Insurance
Copyright Portfolio 1998-2006 Investment Analytics
1
Portfolio Insurance
What is it?
Why use it?
How to apply in practice
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 2
Portfolio Insurance: Rationale
Portfolio
1000 shares stocks A, 1000 shares stock B
Volatility of each stock 30%
Current stocks price of A and B is $50
T-Bill rate is 8%
Scenario:
Expect FED to tighten in next three months
Portfolio value must not fall below $90,000
Calculate all-in costs of protection:
Use the option calculator
Worksheet: Portfolio Insurance Intro
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 3
Cost of Protection
Cost of 3m put option with $45 strike is
$0.79
Need 2,000 options
Hence all-in cost is $0.79 x 2000 = $1,586
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 4
Hedging a Portfolio
Next, assume A and B are uncorrelated:
Corr(A,B) = 0
Compute the volatility of the portfolio
Calculate the all-in cost of hedging the
portfolio
Reminder:
2
2
p = [ w12 1 + w2 2 + 2 w1 w2 1 2 ]
2
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 5
Cost of Hedging Portfolio
Hedging Cost:
SD of Portfolio 21.21%
Black Scholes option price $0.29
Hence cost of portfolio hedge:
2,000 x $0.29 = $579
Cost saving = $1,007
Finding:
Cost of option on portfolio is less than
cost of portfolio of options
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 6
Evolution of Portfolio Insurance
PROBLEM: option on portfolio doesnt exist
Hence create put option synthetically
Replicate hedged portfolio
Dynamic Hedging Strategy
Leyland, OBrien, Rubenstein (LOR)- 1980s
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 7
Po sur
rtf ed
ol
io
In
Po
rtf
ol
io
Fund Value
Portfolio Insurance Illustrated
S&P500
Floor
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 8
Po sur
rtf ed
ol
io
Cost
In
Po
rtf
ol
io
Fund Value
Portfolio Insurance Illustrated
S&P500
Floor
Deductible
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 9
Po sur
rtf ed
ol
io
In
Smaller
Deductible =
Higher Cost
Po
rtf
ol
io
Fund Value
Insurance
Cost
Deductible
S&P500
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 10
Portfolio Insurance as Call Option
Put-Call Parity: Put + Stock = Call + Cash
Portfolio
Insured
Portfolio
Put Option
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 11
Role of Portfolio Insurer
Portfolio can be insured separately:
Fund Manager
Portfolio Insurer
S&P 500
Fund
Hedging
Transactions
Insured
Portfolio
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 12
Lab: S&P500 Tracking Portfolio
Client has $100MM S&P500 Tracking Fund
Wants to ensure fund value > $95MM at yearend
Engages you as insurer
Background:
S&P500 volatility 15%
Tracking error volatility 2.5%
How do you proceed?
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 13
Portfolio Insurance Steps
Estimate volatility of portfolio
Determine option insurance required
Put/call
Strike
Use Option Calculator to price option
Delta gives us proportion we need to
invest in risky assets (the portfolio)
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 14
Constructing Portfolio /1
We need a put option with a $95 strike
From Black Scholes: Put cost = $1.53MM
Problem: Have $100MM fully invested
Cant afford another $1.53MM for option
Scale down portfolio size from $100MM so
that:
Portfolio value + option cost = $100MM
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 15
Next step:
Experiment with various portfolio values
(below $100MM)
Use option calculator to price option
Add portfolio value to option price
If total <>$100 then repeat!
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 16
Constructing Portfolio /2
Solution: Fund $98.12MM, Put option
$1.88MM
But: we want to replicate option, not buy it
How do we replicate it?
