investment guarantees the new science phần 6 pdf

There is, therefore, no survival benefit due, and the guarantee value is increased forthe renewed 10-year contract to the month-end fund value, $158.99.. After the guarantee has been rese

141–142 107.01 0.0446 12.339% 119.91 158.99 39.08142–143 119.91 0.0500 1.251% 121.11 158.99 37.88143–144 121.11 0.0505 1.206% 122.26 158.99 36.73144–145 158.99 0.0662 1.649% 155.98 158.99 3.01145–146 155.98 0.0650 4.362% 162.38 158.99 0

end-0 427 percent,leading to an end-year fund of $99.32 This is still greater than thecurrent guarantee of $80, so there is no guarantee liability for death benefits

0

0

1 1

after expenses is $99.75, which earns a return of I = .

All through the first two years, the fund exceeds the guarantee at the end

of each month At the end of the 24th month the first renewal date applies

In this scenario 158 99, compared with the guarantee of $80 There

is, therefore, no survival benefit due, and the guarantee value is increased forthe renewed 10-year contract to the month-end fund value, $158.99

In the 10 years following the first renewal under this single stock returnscenario, the index rises very slowly After the guarantee has been reset tothe fund value, the fund value drifts below the new guarantee level, leaving

a potential death benefit liability In fact, over the entire 10-year period theaccumulation is only 3.8 percent Since expenses of 0.25 percent per monthare deducted from the fund, by the end of 144 months the fund has fallen

$36.73 below the guarantee that was set at the end of 24 months

At the second renewal, then, the insurer must pay the difference to makethe fund up to the guarantee, provided the policy is still in force Therefore,

at the start of the 145th month the fund has been increased to the guaranteevalue of $158.99

Since the fund was less than the guarantee at the renewal date, theguarantee remains at $158.99 for the final 10 years of the contract Afterthe 145th month the fund is never again lower than the guarantee value,and there is no further liability However, the risk-premium portion ofthe management charge continues to be collected at the start of eachmonth In Table 6.4, we show the liability cash flows under this particularscenario

Each month a negative cash flow comes from the income from therisk-premium management charge The amount from the third column ofTable 6.3 is multiplied by the survival probability for the expectedcash flow

) is greaterthan zero at the month end For example, if the policyholder dies in the

)

$9.05 Since we allow for mortality deterministically, we value this deathbenefit at the month end by multiplying by the probability of death inthe 26th month, , which is an expected payment of $0.00273 Theprobability of the policyholder’s surviving, in force, to the second renewaldate is 0 35212, and the payment due under the survival benefit

is $36.73, leading to an expected cash flow under the survival benefit of

36 73 $12.93

In the final column, the cash flows from the th month are discounted

to the start of the projection at the assumed risk-free force of interest of 6percent per year The management charge income is discounted from thestart of the month, and any death or survival benefit is discounted fromthe end of the month

x t

t

d x

26

( ) 25 ( )

144

( )

144

A death benefit liability arises in months for which (G– F

26th month, the death benefit due at the month end would be (G– F

In-Force Mortality Expected Expected

Probability Probability Death Survival

For this example scenario, the net present value (NPV) of the guaranteeliability is $2.845 The contribution of the death benefit guarantee is $1.338,and the survival benefit expected present value is $6.295 The managementcharge income offsets these expenses by $4.788

In fact, this example is unusual; in most scenarios there is no survivalbenefit at all, and the management charge income generally exceeds theexpected outgo on the death benefit, leading to a negative NPV of theguarantee liability

The NPV of the Liability

STOCHASTIC SIMULATION OF LIABILITY CASH FLOWS

One method of summarizing the output is to look at the simulated NPVsfor the liability under each simulation As an example, we have repeatedthe GMAB example above for 10,000 simulations, all generated using thesame stock return model The range of net liability present values generated

The principle of stochastic simulation is that the simulated empiricaldistribution function is taken as an estimate of the true underlying distribu-tion function This means that, for example, since 8,620 projections out of10,000 produced a negative NPV, the probability that the NPV is negative

is estimated at 0.8620 We can, therefore, generate a distribution function

˜for the NPVs Let ( ) denote the empirical distribution function for theNPV at some value Then

Number of simulations giving NPV

˜ ( )

10 000This gives the distribution function in Figure 6.1

It may be easier to visualize the distribution from the simulated densityfunction The density can be estimated from the distribution using theprocedure:

Partition the range of the NPV output into, say, 100 intervals, indicated

by ( ) The intervals do not have to be equal; for bestresults use wider intervals in the tails and smaller intervals in the center

of the distribution

The estimated density function at the partition midpoints is

˜( ) ˜( )

˜2

is – $24.6 to $37.0 The number of NPVs above zero (implying a raw loss

on the contract) is 1,380 The mean NPV is– $4.0

Simulated distribution function for GMAB NPV example.

