect your investment according to the exchange rate at the day of investment and your liability will be linked to yen. Actually there is no currency risk.

FI - 6 slide 76 Airline company fuel - oil prices and $ purchasing airplanes - $ and Euro salaries - NIS, some $ tickets $ marketing - different currencies payments to airports for services

FI - 6 slide 77 Airline company loans equity callable bonds

FI - 6 slide 78 Airline company Base currency - by major stockholder. Time horizon - by time of possible price change. Earnings at risk, not value at risk, since there is too much optionality in setting prices. One can create a one year cashflow forecast and measure its sensitivity to different market events.

FI - 6 slide 79 Reporting Division of VaR by business units, areas of activity, counterparty, currency. Performance measurement - RAROC (Risk Adjusted Return On Capital).

FI - 6 slide 80 How VaR is used Internal Risk Management Reporting Regulators

FI - 6 slide 81 Backtesting Verification of Risk Management models. Comparison if the model’s forecast VaR with the actual outcome - P&L. Exception occurs when actual loss exceeds VaR. After exception - explanation and action.

FI - 6 slide 82 Backtesting Green zone - up to 4 exceptions Yellow zone exceptions Red zone - 10 exceptions or more OK increasing k intervention

FI - 6 slide 83 Stress Designed to estimate potential losses in abnormal markets. Extreme events Fat tails Central questions: How much we can lose in a certain scenario? What event could cause a big loss?

FI - 6 slide 84 Unifying Approach One number Based on Statistics Portfolio Theory Verification Widely Accepted Easy Comparison

FI - 6 slide 85 Board of Directors (Basle, September 1998) periodic discussions with management concerning the effectiveness of the internal control system a timely review of evaluations of internal controls made by management, internal and external auditors periodic efforts to ensure that management has promptly followed up on recommendations and concerns expressed by auditors and supervisory authorities on internal control weaknesses a periodic review of the appropriateness of the bank’s strategy and risk limits.

FI - 6 slide 86 pluto.mscc.huji.ac.il/~mswiener/ Useful Internet sites Regulators Insurance Companies Risk Management in SEC reports Risk Management resources

FI - 6 slide 87 Risk Measuring Software CATS, CARMA Algorithmics, Risk Watch Infinity J.P. Morgan, FourFifteen FEA, Outlook Reuters, Sailfish Kamacura Bankers Trust, RAROC INSSINC, Orchestra