Financial Risk Management: Techniques and Applications

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Financial Risk Management Techniques and
Applications

• Workshop ICWA 13th November 2009
• Sreejata Banerjee
• Associate Professor
• Madras School of Economics

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Important Concepts

• The concept and practice of risk management
• The benefits of risk management
• The difference between market and credit risk
• How market risk is managed using delta, gamma,
  vega, and Value-at-Risk
• How credit risk is managed, including credit
  derivatives
• Risks other than market and credit risk

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• Definition of risk management the practice of
  defining the risk level a firm desires,
  identifying the risk level it currently has, and
  using derivatives or other financial instruments
  to adjust the actual risk level to the desired
  risk level.

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Why Practice Risk Management?

• The Impetus for Risk Management.
• Growth in use of derivatives have been subject to
  a fare amount of suspicion I, distrust and
  outright fear of derivatives. But eventually
  firms began to realize that they were the best
  tools for coping with volatile market
• Firms practice risk management for several
  reasons
• Interest rates, exchange rates and stock prices
  are more volatile today than in the past.
• Significant losses incurred by firms that did not
  practice risk management
• Improvements in information technology
• Favorable regulatory environment
• Sometimes we call this activity financial risk
  management.

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Why Practice Risk Management? (continued)

• The Benefits of Risk Management
• What are the benefits of risk management, in
  light of the Modigliani-Miller principle that
  corporate financial decisions provide no value
  because shareholders can execute these
  transactions themselves?
• Firms can practice risk management more
  effectively.
• There may be tax advantages from the progressive
  tax system.
• Risk management reduces bankruptcy costs.
• Managers are trying to reduce their own risk.

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Why Practice Risk Management? (continued)

• Firms manage risk to reduce taxes, to lower
  bankruptcy costs, because managers are concerned
  about their own wealth, to avoid under-investing,
  to take advantage of speculative positions
  occasionally, to earn perceived arbitrage profits
  or to assume credit risks lower borrowing costs

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Why Practice Risk Management? (continued)

• The Benefits of Risk Management (continued)
• By protecting a firms cash flow, it increases
  the likelihood that the firm will generate enough
  cash to allow it to engage in profitable
  investments.
• Some firms use risk management as an excuse to
  speculate.
• Some firms believe that there are arbitrage
  opportunities in the financial markets.
• Note The desire to lower risk is not a
  sufficient reason to practice risk management.

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Managing Market Risk

• Market risk the uncertainty associated with
  interest rates, foreign exchange rates, stock
  prices, or commodity prices.
• Example A dealer with the following positions
• A four-year interest rate swap with 10 million
  notional principal in which it pays a fixed rate
  and receives a floating rate.
• ( A swap contract allows a debt holder to pay a
  fixed interest rate and receive a floating
  interest rate)
• A 3-year interest rate call with 8 million
  notional principal. The dealer is short and the
  exercise rate is 12.
• ( A call option allows the holder of the contract
  to buy a specific underlying asset at a
  predetermined price the exercise price which can
  below the spot rate)
• See Table 15.1, p. 546 for current term structure
  and forward rates. We obtain the call price as
  73,745 and the swap rate is 11.85.

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Managing Market Risk (continued)

• Delta Hedging
• The Delta is approximately the change in the
  option price for a small change in the stock
  price/ underlying asset. In our case it is
  interest rate.
• We estimate the delta by repricing the swap and
  option with a one basis point move in all spot
  rates and average the price change.
• See Table 15.2, p. 547 for estimated swap and
  option deltas.
• We are long the swap so we have a delta of
  2,130.5, round to 2,131.
• We are short the option so we have a delta of
  -244.
• Our overall delta is 1,887.

