Zvi Wiener

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Fixed Income 6

• Zvi Wiener
• 02-588-3049
• http//www.tfii.org

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IR derivatives

• Forward and Futures
• Options
• Caps, Floors
• Swaps
• Structured notes
• Hedging
• Mathematical models

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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

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Margin requirements

• Initial margin
• Maintenance margin - margin call trigger
• Variation margin - after a margin call
• Mark to market procedure reduces counterparty
  risk!

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Marking to Market
Your balance
Initial margin
Maint. margin
margin call
time
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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

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Options

• This type of contract is an obligation of one
  side only, but it requires a payment to purchase
  the right to choose.

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Call Value before Expiration
E. Call
X Underlying
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Put Value before Expiration
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IR Options

• Call
• Put
• European
• American
• Bermudian
• Exotic Asian, Digital, Knock-In, Knock-Out, path
  dependent and multiple asset options.

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Various IR Options

• Futures options
• Caps
• Floors
• Exchange options
• Swaptions

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Cap

• Is priced as a sequence of caplets

cap
time
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Floor
floor
time
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Collar
cap
floor
time
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Option pricing

• Time value, intrinsic value
• underlying
• time to maturity
• interest rates
• strike
• coupons
• volatility

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Swaps

• Currency swap
• Interest rate swap
• Amortizing swap
• Swaption

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Currency swap
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Currency Swap
100Y
3Y 3Y 3Y 3Y 3Y 3Y
5 5 5 5 5 5
130
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IR swap
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IR Swap
100
3 3 3 3 3 3
L1 L1 L1 L1 L1 L1
100
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IR Swap
L1 L1 L1 L1 L1 L1
100
a regular LIBOR loan for one year!
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Term Structure Models

• Binomial trees
• Short-term based analytical models
• LIBOR based analytical models
• Multi-factor models
• Simulations

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Binomial Trees
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Interest rates
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Bond prices
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Typical yield curves
yield
time to maturity
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Analytic Term Structure Models
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Analytic Term Structure Models

• Hull, White
• Black-Karasinsky
• Black-Derman-Toy
• Heath-Jarrow-Morton
• Affine TS modles
• Gaussian models

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Arithmetic BM dX ? dt ? dW
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Geometric BM dX ?Xdt ?XdW
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Mean Reverting Process dX ?(?-X)dt ?X?dW
X
?
time
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Ho and Lee Model

• Rates are normally distributed.
• All rates have the same variability.
• The model has an analytic solution.

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Bond Prices under Ho and Lee

• Where

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Option Prices under Ho and Lee

• A discount bond matures at s, a call option
  matures at T

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Monte Carlo
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Monte Carlo Simulation
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Callable Bond
Straight Debt
Payoff
Callable Bond
Debt
Value of the firms call option
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Convertible Bond
Stock
Payoff
Convertible Bond
Straight Bond
Stock
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Protective Put
Payoff
Protective Put
X
Put
Stock
X Underlying
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Covered Call
Stock
Payoff
X
Covered Call
X
Written Call
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Straddle
Payoff
X
Call
Straddle
Put
X
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DAC

• Zvi Wiener
• 02-588-3049
• http//pluto.mscc.huji.ac.il/mswiener/zvi.html

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Life Insurance

• yearly contribution 10,000 NIS
• yearly risk premium 2,000 NIS
• first year agents 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.

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10,000 NIS

• insufficient funds if the client leaves
• insufficient profits

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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

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Obligations

• The most important question is what are the
  obligations?
• The Ministry of Finance should decide
• Transparent to a client
• Accounted as a loan

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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 P0, fees lost L0, fixed fee a of
  the final value P1, participation fee b of real
  profits (we ignore real).
• Investment policy TA-25 (MAOF).

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Liabilities (no actual loan)
Assets (no actual loan)
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TotalAssets-Liabilities
Fair value
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Liabilities (actual loan)
Assets (actual loan)
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TotalAssets-Liabilities (loan)
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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).
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2 years fair value, no loan
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2 years liabilities (with a loan)
2 years assets (with a loan)
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10 years, L07
With a loan
No loan
Profit
Stock index
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Partial loan - portion q
Theoretically q can be negative.
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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.

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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.

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Risk Management

• Zvi Wiener
• 02-588-3049
• http//pluto.mscc.huji.ac.il/mswiener/zvi.html

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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

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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

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Types of Assets and Risks

• Real projects - cashflow versus financing
• Fixed Income
• Optionality
• Credit exposure
• Legal, operational, authorities

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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

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How to measure VaR

• Historical Simulations
• Variance-Covariance
• Monte Carlo
• Analytical Methods

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Historical Simulations

• Fix current portfolio.
• Pretend that market changes are similar to those
  observed in the past.
• Calculate PL (profit-loss).
• Find the lowest quantile.

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Returns
year
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VaR
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Variance Covariance

• Means and covariances of market factors
• Mean and standard deviation of the portfolio
• Delta or Delta-Gamma approximation
• VaR1 ?P 2.33 ?P
• Based on the normality assumption!

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Variance-Covariance
?-2.33?
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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.

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Monte Carlo

• Distribution of market factors
• Simulation of a large number of events
• PL for each scenario
• Order the results
• VaR lowest quantile

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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?

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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.

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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.

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Airline company

• fuel - oil prices and
• purchasing airplanes - and Euro
• salaries - NIS, some
• tickets
• marketing - different currencies
• payments to airports for services

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Airline company

• loans
• equity
• callable bonds

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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.

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Reporting

• Division of VaR by business units, areas of
  activity, counterparty, currency.
• Performance measurement - RAROC (Risk Adjusted
  Return On Capital).

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How VaR is used

• Internal Risk Management
• Reporting
• Regulators

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Backtesting

• Verification of Risk Management models.
• Comparison if the models forecast VaR with the
  actual outcome - PL.
• Exception occurs when actual loss exceeds VaR.
• After exception - explanation and action.

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Backtesting
OK increasing k intervention

• Green zone - up to 4 exceptions
• Yellow zone - 5-9 exceptions
• Red zone - 10 exceptions or more

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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?

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Unifying Approach

• One number
• Based on Statistics
• Portfolio Theory
• Verification
• Widely Accepted
• Easy Comparison

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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
  banks strategy and risk limits.

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pluto.mscc.huji.ac.il/mswiener/
Risk Management resources

• Useful Internet sites
• Regulators
• Insurance Companies
• Risk Management in SEC reports

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Risk Measuring Software

• CATS, CARMA
• Algorithmics, Risk Watch
• Infinity
• J.P. Morgan, FourFifteen
• FEA, Outlook
• Reuters, Sailfish
• Kamacura
• Bankers Trust, RAROC
• INSSINC, Orchestra