Stochastic Simulation of Liability Cash Flows

Altering the partition will give more or less smoothness in the function Thesimulated density function for the 10,000 simulations of the GMAB NPV ofthe liability is presented in the first diagram of Figure 6.2; in the right-handdiagram we show a smoothed version

The density function demonstrates that although most of the tion lies in the area with a negative liability value, there is a substantial righttail to the distribution indicating a small possibility of quite a large liability,relative to the starting fund value of $100 We can compare the distribution

distribu-of liabilities under this contract with other similar contracts—for example,with a two-year contract with no renewals, otherwise identical to thatprojected in Figures 6.1 and 6.2

A set of 10,000 simulations of the two-year contract produced a range

when renewals are taken into consideration Thus, at first inspection itlooks advantageous to incorporate the renewal option—after all, if thecontract continues for 20 years, that’s a lot more premium collected withonly a relatively small risk of a guarantee payout But, when we take riskinto consideration, the situation does not so clearly favor the with-renewal

of outcomes for the NPV of the liability of – $1.6 to $37.1, comparedwith – $24.6 to $37.0 for the contract including renewals The mean ofthe NPVs under the two-year contract is – $0.30 compared with – $4.00

0.05 0.10 0.15

In addition to the NPV, which is a summary of the nonfund cash flows forthe contract, we can use simulation to build a picture of the pattern of cashflows that might be expected under a contract In the GMAB example, thenonfund cash flows are the management charge income, the death benefitoutgo, and the maturity benefit outgo Any picture of all three sources isdominated by the rare but relatively very large payments at the renewaldates In Figure 6.3, we show 40 example projections of the cash flowsfor the GMAB contract The income and the death benefit outgo are onthe same scale, but the maturity benefit outgo is on a very different scale.For this contract, the death benefit rarely exceeds the management charge

An interesting feature of the death benefit outgo is the fact that the largerpayments increase after each renewal As the guarantee moves to the fundlevel, both the frequency and severity of the death benefit liability increase

In most projections there is no maturity benefit outgo, but when there is aliability, it may be very much larger than the management charge income.The cash flows plotted allow for survival and are not discounted

This type of cash-flow analysis can help with planning of appropriateasset strategy, as well as product design and marketing We can also examinethe projections to explore the nature of the vulnerability under the contract.For a simple GMMB with no resets or renewals, the risk is clearly thatreturns over the entire contract duration are very low For the GMAB, there

is an additional risk that returns start high but become weaker after thefund and guarantee have been equalized at a renewal date By isolating the

MODELING THE GUARANTEE LIABILITY

0 50 100 150 200 250 0.0

0.10

0 50 100 150 200 250 0.0

5 15

Simulated projections of nonfund cash flows for GMAB contract

Stochastic Simulation of Liability Cash Flows

stock return projections for those cases where a maturity benefit was paid,

we may be able to identify more accurately what the risks are in terms ofthe stock returns

In Figure 6.4, we show the log stock index for the simulations leading

to a maturity benefit at the first, second, and third renewal date In the finaldiagram we show 100 paths where there was no maturity benefit liability.The risk for the two-year maturity benefit is, essentially, a catastrophicstock return in the early part of the projection This is simply a two-yearput option, well out-of-the-money because at the start of the projection theguarantee is assumed to be only 80 percent of the fund value For the secondand third maturity benefits, the stock index paths are flat or declining,

on average, from the previous renewal date to the payment date For thiscontract the 10-year accumulation factor has a substantial influence onthe overall liability In addition, the two-year accumulation factor plays themajor role in the liability at the first renewal date The calibration procedurediscussed in Chapter 4 considers accumulation factors between 1 and 10years to try to capture this risk However, the right-tail risk is not tested inthat procedure

Stock Index for Early Maturity Benefit

0 50 100 150 200 250 0

2 4 6 Stock Index for Middle Maturity Benefit

2 4 6 Stock Index, no Maturity Benefit

is in four years Stocks have performed well, and the separate fund isnow worth, say, 180 percent of the guarantee If the same contract is stilloffered, the policyholder could lapse the contract, receive the fund value, andimmediately reinvest in a new contract with the same fund value but withguarantee equal to the current fund value The term to the next rolloverunder a new contract would generally be 10 years, so the policyholderreplaces the rollover in 4 years with another in 10 years with a higherguarantee

MODELING THE GUARANTEE LIABILITY

The Voluntary Reset

Reset Assumption Threshold 5% 25% 50% 75% 95%

Perhaps in order to avoid the lapse and reentry issue, many insurerswrote the option into the contract A typical reset feature would allowthe policyholder to reset the guarantee to the current fund value; the nextrollover date is, then, extended to 10 years from the reset date The number

of resets per year may be restricted, or the option may be available only oncertain dates

