FASB Statement No. 133 Refresher Valuation Regression

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FASB Statement No. 133 RefresherValuationRegre
ssion

• Speakers
• Daniel Kahn, Ernst Young
• Scott Underberg, Ernst Young

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• Valuation

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Background on Swaps

• First energy swaps were traded in October 1986
  (Chase Manhattan Bank vs. Cathay Pacific Airways
  and Koch Industries, oil-indexed price swap)
• Swaps are also known as Contracts-for-Differences
  or Fixed-for-Floating contracts
• Used to lock in a fixed price for a certain
  predetermined but not necessarily constant
  quantity
• An agreement in which counterparties exchange a
  floating energy price for a fixed energy price

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Numerical Example

• Single commodity swap
• Type Pay Fixed, Receive Floating
• Commodity Crude Oil WTI
• Number of Contracts 1 per month
• 1 Contract 1,000 barrels of oil
• Valuation Date 09/01/2008
• End Date 03/01/2009
• Contractual Fixed Price 98.23
• Discount Curve USD Swaps (30/360, S/A)
• Forward Curve Market rate extracted from traded
  oil futures

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Present Value of Cash Flows Before Adjusting for
Credit Risk
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Own and Counterparty Credit Risk

• The company has credit risk exposure to its
  counterparty at a given settlement date when the
  company is in a receivables position.
• The counterparty has credit risk exposure to the
  company on a future settlement date when the
  company is expected to be in a payables position.
• This is the notion of own and counterparty credit
  risk.

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Own and Counterparty Credit Risk (continued)

• To incorporate own and counterparty credit risk
  in a swap valuation, projected cash flows should
  be discounted at risk adjusted discount rates.
• Whenever a future payment is a liability to the
  company, the cash flow is discounted using a
  credit adjusted rate reflecting the companys
  credit risk. Whenever a future payment is an
  asset to the company, the cash flow is discounted
  using a credit adjusted rate reflecting the
  counterpartys credit risk.

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Present Value of Cash Flows After Adjusting for
Credit Risk Simplified Example
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How Can Credit Spreads Be Estimated?

• By observing publicly traded instruments like
  debt and credit default swaps
• Option Adjusted Spreads measure the incremental
  return over the risk free rate of a fixed-income
  security, adjusted for embedded options (if any).
  
• A credit default swap (CDS) is a credit
  derivative contract between two counterparties,
  whereby the buyer or fixed rate payer pays
  periodic payments to the seller or floating
  rate payer in exchange for the right to a payoff
  if there is a default or credit event with
  respect to the debt of a third party or
  reference entity

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Current Exposure vs. Potential Future Exposure

• The examples provided are based on the current
  exposure of the swap. The current exposure is
  equal to the swaps termination, or settlement,
  value.
• The true credit exposure of a derivative includes
  the current exposure as well as the derivatives
  potential future exposure.
• The potential future exposure of a derivative
  focuses on expected exposures at future dates
  rather than expected cash flows. Expected
  exposure is a function of potential movements in
  prices over time and their related
  probability-weighted fair values.

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Other Factors to Consider

• Collateral arrangements
• Netting agreements
• Threshold amounts
• Terms structures

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Options

• An Option is the right, but not the obligation,
  to buy or sell a security at a fixed price.
• The right to buy is called the Call Option and
  the right to sell is called the Put Option.
• The price in the contract is called the Exercise
  or Strike price and the date in the contract is
  called the Expiration date or maturity.
• American options can be exercised at any time up
  to the expiration date.
• European options can be exercised only on the
  expiration date itself. Since the holder is never
  required to exercise an option, the holder can
  never lose more than the purchase price of the
  option (i.e., option premium).
• Hence, an Option is a limited liability
  instrument.

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Option payoffs
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Major factors influencing the price of an option

• CONTRACTUAL TERMS
• Type of Option, i.e., Long / Short Put / Call
  American / European etc
• Exercise or Strike Price
• Maturity or Expiration Date
• MARKET DATA/ASSUMPTIONS
• Price of underlying security, currency,
  commodity, or financial instrument
• Risk-free interest rate
• Yield or return on the underlying security,
  currency, commodity, or financial instrument
• Volatility of the underlying asset

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A call option on the underlying asset price
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Option Pricing Summary
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Analytical formula The Black-Scholes option
pricing model

Given C e r T E Max ( ST K, 0) and
assuming the Geometric Brownian motion, the value
of a call option is given by
C e r T SN (d1)e r T KN (d2)

• C Call premium
• S current stock price
• T time until expiration
• K strike price
• r risk-free rate
• N cumulative standard normal distribution
• E risk-neutral expectation

e exponential function volatility of the
underlying ln natural logarithm
s
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Assumptions of the Black-Scholes model

• The stock prices follow Geometric Brownian motion
  
• There are no dividends during the life of the
  derivative
• Short selling of securities with full use of
  proceeds is permitted
• There are no transaction costs or taxes
• There are no riskless arbitrage opportunities
• Security trading is a continuous process
• Risk-free interest rate, r, is constant
  throughout the life of the option
• Markets are efficient
• Options are European type exercised only at
  maturity
• Returns are normally distributed