Steps:
Use Option Calculator to derive option delta
Proportion to invest in portfolio: (1 - delta)
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 17
Constructing Portfolio /3
Option delta is -0.208
So invest (1 - 0.208) x $98.12MM =
$77.7MM in portfolio
$22.3MM in T-Bills
TOTAL $100MM
Copyright 1998-2006 Investment Insurance
Slide: Analytics
Portfolio 18
Constructing Portfolio /4
Need equiv. of $77.7MM in portfolio, $22.3 in TBills
BUT: already have $100MM in portfolio
So use S&P500 futures to create equiv. position
STEPS:
Need to sell equiv. $22.3MM of portfolio
Current S&P500 index level is 630
Value of 1 S&P contract?
How many contracts to sell?
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 19
Constructing Portfolio /5
Value of 1 S&P500 future is
$630x 500 = $315,000
So, sell $22.3MM / $315,000 = 71 contracts
Fund Manager
S&P 500
Fund
$100MM
Portfolio Insurer
Sells 71 S&P
500 Futures
Insured Portfolio
with $95MM
Floor
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 20
Lab: Checking Insurance Policy
What happens if S&P index falls to 617?
Work out new value of portfolio:
Original fund value $100MM
How many S&P futures is this equiv to?
Multiply this no. by 600 to get new fund value
Work out gain on short S&P500 futures
Dont forget T-Bill interest!
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 21
Checking Insurance
Fund Value $100MM
Was equiv to $100MM / (630 x 500) = 317 Futures
Value Now: 617 x 317 x $500 = 97.79MM
Gain on Short Futures
71 x (630-617) x $500 = $0.46 MM
Interest on T-Bills
$22.3MM x 8% = $1.78MM
TOTAL VALUE
$97.79MM + $0.46MM + $1.78MM = $100.04MM
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 22
Limits to Insurance:
What happens if S&P500 index falls to
550?
Portfolio Value is 550x317x$500 = $87.18MM
Gain on Short Futures Hedge =
71x(631-550)x$500 = $2.83MM
Interest = $1.78MM
Total Value is $91.79MM
Floor was supposed to be $95MM
What went wrong?
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 23
Delta Revisited
Put Value
Delta is the rate of change of option value
S2
S1
Stock Price
Delta is the slope of the tangent at stock price
- Slope changes (more negative) from S1 to S2
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 24
Rebalancing
The Delta of a put option changes as the
S&P500 index moves
We need to reflect this by changing the
proportions held in T-bills & risky asset
As index falls, delta becomes more negative
So we need to sell more stock as stock falls,
buy more stock when it rises
Buy High, Sell Low
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 25
Lab: Rebalancing
Suppose after 6 months, S&P index falls to 610
Calculate the new value of the fund
Use the Option Calculator to find the new delta
Compute the amount to be held in the Fund (and
the amount in T-Bills)
How many S&P Futures do we need to sell?
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 26
Solution: Rebalancing
New Fund Value :
317 x 610 x $500 = $96.69MM
Delta is now 0.278
Hence need only $96.69MM x (1-0.278) = $70.87MM in
Fund
Need to sell an equiv. of:
($77.74-$70.87) = $6.87MM of Fund
Current Futures Value is $500 x 610 = $305,000
Hence sell $6.87MM/0.305MM = 23 contracts
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 27
Rebalancing Filter Rules
Continuous rebalancing too expensive
In practice, rebalance:
at specified time intervals
when portfolio value changes by specified
amount
by using band around correct hedge ratio
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 28
1987 and All That
Bull market was well into 5th year
Equity values at historical extremes
Many P/Es over 20, dividend yield below 3%
Investors regarded portfolio insurance as
cash substitute:
why sacrifice upside?
Fund under management increased 4x in 87
Over-estimated liquidity of stock market
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 29
October 19, 1987
Market fell by over 20%
$1trillion wiped off market value
As prices fell, insurers sold index futures
Futures were at massive discount to cash
More selling in the cash market by program
traders
Not enough liquidity in cash or futures markets
Hedging became impossible
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 30
Portfolio Insurance Today
Portfolio Insurance was blamed by
investors
Brady Commission said PI contributed
to severity
Much less popular today, although still
very widely used
Copyright 1998-2006 Investment Analytics
Portfolio Insurance
Slide: 31