CPPI strategies and guarantees on ULIPs

Avijit Chatterjee November 23, 2010

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Agenda

Simple CPPI strategies

More sophisticated algorithms

Reserving for guaranteed ULIPs

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Introduction

Highest NAV guaranteed ULIP structures have gained in popularity over last couple of years

Guarantee is often manufactured using the Constant Proportion Portfolio insurance (CPPI) algorithm

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Guarantee structure

NAV paid to policyholder

Guaranteed NAV

Guarantee increases as fund performs

Capital guarantee

Time

Year 7 Premium Payment Period Year 10

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Highest NAV during first 7 years

Closing NAV

Initial NAV 10

20

Year3

5

Equity + Debt = Assets

Equity / x + Debt = PV of guaranteed NAV

Investment strategy: CPPI

Current value Ensure

acceptable credit quality

Invest in liquid stocks

X = maximum sustainable fall between rebalancing

days

Duration management for reinvestment risk

Discounted at acceptable credit

curve

x/(x-1) = multiplier indicates level of leverage & speed of required reallocation; possible

range of 2-5 depending on risk appetite

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Equity + Debt = Assets

Equity / x + Debt = PV of guaranteed NAV

Investment strategy: CPPI

Daily monitoring and rebalancing of asset allocation Implication : As markets fall, equities are sold : With time, equity allocation reduces

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Managing CPPI - illustrations

Multiplier : 3 Interest rate : 8%

Remaining maturity (yrs) 10 5 5 Guarantee (Highest NAV) 10.00 25.00 25.00 Current NAV 10.00 25.00 17.50 PV of guarantee 4.63 17.01 17.01 Current NAV – PV of guarantee

5.37 7.99 0.49

Equity allocation 10.00 23.95 1.47

Scenario A

Scenario B

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Key risks and PRE

Mis-selling risk Product can be perceived as an equity product though the CPPI strategy exposes the customer to the risk of complete or partial lock-in to debt

Path dependence

If the strategy allocates 100% to debt due to sharp equity market fall, it cannot gain from any subsequent recovery High path dependence

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Key risks - insurer

Gap risk : If impact of falling equities and interest rates is more than the assumed sustainable fall, guarantee cannot be met

Key risk is drying up of liquidity in equity markets extending the exit period and increasing equity VaR Interest rate risk may be amplified by illiquidity of duration matched debt securities, especially if strategy assumes use of corporate bonds

Persistency Risk – Mass discontinuance if fund gets locked in debt

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Key risks – macro economic Estimated ULIP AUM under CPPI strategies

Rs 200 000 million as of today Rs 500 000 million in FY2013 (including renewal premium & market returns, excluding future new business)

Assuming average multiplier of 3 for these funds, a 10% fall in equities would trigger CPPI driven sales of Rs 90 000 million in FY2013

Average daily turnover in Nifty stocks is Rs 100 000 million

Large AUM in CPPI can potentially destabilize the markets by creating a self fulfilling cycle

Large fall triggers further CPPI driven sales causing further fall and further sales pressure Source : public disclosures by insurers,

internal estimates

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Agenda

Simple CPPI strategies

More sophisticated algorithms

Reserving for guaranteed ULIPs

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CPPI with volatility cap

Algorithm Hold a mix of equities and cash, and maybe bonds The mix is decided such that the volatility of the fund is capped at some pre-determined level Weight of equities and cash changes with underlying equity volatility

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CPPI with volatility cap

Underlying assumptions Equity market crashes are generally preceded or accompanied by heightened volatility Reducing equity exposure during such periods would reduce probability of shortfall While reducing the cost of guarantees, volatility capping strategy still allows subsequent exposure to equities when market conditions allow

Possible fallacies

A market crash triggered by an event like 9/11 or the 1992 Harshad Mehta scam may not be preceded by increased volatility

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CPPI with volatility cap

Macroeconomic consequences Can be further exaggerated as the speed of equity de-allocation increases as the volatility increases and as market prices fall Any trading algorithm is dangerous if too many sections of market follow it

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Black Monday - October 19, 1987

Crash was the climactic culmination of a market decline that had begun five days earlier DJIA fell 3.81% on Oct 14, 4.60% on Oct 16, 22.6% Oct 19;

cumulatively losing over 31%

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Black Monday - October 19, 1987

Apart from fundamental factors of overvaluation, programme trading & portfolio insurance were blamed

Portfolio insurance, like a put option, was supposed to limit the losses from a declining market by reducing weight on stocks during falling markets

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Black Monday - October 19, 1987

Portfolio insurers started Monday with an overhang of unexecuted sell orders from the accelerating decline of previous week

As the decline deepened, this backlog also deepened, cascading the sales Despite the fact that all the model generated sales could not be executed, 40% of non-market-maker sales in the futures market were conducted by portfolio insurers

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More alternatives

CPPI with a dynamic multiplier where the multiplier falls as the cushion falls Option replication strategies

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Agenda

Simple CPPI strategies

More sophisticated algorithms

Reserving for guaranteed ULIPs

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Traditional Method

Traditional method of calculating cost of guarantee Assume investment return with risk premiums Generate stochastic simulations Assess cost of guarantee in each simulation Discount

The cost of guarantees (CoG) is the expected liability

Shortfall in each simulation has a floor of zero Capital needs to be held for any mismatch

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CoG Example – Traditional Method

Unit Linked Product Single Premium of Rs 1,000 5 year period Assuming no charges or expenses Guaranteed return of 5% annually compounded, applicable at maturity Guarantee backed by same asset as unit reserve

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CoG Example – Traditional Method

Model parameters Equity Model – Lognormal with Geometric Brownian Motion Interest Rate Model – Hull White one factor model Risk Free Rate = 8% Equity Volatility = 15% Debt Volatility = 4%