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Alternatives to Black-Scholes

• American and path-dependent options cannot be
  valued using the Black-Scholes Option Pricing
  Model
• Two numerical procedures commonly used for such
  valuations are
• Binomial Option Pricing Model
• Monte Carlo Simulation

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Binomial option pricing model

• The Binomial model breaks down the time to
  expiration into potentially a large number of
  intervals, or steps
• A tree of stock prices is initially produced
  working forward from present to expiration, and
  at each step it is assumed that the stock price
  will move up or down by an amount calculated
  using volatility and time to expiration
• This produces a Binomial tree of underlying stock
  prices
• At the end of the tree (at expiration of the
  option) all the terminal option values are known
  as they simply equal their intrinsic values
• Next, the option prices at each step of the tree
  are calculated working back from expiration to
  the present using the risk-free rate

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Valuation of American options using Binomial
pricing model
D
24.2 0.00
u 1.1 d 0.9 r 0.12 T 0.25 p 0.6523
B
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Max (0.405, 0.00)
19.8 1.2
E
A
20
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Put option Strike 21
Max (1.2684, 1.00)
C
16.2 4.8
early exercise
Max (2.3792, 3.00)
F
The above example illustrates the valuation of
American Options using a Binomial Tree. The
procedure is to work back through the tree from
the end to the beginning, testing at each node
whether early exercise is optimal.
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Simulation models Monte Carlo simulation

• The goal of Monte Carlo simulation is to generate
  potential paths of the underlying asset over a
  specified time horizon.
• The behavior of the underlying asset will be
  governed by an assumed diffusion process (i.e.,
  Geometric Brownian Motion), which describes the
  change in the underlying as a combination of a
  deterministic and a random process.
• The distribution of the asset value at the end of
  the time horizon will be influenced by the
  probability distribution associated with the
  random process.
• Potential paths will differ from one another by
  draws of random numbers from the specified
  probability distributions.
• At the end of each path, the option may be
  valued, and the mean option value across all the
  paths is calculated. The final option value
  equals the discounted mean option value.

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Simulation models Monte Carlo simulation (cont.)
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Option Pricing Summary

• Analytical Solution
• Computationally efficient
• Assumptions on price distributions may be limited
• Generally handles European options only
• Lattice
• Generally handles early exercise options
• May be difficult to implement or computationally
  intensive
• Monte Carlo Simulation
• Generally handles path dependent options
• Can incorporate modelling of market behaviour
  such as jumps
• Computationally intensive

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• Regression

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Long Haul Statistical Analysis - Linear
Regression

• Measures changes in the hedging instrument
  relative to the hedged item.
• Measurement indicator- Beta (Slope Coefficient)

• The relationship indicates the optimal hedge
  ratio and indicates the proportions of the
  hedging instrument to use relative to the hedged
  item.
• Can be used to assess how effective a derivative
  will be at offsetting changes in the fair value
  or cash flows of the hedged item
• Beta should generally be in the range from .8 Beta

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Long Haul Statistical Analysis - Linear
Regression (cont.)

• Measurement indicator- R Squared (Co-efficient of
  determination) Indicates the percent of variance
  in the hedged item that is explained by the
  hedge.
• For example, an R-squared of .85 indicates that
  85 of the movement in the dependant variable is
  explained by variation if the independent
  variable
• R-squared should generally be in the range of
  80-100.

• PROS Smoothes variations in data points as it
  creates the line that best fits.
• CONS Can be complex to implement, but provides
  the best representation of the hedging
  relationship.

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• Sample Regression Analysis

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Data to be Regressed
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Correlation of Monthly Changes

• Correlation of monthly changes
• CORREL(X,Y).9608
• Is a high correlation sufficient to assess
  effectiveness?
• No!
• Multiply each Monthly Change for Chicago
  Citygate Price by 1000
• Correlation remains .9608
• Thus, we want to also look at the Hedge Ratio
  when
• assessing effectiveness
• The Hedge ratio takes into account the relative
  standard deviation of each time series
• The Hedge Ratio is the slope of the regression
  line

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Performing a Regression Analysis
Select Tools ? Data Analysis ? Regression
Select Henry Hub and Chicago Citygate daily
changes as the X and Y input ranges
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Performing a Regression Analysis
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Regression Output Explained

• R Square is the Correlation squared
• (.9608)2 0.92305
• R2 0.92305
• Slope of the regression line is the Hedge Ratio
• Should generally lie between 0.8 and 1.2
• T statistic output by Excel tests the null
  hypothesis of
• whether the slope equals 0
• Look at the 95 confidence interval of the Hedge
  Ratio to see its Range
• Lower 95 0.8815
• Upper 95 1.0787

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Regression Inputs Historical Data

• Two types of historical data available
• Historical Settlements
• Historical Forward Curves
• Historical Settlements
• Regression analysis will assess how much of the
  final settlement of one price can be explained by
  the final settlement of another How closely
  together prices settled (static basis prices
  could in a perfect regression)
• Historical Forward Curves
• Regression analysis will assess how much of the
  daily change in one price can be explained by the
  daily change in another How closely do prices
  move over time

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Thank you