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Managing Market Risk (continued)

• Delta Hedging (continued)
• A delta hedged position is one on which the
  combined spot and derivatives positions have a
  delta of zero. The portfolio would then have no
  gain or loss is value from very small change in
  the underlying source of risk(in our case
  interest rate)
• Instead of executing a swap which pays floating
  and receives fixed rate, and buying an option
  with exercise price of 12, a typical trade
  would be to go long Eurodollar futures. That is
  because Eurodollars are like bonds that move in
  the opposite direction of interest rate. The
  delta is -25 computed as follows-
• -1,000,000(0.0001)(90/360)

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Why Practice Risk Management? (continued)

• We need a Eurodollar futures position that gains
  1,887 if rates move down and loses that amount
  if rates move up. Thus, we require a long
  position of 1,887/25 75.48 contracts. Round
  to 75. Overall delta
• 2,131 (from swap)
• -244 (from option)
• 75(-25) (from futures)
• 12 (overall)
• This means that the portfolio value will go up
  12 if rates move up one basis point. This is
  basically a perfect hedge

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Managing Market Risk (continued)

• Gamma Hedging
• Delta hedging is effective if the movements are
  small like one basis point, if the movement is
  larger say 50 basis points then the delta hedging
  is not enough. We now need to gamma hedge.
• Here we deal with the risk of large price moves,
  which are not fully captured by the delta.
• See Table 15.3, p. 549 for the estimation of swap
  and option gammas. Swap gamma is -12,500, and
  option gamma is 5,000. Being short the option,
  the total gamma is -17,500.
• Eurodollar futures have zero gamma so we must add
  another option position to offset the gamma. We
  assume the availability of a one-year call with
  delta of 43 and gamma of 2,500.

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Managing Market Risk (continued)

• Gamma Hedging (continued)
• We use x1 Eurodollar futures and x2 of the
  one-year calls. The swap and option have a delta
  of 1,887 and gamma of -17,500. We solve the
  following equations
• 1,887 x1(-25) x2(43) 0 (zero delta)
• -17,500 x1(0) x2(2,500) 0 (zero gamma)
• Solving these gives x1 87.52 (go long 88
  Eurodollar futures) and x2 7 (go long 7 times
  1,000,000 notional principal of one-year option)

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Managing Market Risk (continued)

• A delta and gamma hedge is one in which the
  combined spot and derivatives positions have a
  delta of zero and gamma of zero.
• Let us check the delta is 1887 88(-25) 4(73)
  -12.00 and the gamma is -175007(2500) 0
• Unfortunately the use of options introduces a
  risk associated with changes in volatility. Hence
  a portfolio of derivatives that is both delta and
  gamma hedged can incur a gain or lass from a
  change in the volatility. Most options are highly
  sensitive to volatility, so it is important to
  hedge vega risk.

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Managing Market Risk (continued)

• Vega Hedging
• Swaps, futures, and FRAs do not have vegas.
• We estimate the vegas of the options
• On our 3-year option, if volatility increases
  (decreases) by .01, option will increase
  (decrease) by 42 (-42). Average is 42. We
  are short this option, so vega -42.
• One-year option has estimated vega of 3.50.
• Overall portfolio has vega of (3.50)(7 million)
  - 42 -17.50.

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Managing Market Risk (continued)

• Vega Hedging (continued)
• We add a Eurodollar futures option, which has
  delta of -12.75, gamma of -500, and vega of
  2.50 per 1MM.
• Solve the following equations
• 1,887 x1(-25) x2(43) x3(-12.75) 0
  (delta)
• -17,500 x1(0) x2(2,500) x3(-500) 0
  (gamma)
• -42 x1(0) x2(3.50) x3(2.50) 0 (vega)
• The coefficients are the multiples of 1,000,000
  notional principal we need.
• Solutions are x1 86.61, x2 8.09375, x3
  5.46875.

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Managing Market Risk (continued)

• Vega Hedging (continued)
• Any type of hedge (delta, delta-gamma, or
  delta-gamma-vega) must be periodically adjusted.
• In spite of a dealers efforts at achieving a
  delta-gamma-vega neutral position, it is really
  impossible to achieve an absolute perfect hedge.
• It is apt to remember a famous expression The
  only perfect hedge is in a Japanese garden

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Managing Market Risk (continued)

• Value-at-Risk (VAR)
• A dollar measure of the minimum loss that would
  be expected over a given time with a given
  probability. Example
• VAR of 1 million for one day at .05 means that
  the firm could expect to lose at least 1 million
  over a one day period 5 of the time.
• Widely used by dealers and increasingly by end
  users.
• See Table 15.4, p. 553 for example of discrete
  probability distribution of change in value. VAR
  at 5 is 3 million loss.
• See Figure 15.1, p. 554 for continuous
  distribution.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• VAR calculations require use of formulas for
  expected return and standard deviation of a
  portfolio
• where
• E(R1), E(R2) expected returns of assets 1 and 2
• ?1, ?2 standard deviations of assets 1 and 2
• ? correlation between assets 1 and 2
• w1, w2 of ones wealth invested in asset 1 or
  2

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• Three methods of estimating VAR
• Analytical method Uses knowledge of the
  parameters (expected return and standard
  deviation) of the probability distribution and
  assumes a normal distribution.
• Example 20 million of SP 500 with expected
  return of .12 and volatility of .15 and 12
  million of Nikkei 300 with expected return of
  .105 and volatility of .18. Correlation is .55.
  Using the above formulas, the overall portfolio
  expected return is .1144 and volatility is .1425.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) Analytical Method (continued)
• For a weekly VAR, convert these to weekly
  figures.
• Expected return .1144/52 .0022
• Volatility .1425/Ö52 .0198.
• With a normal distribution, we have
• VAR .0022 - 1.65(.0198) -.0305
• So the VAR is 32,000,000(.0305) 976,000.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) Analytical Method (continued)
• Let us use an extremely risky portfolio, a short
  call on a stock index.
• Example using options 200 short 12-month calls
  on SP 500, which has volatility of .15 and price
  of 14.21 (as per Black Scholes model). The spot
  of the index is 720 and the wxercise price is 720
  as well.
• The index option has a multiplier contract of
  500. So 500 14.20 Total value for 200 calls
  will be 1,421,000.
• Based on monthly data, expected return is
  1144/12.0095 and volatility is 0.4125/3.46
  .0412.
• Upside 5 is .0095 1.65(.0412) .0775, which
  is 720(1.0775) 775.80.
• Option would be worth 775.80 - 720 55.80 so
  loss is 55.80 - 14.21 41.59 per option.
• Total loss 200(500)(41.59) 4.159 million.
  This is the VAR.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) Analytical Method (continued)
• One assumption often made is that the expected
  return is zero. This is not likely to be true.
• Sometimes rather than use the precise option
  price from a model, a delta is used to estimate
  the price. This makes the analytical method be
  sometimes called the delta-normal method.
• Volatility and correlation information is
  necessary. See the web site www.riskmetrics.com
  , where data of this sort are provided free.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• Historical method Uses historical information
  on the users portfolio to obtain the
  distribution.
• Example See Figure 15.2, p. 557. For portfolio
  of 15 million, VAR at 5 is approximately a loss
  of 10 or 15,000,000(.10) -1,500,000.
• Historical method is subject to limitation that
  the past holdings of the portfolio may not have
  the same distributional properties as the future
  holdings. It also is limited by the results of
  the chosen time period, which might not be
  representative of the future.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• A Comprehensive Calculation of VAR
• We do an example of a portfolio of 25 million in
  the SP 500. We want a 5 1-day VAR using each
  method. We collect a sample of daily returns on
  the SP 500 for the past year and obtain the
  following parameter estimates Average daily
  return .0457 and daily standard deviation
  1.3327. These result in annual figures of

