INSURANCE 811 DERIVATIVES

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INSURANCE 811 DERIVATIVES

• Neil Doherty

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Why are we interested in OPTIONS?

• Options and other derivatives are useful for
  hedging risk
• We will show that the value of the claims on a
  firm, debt and equity, can be represented as
  options
• We will also show that options increase in value
  as risk increases
• Thus, increases in the risk of the firm transfer
  wealth between shareholders and creditors
• This can lead to governance problems and to
  decisions that destroy corporate value
• Thus we need to manage risk to restore governance
  and to remove these distortions in decision
  making

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RISK MANAGEMENT INSTRUMENTS

• OPTIONS
• Option to buy (call) or sell (put) a security
• Can be used to hedge (short call or long put)
• Can be used to speculate (bare position)
• FUTURES AND FORWARDS
• Futures are exchange traded and marked to market
• Enables hedger to avoid price risk until maturity
• SWAPS
• Exchange one cash flow (often an unbundled cash
  flow from another security) for another

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BASIC OPTION POSITIONS
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Basic Options Positions

• Long Call
• Open profit w/ price rise
• Fixed loss w/price fall
• Long put
• Open profit w/ price fall
• Fixed loss w/price fall
• Short Call
• Fixed gain with price fall
• Open ended loss w/ price rise
• Short put
• Fixed gain with price rise
• Open ended loss w/ price fall

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SIMPLE HEDGES WITH OPTIONS
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HEDGING TAIL RISK AND HEDGING NORMAL RISK
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BINOMIAL PRICING MODEL. call option

• Current share price 50
• End of year price either 23 or 77
• Strike price 52 out of the money
• REPLICATE OPTION PAYOFF
• Purchase 0.46296 of a share
• Borrow 9.68 - paid back with 10 interest
• Portfolio has year end value opposite.
• Portfolio has same payoffs as the option
• So must sell for the same price.
• Value of Call
• 0.46296 (price share) - 9.68 loan
• 0.46296(50) - 9.68
• 13.468

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HEDGE RATI0

• Since the payoffs to the option can always be
  replicated by combining a risk free position
  (lending or borrowing) with a position in the
  underlying asset, this formula enables you to
  value the option. The trick to doing this is to
  find how much of the underlying asset needs to
  included in this portfolio. This is the HEDGE
  RATIO

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BINOMIAL PRICING MODEL. put option

• Current share price 50
• End of year price either 23 or 77
• Strike price 50 - at the money
• REPLICATE OPTION PAYOFF
• Short 0.5 of a share
• Lend 35 - paid back with 10 interest
• Hedge ratio, (d) is (0-27)/(77-23) -0.5.
• Portfolio has same payoffs as put
• So must sell for the same price.
• Value of put
• -0.5 (price share) 35 loan
• -0.5(50) 35
• 10

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PUT CALL PARITY

• Value of call - strike price 52 13.468
• Value of put - strike price 50 10.
• Value of call - strike price 50 ( not
  shown) 14.5455
• PUT CALL PARITY relationship between prices of
  call and put with SAME strike price
• C - P S - PV(E)
• Where C and P are the prices of the call and put,
  S is the share price and E is the strike price
• Thus, if both options have an exercise price of
  50
• 14.5455 - 10 50 - 50/(10.1) 4.5455

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Factors determining options prices

• Option value affected by several factors. Signs
  of these affect opposite
• Other factors (not shown) are
• maturity
• interest rate
• dividends
• Note importance of volatility on option values
  important for risk management

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THE DEFAULT PUT OPTION

• VALUE OF FIRM
• VF VD VE
• DEBT RISK FREE DEBT, D minus PUT OPTION
  written on VF and D
• VD D - P(VF D)
• EQUITY IS CALL OPTION written on VF and D
• VE C(VF D)
• VF D - P(VF D) C(VF D)
• RE-ARRANGE
• VE C(VF D)
• VE VF - D P(VF D)

• DEFAULT PUT

D
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The Default Put Option

• Equity is a CALL option on the value of the firm
• Value of equity INCREASES as risk of firm
  increases
• Alternatively VE VF - D P(VF D)
  Equity owns default put option
• Since value of default put increases with risk
  so value of equity increase as risk of firm
  increases.
• But value of debt falls as firm risk increases.
• VD D - P(VF D)
• Shareholders like risk choose more risky
  investments
• Creditors dislike risk need to monitor
  shareholders
• Risk creates conflicts of interest and leads to
  sub-optimal investment decisions

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The Default Put Option

• Current firm value 132
• End of year value either 100 or 200
• Debt Strike price 120
• REPLICATE OPTION PAYOFF
• .
• Value of default put
• (-0.2)(132) 36.36 9.96
• Value of Debt
• 120/1.1 - 9.96 99.13
• Value of Equity
• 32.87
• Value of firm 132

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RISK MANAGEMENT AND THE DEFAULT PUT OPTION

• Because shareholders own the default option,
  equity values tend to rise with increases in firm
  risk.
• This can divert shareholders from maximizing the
  value of the firm to extract wealth from
  creditors
• Increasing leverage
• Choosing high risk projects
• Creditors anticipate this distortion and this
  reduces value of debt
• Thus, in long, shareholders are worse off
• Higher cost of debt
• Firm not maximizing value
• NEED TO MANAGE RISK TO REMOVE DISTORTION

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ILLUSTRATION OF INTEREST SWAP

• Property owner has portfolio (leases) that yields
  5m per year in perpetuity. Interest rates
  currently 5 so value is
• V(A) 100 m
• The value of this asset is sensitive to changes
  in interest rates. With a change in rates ?r,
• Firm also has floating rate debt with a value of
  40m.
• V(D1) F where F
  is the face value
• Also, the firm has 40m debt paying a fixed
  interest of 2m (r5). As with asset, the value
  of this fixed debt will change with interest
  rates as follows

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Firm value and swap

• The TOTAL FIRM VALUE IS
• V(A) 100m - (5m/r2) ?r
• minus V(D1) 40m
• minus V(D1) 40m - (2m /r2) ?r.
• Net worth 20m - (3m/r2) ?r
• SWAP
• Floating debt exchanged for fixed rate debt at
  cost c
• V(D1) 40
• Minus V(D1) 40m - (2m /r2) ?r
• DIFFERENCE (2m /r2) ?r c
• RED INTEREST SENSITIVITY

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Swap dampens interest sensitivity

• VALUE OF FIRM WITH SWAP
• V(A) 100m -
  (5m/r2) ?r
• minus V(D1) 40m
• minus V(D1) 40m - (2m /r2) ?r.
• Plus SWAP (2m /r2) ?r.- c
• NET WORTH 20m - (1m/r2) ?r - c
• which is much less sensitive to interest rate
  changes than before the swap.

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Scaling the swap for perfect hedge

• Notice that, if the firm has undertaken the swap
  on a notional amount of 60m (i.e., 1.5 times
  as big as the previous swap)
• For 1.5(-c (2m /r2) ?r)
• Then the firm would have been perfectly hedged
• V(A) 100m -
  (5m/r2) ?r
• minus V(D1) 40m
• minus V(D1) 40m - (2m /r2) ?r.
• Plus SWAP -1.5c (3m /r2) ?r.
• NET WORTH 20m - 1.5c