Asset Class CoG Unit Reserve invested in Equity Equity Risk Premium = 3%

51.03

Unit Reserve invested in Debt 31.78 Unit Reserve invested in Equity Equity Risk Premium = 5%

29.33

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Market Consistent Approach

Principle is ‘arbitrage free pricing’ Construct a portfolio to replicate the payout under the guarantee, in all circumstances Value the guarantee as the value of the replicating portfolio Same principles as are used in option pricing

In developed countries with liquid and deep financial markets, the match can be nearly perfect Capital needs to be held for any mismatch

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CoG Example – MC Method

Using same volatilities and same risk free rate as before Cost is dependent on risk free rate and volatility Can in theory calibrate these to observed market parameters =>less subjectivity Will always show that if the underlying asset is debt, the CoG is lower than the case where underlying asset is equity

Assuming debt is less volatile than equity

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CoG Example – MC Method

106.55 29.33 Unit Reserve invested in Equity Equity Risk Premium = 5%

31.78 31.78 Unit Reserve invested in Debt

106.55 51.03 Unit Reserve invested in Equity Equity Risk Premium = 3%

MC CoG Traditional CoG

Asset Backing Guarantee

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CoG Example – MC Method

Asset Backing Guarantee MC CoG Unit Reserve invested in Equity Volatility = 15%

106.55

Unit Reserve invested in Equity Volatility = 30%

382.63

There remains considerable scope for judgement

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CoG reserves - conclusions

MC methods should be more objective though there remains considerable scope for judgement

MCEV, IFRS 4 Phase II and Solvency II adopting some form of fair value for liability measurement, in particular for embedded derivatives In the Indian context, with no observable long term implied rates, parameterisation is quite subjective

MC COG recognises risks and is therefore good guide for management actions Black Scholes equation relies on assumption of continuity, i.e. no ‘gap risk’

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CoG reserves - conclusions

In the context of CPPI strategies, continuous models will reveal zero cost of guarantee To get a cost of guarantee, need to use a jump model Question: how to calibrate it? Or should you use deterministic jump stresses to assess a capital requirement?

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CoG reserves - conclusions Insurer needs to recognise that large scale hedging will have destabilising effect on markets

Measuring the liability as an option will encourage hedging While improving the risk management and therefore policyholder protection at the insurer, it brings more risks into financial markets, potentially destabilizing it

Macro economic paradox – more risk management leads to more risk

This is traditionally very different from the macroeconomic function of life funds

Move from buy and hold approach to trading approach

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Thank you

All the views and opinions expressed here are personal views of the speaker and may not be the views or policies of ICICI Prudential Life Insurance Company.

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One period Binomial Model

u is the up factor and d is the down factor, u>d

S0

S1(H)=uS0

S1(T)=dS0

u=S1(H)/S0 d=S1(T)/S0

Shreve S. E. (2004). Stochastic Calculus for Finance I - The Binomial Asset Pricing Model

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One period Binomial Model (contd.)

Let r be the risk free interest rate The model is based on arbitrage free pricing Therefore, we may assume 0<d<1+r<u

Let K be the strike price A derivative security is a security that pays some amount V1(H) in the up movement and V1(T) in the down movement European put option

V1(T) = Maximum (0, K-S1(T)), i.e. Maximum (0, K-dS0) V1(H) = 0

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Arbitrage free pricing

)SX)(r1(SX 000101 Δ−++Δ=We want to choose X0 and such that

X1(H) = V1(H) and X1(T) = V1(T) Replication of the derivative security thus requires that

0Δ

)())(1()( 100010 HVSXrHS =Δ−++Δ

)())(1()( 100010 TVSXrTS =Δ−++Δ

S1 and V1 are known in terms of K, u and d. Solve for X0 and Δ0

Replicate the option by trading in stock and money markets Let X0 be initial wealth, and let be the shares of stock to be bought The value of our portfolio of stock and money market at time one is

0Δ

and

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Arbitrage free pricing (contd.)

Let )p1(qand

dudr1p

~~~−=

−−+

=

⎥⎦⎤

⎢⎣⎡ +

+= )T(Vq)H(Vp

r11X 1

~

1

~

0

)T(S)H(S)T(V)H(V

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110 −

−=Δ

and are ‘risk neutral’ probabilities Delta hedging formula

The Delta hedge will be negative for a put option, i.e. hold short position in underlying stock

~p

~q

Then

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Multi-period Binomial Model

VN be a derivative security paying off at time N VN is dependent on the first N steps N54321 ...ωωωωωω

⎥⎦⎤

⎢⎣⎡ ωωωω+ωωωω

+=ωωωω ++ )T...(Vq)H...(Vp

r11)...(V n3211n

~

n3211n

~

n321n

)T...(S)H...(S)T...(V)H...(V

)...(n3211nn3211n

n3211nn3211nn321n ωωωω−ωωωω

ωωωω−ωωωω=ωωωωΔ

++

++

)SX)(r1(SX nnn1nn1n Δ−++Δ= ++

The wealth equation:

If we set X0=V0

)...(V)...(X n321nn321n ωωωω=ωωωω

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Continuous time model

)(xΦ

Black Scholes formula: V0=V(s,T)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

σ

⎟⎠⎞

⎜⎝⎛ σ++−

Φ−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

σ

⎟⎠⎞

⎜⎝⎛ σ−+−

Φ= −

T

)T21r

ks(log

sT

)T21r

ks(log

ke)T,s(V

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rT

Where is the probability that N(0,1) has value less than x s = S0 , k = strike price of the option, T = exercise date, r = risk-free rate and is the volatility σ

Cash Stock

Baxter M., & Rennie A. (1996). Financial Calculus – An introduction to derivative pricing