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• A Comprehensive Calculation of VAR (continued)
• Analytical method We have 0.0457
  - (1.65)1.3327 -2.1533. So the VAR is
• .021533(25,000,000) 538,325
• The .21 standard deviation is historically a bit
  high. Re-estimating with a standard deviation of
  .15 gives us a daily standard deviation of
  0.9430. Then we obtain 0.0474 - 1.65(0.9430)
  -1.5086 and a VAR of
• .015086(25,000,000) 377,150
• Are our data normally distributed? Observe
  Figure 15.3, p. 559.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• A Comprehensive Calculation of VAR (continued)
• Historical method Here we rank the returns from
  worst to best. For 253 returns we obtain the 5
  worst by observing the .05(253) 12.65 worst
  return. We shall make it the 13th worst. This
  would be -2.0969. Thus, the VAR is
• .020969(25,000,000) 524,225

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• A Comprehensive Calculation of VAR (continued)
• Monte Carlo simulation method We shall use an
  expected return of 12, a standard deviation of
  15 and a normal distribution.
• We generate 253 random returns (this number is
  arbitrary and should actually be much larger) by
  the following method
• where ? is a standard normal random number.

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• A Comprehensive Calculation of VAR (continued)
• Monte Carlo simulation method (continued) We do
  this 253 times, rank the returns from worst to
  best and obtain the 13th worst return, which is
  -1.3942. Then the VAR is
• .013942(25,000,000) 348,550

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• A Comprehensive Calculation of VAR (continued)
• So VAR is estimated to be either
• 538,325
• 377,150
• 524,225
• 348,550
• Key considerations wide ranges such as this are
  common, real-world portfolios are more
  complicated than this, ex post evaluation should
  be done

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• Benefits and Criticisms of VAR
• Widely used
• Facilitates communication with senior management
• Acceptable in banking regulation
• Used to allocate capital within firms
• Used in performance evaluation

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Managing Market Risk (continued)

• Value-at-Risk (VAR) (continued)
• Extensions of VAR
• Stress testing
• Conditional VAR expected loss, given that loss
  exceeds VAR. In SP example, this is 719,450.
• Cash Flow at Risk (CAR or CFAR), which is more
  appropriate for firms with assets that generate
  cash but for which a market value cannot easily
  be assigned.
• Example expected cash flow of 10 million with
  standard deviation of 2 million. CAR at .05
  would be 1.65(2 million) 3.3 million because
• Prob10 million - 3.3 million gt Actual Cash
  Flow .05.
• Note CAR is a shortfall measure.
• Earnings at Risk (EAR) measures earnings
  shortfall.

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Managing Credit Risk

• Credit risk or default risk is the risk that the
  counterparty will not pay off in a financial
  transaction.
• Credit risk is the uncertainty and potential for
  loss due to a failure to pay on the part of the
  counterparty
• Credit ratings are widely used to assess credit
  risk.
• Credit ratings are provided by firms most notable
  Standard and Poors, Moodys and Fitchs.
• Each have their own labels terms like Triple A
  and Double B you see ratings like AAA, AA,
  BBB, BB B and so on to D. These ratings are based
  on a variety of factors like financial health of
  the firm, sates of th4e economy and the state of
  the industry.

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Managing Credit Risk (continued)

• Option Pricing Theory and Credit Risk
• Here we use option pricing theory to understand
  the nature of credit risk. Consider a firm with
  assets with market value A0, and debt with face
  value of F due at T. The market value of the
  debt is B0. The market value of the stock is S0.
• See Table 15.5, p. 563 for the payoffs to the
  suppliers of capital. Note how the stock is like
  a call option on the assets. Its payoff is
• Max(0,AT F)
• Using put-call parity, we have
• P0 S0 A0 F(1 r)-T

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Managing Credit Risk (continued)

• Option Pricing Theory and Credit Risk (continued)
• This put is the value of limited liability to the
  stockholders. Rearranging, we obtain S0 A0
  P0 F(1 r)-T. But, by definition, S0 A0
  B0. Setting these equal, we obtain
• B0 F(1 r)-T P0.
• Thus, the bond subject to default risk is
  equivalent to a risk-free bond and a put option
  written by the bondholders to the stockholders.
• The Black-Scholes formula adapted to price the
  stock as a call would be
• Using B0 A0 S0, we can obtain the value of
  the bonds.

