INSURANCE PORTFOLIO. CSc.

Transcription

1 Ekonomická univerzita, Fakulta hospodárskej informatiky Dolnozemská cesta, Bratislava INSURANCE PORTFOLIO Doc. RNDr. Ľudovít t Pinda, CSc. FHI EU, Katedra matematiky mail: pinda@euba.sk Marec 00 Bratislava

2 Call option ( long and short position ) Investor Maximum profit Maximum loss Profit Loss Writter Fig. Put option ( long and short position ) Investor Profit Loss Maximum profit Maximum loss Writer Fig.

3 The Generalized Black Scholes Option Pricing Formula c GBS p GBS N( d ) S X Price of Europen call option, Price of Europen put option, The cumulative normal distribution function, Stock price, Strike price of option, T σ r Time to expiration in year, Volatility of the relative price change of the underlying stock price, Risk free interest rate. c GBS = S e ( b r ) T r T N( d ) X e N( ) d ()

4 p GBS = X e N ( b r ) T ( d ) S e N ( ) r T d () b = r Black-Scholes (973) stock option model, b = r-q Merton (973) stock option model with continuous dividend yield, b = 0 Black (976) futures option model, b = r-r f Garman and Kohlhagen (983) currency option model. The cumulative normal distribution function N ( x) = e dz π x z (3) N ( x) n = N 3 ( x)( a k + a k + a k ), ( x), 3 for x 0, for x < 0, k = + 0,3367 x a = , a = , a3 =

5 ( )., ln, T d d T T b X S d e d n d σ σ σ π = + + = = Portfolio Portfolio insurance insurance Value of no insure portfolio M K L Value of insure portfolio Fig. 3

6 Pro bab ilit y Preferred distribution Normal distribution F % Expected return Fig. 4 SITUATION British fund - 0 mil. diversified portfolio, FTSE Index 000 index points, index point - 0 Short term interest rate 0 %,

7 Problem: To protect the fund from a fall in FTSE below 800 index points. The simple solution: To buy put options on the index at an exercise price of 800 index points Investor need: / ( 0 000) = 500 put contracts If index fall to 700 index poins: The total fund value = ( 700/ 000) 0 mil = = 8.5 mil mil. Insurance is not free: The price of an 800 put option = 40 index poins Total insurance cost: = 0. mil. Manager need: 0. mil. to implement the strategy rather than a 0 mil. To rescale by a factor 0/0. = then a 0 mil. fund need 490 put options, the fund = insurance + invest in shars = mil. = = 0.96 mil mil. The fund would garantee a terminal value: ( 800/000) = 8.83 Level All-share Insured portfolio of index portfolio value Value of shares Value of puts at index 800 Total value of fund , , , , Tab.

8 3 Por tfol io val ue Shares only Insured portfolio Index level Put-call parita p S = c + PV (E) Fig. 5 + (4) p - Price of Europen call option, c - Price of Europen put option, S - Share, PV(E) - Present value of lending equity.

9 The garanteed minimum The amount at 0 % /. = 8.0 and =.979 mil. 800 c = p + S PV( E) = = 403.6, The number of calls ( 0 403,6) = 490. index points Level of Value of 490 calls Total value of Value of T-bills index at E = 800 fund 600 8,83 0 8, ,83 0 8, ,83 0 8, ,83 0,49 9, ,83 0,98 9, ,83,47 0,93 Tab. Assume: Investor was not willing to accept any loss a one year horizont He invested at risk free rate 0 mil /. = 9.09 mil. 0 mil = mil. in calls. Let calls cost an exercise price 000 index points 50 points for one year Buy / ( 0 50 ) = calls

10 For each point exceeded 000 the manager would gain = The manager participate in / = 7.7 % of any rise in market ( = / 000 ). The higher the guaranteed value of the fund => the smaller participation in any rise 5 % gain => 0.5 mil. => equivalent index level 00 => /. = = invested in the risk free assets and mil. in calls with price 60 points, / ( 0 60 ) = 84.4 calls The manager participate in 844 / = % of any rise in market ( = / 000 ). The highest guarantee is mil. at the 0 % risk free rate => the participation in any rise is 0 %. Fund Guarantee Exercise Calls T-bills Participation size price Number Cost cost rate (%) e d c b a Tab. 3

11 Portfolio value 0 a b c 9 d Index level Fig. 6 Portfolio strategy with respect to insurance: the floor, the participation rate in any rise of the index above the floor.

12 Portfolio value = Floor + max {0, w [ ( g Index ) - Floor]} g initial portfolio value per index point, 0.0 / 000 = 0.005, w participation rate, Let a floor is 0.5 mil. and index level is 00 Portfolio value = Floor + max {0, w [ ( g Index ) - Floor]} = = max { 0, [ ( ) 0.5]} = = max {0, 0.844} = mil.

13 Ďakujem za pozornosť Doc. RNDr. Ľudovít Pinda, CSc. mail: