©K.Cuthbertson and D.Nitzsche 1 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE Portfolio Insurance 1/9/2001

©K.Cuthbertson and D.Nitzsche 2 Topics Static Hedge, Stock-Put (Protective Put) Dynamic Portfolio Insurance

©K.Cuthbertson and D.Nitzsche 3 Static Hedge: Stock+Put (Protective Put)

©K.Cuthbertson and D.Nitzsche 4 Static Hedge, Stock+Put (Protective Put) stock+put = {+1, +1} plus {-1, 0} = {0, +1} September: S&P500, S 0 = 280 (z p = $500) December-280, S&P500 put option, P 0 = 2.97 (index units) Cost of one put option = z p P = $500 (2.97) = $1,485 Value of stocks underlying put option = z p S =$140,000 For $1,250 the put is a claim on $140,000 of the index (ie. P 0 /S 0 = 0.89%). Number of units of the index held is : [13.1]N 0 * = V 0 /S 0 = 2,000 (ie. 2,000-dollars per unit of the index)

©K.Cuthbertson and D.Nitzsche 5 Static Hedge, Stock+Put (Protective Put) Number of units of the index held in stocks: [13.1]N 0 * = V 0 /S 0 = 2,000 (ie. $2,000 per unit of the index) We need to pay for the puts We hold an equal number N 0 (index units) in stocks and puts: [13.2]V 0 = N 0 (S 0 + P 0 )

©K.Cuthbertson and D.Nitzsche 6 Static Hedge, Stock+Put (Protective Put) At expiration: [13.6a] V T = N 0 S T if S T K(Upside potential) [13.6b] V T = N 0 S T + N 0 (K - S T ) = N 0 K if S T < K(Insured level) Net profit = V T - N 0 P 0 ‘Insured’ lower value at: [13.7]V min = N 0 K = 1,979 (280) = $554,120 Let S T = 310 Value of insured portfolio = V T = N 0 S T = 1,979 (310) = $614,490 Value of uninsured portfolio = N 0 *S = 2,000 (310) = $620,000 Upside capture is 98.95% (= $614,513/$620,000)

©K.Cuthbertson and D.Nitzsche 7 Dynamic Portfolio Insurance

©K.Cuthbertson and D.Nitzsche 8 Dynamic Portfolio Insurance Matches the price movement of a Stock+Put (protective put) with either Stock + Futures portfolio or Stock+T-Bill portfolio these are ‘replication portfolios’

©K.Cuthbertson and D.Nitzsche 9 Why use the Replication Portfolios ?  the hedging horizon of pension funds and mutual funds may be long stock index puts then have to be rolled over may be costly  most traded stock index options are American and their premia reflect the early-exercise premium Portfolio managers with fixed hedging horizons are reluctant to bear this cost  position limits (set by the clearing house) may prevent portfolio managers setting up their optimal hedged position in stock index puts.

©K.Cuthbertson and D.Nitzsche 10 Dynamic Portfolio Insurance: Algebra Stock+Put [13.15] N 0 = V 0 /(S 0 +P 0 ) N 0 is fixed throughout the hedge. At t > 0 ‘stock+put’ portfolio: [13.16] V s,p = N 0 (S + P), and Stock + Futures N 0 * = V 0 / S 0 N 0 * is also held fixed throughout the hedge. At t > 0: [13.19] V S,F = N 0 * S + N f (F z f ) and Hence:

©K.Cuthbertson and D.Nitzsche 11 Stock+T-Bill Portfolio [13.26]V S,B = N S S + N B B Equate dV of stock+put with stock+T-bill Dynamic Portfolio Insurance: Algebra

©K.Cuthbertson and D.Nitzsche 12 Example : Portfolio Insurance Value of stock portfolioV 0 = $560,000 S&P500 indexS 0 = 280 Maturity of Derivatives T - t = 0.10 Risk free rater = 0.10 p.a. (10%) Compound\Discount Factore r (T – t) = 1.01 Standard deviation S&P  = 0.12 Put PremiumP 0 = 2.97 (index units) Strike PriceK = 280 Put delta  p = (Call delta)(  c = 1 +  p = ) Futures Price (t=0)F 0 = S 0 e r(T – t ) = Price of T-BillB = Me -rT = 99.0

©K.Cuthbertson and D.Nitzsche 13 Example : Portfolio Insurance Hedge Positions Number of units of the index held in stocks = V 0 /S 0 = 2,000 index units Stock-Put Insurance N 0 = V 0 / (S 0 + P 0 ) = 1979 index units Stock-Futures Insurance N f = [(1979) (0.6112) - 2,000] (0.99/500) = (short futures) Stock+T-Bill Insurance No. stocks = 1979 (0.612) = 1,209.6 (index units) N B = 2,235.3 (T-bills)

©K.Cuthbertson and D.Nitzsche 14 Example : Portfolio Insurance Scenario : Fall of 1 unit in the index,  S = -1 hence S 1 = 280 – 1 =270 New Derivatives Prices Put premium P = (using Black Scholes) Futures F 1 = S 1 e r(T-t) = 279 ( 1.01) = (Approx)Change in Put Premium DP =  p (  S ) = Change in Futures Price DF = – = -1.01

©K.Cuthbertson and D.Nitzsche 15 Example : Portfolio Insurance 1) Stock+Put Portfolio Gain on Stocks = N 0.DS = 1979 ( -1) = -1,979 Gain on Puts = N 0 DP = 1979 ( 0.388) = Net Gain = -1, ) Stock + Futures Dynamic Hedge Gain on Stocks = N s,o DS = 2000 (-1) = -2,000 Gain on Futures = N f.DF.z f = (-1.56) (-1.01) 500 = Net Gain = -1,209.6

©K.Cuthbertson and D.Nitzsche 16 Example : Portfolio Insurance 3) Stock + T-Bill Dynamic Hedge Gain on Stocks = N s DS = (-1) = -1,209.6 Gain on T-Bills= 0 (No change in T-bill price) Net Gain= -1,209.6 The loss on the replication portfolios is very close to that on the stock-put portfolio (over the infinitesimally small time period).

©K.Cuthbertson and D.Nitzsche 17 Example : Portfolio Insurance Problems We are only “delta hedging” and hence if there are large changes in S or changes in  then our calculations will be inaccurate (ie. Actual change in put premium will not equal that calculated using  P =  p (  S ) = ) When there are large market falls, liquidity may “dry up” and it may not be possible to trades quickly enough at quoted prices (or at any price !).

©K.Cuthbertson and D.Nitzsche 18 End of Slides