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Managing Credit Risk (continued)

• The Credit Risk of Derivatives
• Current credit risk is the risk to one party that
  the other will be unable to make payments that
  are currently due.
• Potential credit risk is the risk to one party
  that the counterparty will default in the future.
• In options, only the buyer faces credit risk.
• FRAs and swaps have two-way credit risk but at a
  given point in time, the risk is faced by only
  one of the two parties.

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Managing Credit Risk (continued)

• The Credit Risk of Derivatives (continued)
• Potential credit risk is largest during the
  middle of an interest rate swaps life but due to
  principal repayment, potential credit risk is
  largest during the latter part of a currency
  swaps life.
• Typically all parties pay the same price on a
  derivative, regardless of their credit standing.
  Credit risk is managed through
• limiting exposure to any one party (primary
  method)
• collateral
• periodic marking-to-market
• (by dealers) captive derivatives subsidiaries
• netting (see next)

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Managing Credit Risk (continued)

• Netting
• Netting several similar processes in which the
  amount of cash owed by one party to the other is
  reduced by the amount owed by the latter to the
  former.
• Bilateral netting netting between two parties.
• Multilateral netting netting between more than
  two parties essentially the same as a
  clearinghouse.
• Payment netting Only the net amount of a
  payment owed from one party to the other is paid.
• Cross-product netting payments from one type of
  transaction are netted against payments for
  another type of transaction.

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Managing Credit Risk (continued)

• Netting (continued)
• Netting by novation net value of two parties
  mutual obligations is replaced by a single
  transaction often used in FX markets.
• Closeout netting netting in the event of
  default, where all transactions between two
  parties are netted against each other.
• The OTC derivatives market has an excellent
  record of default. Note the Hammersmith and
  Fulham default where it was found that a town had
  no legal authority to engage in swaps. The town
  was able to get out of paying its losses.

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Managing Credit Risk (continued)

• Credit Derivatives A family of derivative
  instruments that have payoffs contingent on the
  credit quality of a particular party. Types
  include
• Total return swaps See Figure 15.4, p. 570.
  Credit derivative buyer purchases swap from
  credit derivative seller in which it pays the
  total return on a specific bond. If that return
  is reduced by some credit event, this loss is
  passed through automatically in the swap.

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Managing Credit Risk (continued)

• Credit Derivatives (continued)
• Credit swap A swap in which the credit
  derivatives buyer pays a periodic fee to the
  credit derivatives seller. If the buyer sustains
  a credit loss from a third party, it then
  receives payments from the credit derivatives
  seller to compensate. See Figure 15.5, p. 571.
  This is really more like an option.
• Credit spread option An option in which the
  underlying is the yield spread on a bond.

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Managing Credit Risk (continued)

• Credit Derivatives (continued)
• Credit linked security This is a bond or note
  that pays off less than its face value if a
  credit event occurs on a third party.
• The credit derivatives market is small but
  growing rapidly. The notional principal of
  credit derivatives at U. S. banks was estimated
  at about 573 billion in 2002.

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Other Types of Risks

• operational risk (including inadequate controls)
• model risk
• liquidity risk
• accounting risk
• legal risk
• tax risk
• regulatory risk
• settlement risk
• systemic risk

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Summary

• We looked at some techniques for managing market
  risk, including delta, vega and gamma hedging.
• We also examined the concept of Value-at-Risk
  (VAR), which measures the minimum loss expected
  over a period of time with a given probability.
  VAR is widely used in the financial world.
• We examined some variations of VAR including Cash
  Flow at Risk, Earnings at Risk.
• We also looked at credit risk, noting how credit
  risk entails elements of option pricing theory.
• Procedures of reducing credit risk through
  netting of payments owed by parties. And the use
  of credit derivatives.

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