The reset feature can be incorporated in the liability modeling withouttoo much extra effort, although we need to make some somewhat speculativeassumptions about how policyholders will choose to exercise the option.The assumptions used to produce the figures in this section are describedbelow, but it should be emphasized that modeling policyholder behavior is

an enormous open problem

So, we adapt the GMAB contract described in the previous section toincorporate resets We assume the same true term for the contract, andthat the policyholder does not reset in the final 10 years We assumealso that the policyholder will reset when the ratio of the fund to theguarantee hits a certain threshold—we explore the effect of varying thisthreshold later in this section We also assume the effect of restricting themaximum number of resets each year The figures given are for a GMABwith a 10-year nominal term (between rollover terms, if the policyholderdoes not reset) and a 30-year effective term The starting fund to guaranteeratio is 1.0

In Table 6.5, some quantiles of the NPV distributions are given forthe various reset assumptions These result from identical sets of 10,000scenarios Figures are per $100 starting fund

This table shows that the effect of the reset option is not very large,although the right-tail difference is sufficiently significant that it should

be taken into consideration This will be quantified in Chapter 9 Theeffect of different threshold choices is relatively small, as is the choice inthe policy design of restricting the number of resets permitted per year,although that will clearly affect the expenses associated with maintainingthe policy Having a restricted number of possible resets does not mattermuch because infrequent use of the reset appears to be the best strategy

0 50 100 150 200 250 0

10 20 30

Simulated cash flows, with and without resets

Resetting every time the fund exceeds 105 percent of the guarantee may lead

to lost rollover opportunities, so that the contract may pay out less than thecontract without resets

From these figures it does not appear that the reset feature is all thatvaluable, on average, but the tail risk is significantly increased (as repre-sented by the 95th percentile) In addition, the reset will constrain the riskmanagement of the contract, for two major reasons The first is a liquidityissue—without the reset option, the maturity benefit is due at dates set atissue Allowing resets means that the maturity benefit dates could arise atany time after the first 10 years of the contract have expired This will makeplanning more difficult For example, in Figure 6.5 we show 50 simulatedcash flows from a contract without resets; then, with everything else equal,the same contract cash flows are plotted if resets are permitted, and athreshold of 105 percent is used as a reset threshold

The other problem with voluntary resets is that the option has theeffect of concentrating risk across cohorts Consider a GMAB policy written

in 2000 and another written in 2003 Without resets, there is a certainamount of time diversification here, because the first rollover dates for thesecontracts are 2010 and 2013, respectively, and it is unlikely that very poorstock returns will affect both contracts Now assume that both policies carrythe reset option and that stocks have a particularly good year in 2004 Bothpolicyholders reset at the end of 2004, which means that both now haveidentical rollover dates at the end of 2014, and the time diversification islost In the light of these problems, the voluntary reset feature is becomingless common in new policy design

For a more technical discussion of the financial engineering approach torisk management for the reset option see Windcliff et al (2001) and (2002)

MODELING THE GUARANTEE LIABILITY

n Chapter 1 we discussed how the investment guarantees of equity-linkedinsurance may be viewed as financial options Since the seminal work

of Black and Scholes (1973) and Merton (1973), the theory and practice ofoption valuation and risk management has expanded phenomenally Ac-tuaries in some areas have been slow to fully accept and implement theresulting theory Although some actuaries feel that the no-longer-new the-ory of option pricing and hedging is too risky to use, for contracts involvinginvestment guarantees it may actually be more risky not to use it

In this chapter, we revise the elementary results of the financial nomics of option or contingent claims valuation Many readers will knowthis well, and they should feel free to skip to the next chapter For read-ers who have not studied any financial economics (or who may be alittle rusty), the major assumptions, results, and formulae of the theory

eco-of Black, Scholes, and Merton are all discussed We do not prove any

of the valuation formulae; there are plenty of books that do so Boyle et

al (1998) and Hull (1989) are two excellent works that are well known

to actuaries

This chapter will demonstrate the crucial concepts of no-arbitragepricing with a simple binomial model Using this very simple model all ofthe major, often misunderstood, results of financial economics can be clearlyderived and discussed, including:

The ideas of valuation through replication

The difference between the true probability distribution for the riskyasset outcome (the -measure), and the risk-neutral distribution (the-measure), and why it is correct to use the latter when it is clearly notrealistic

The idea of rebalancing the replicating portfolio without cost

THE GUARANTEE LIABILITY AS A DERIVATIVE SECURITY

REPLICATION AND NO-ARBITRAGE PRICING

In the section on the Black-Scholes-Merton assumptions, later in thischapter, we write down the important assumptions underlying the theory

We then show how to determine the valuation and replicating portfolio for

a general uncertain liability, based on an underlying risky asset

In the final sections of the chapter, we give the formulae and methodsfor the options that arise in the context of equity-linked insurance Wefind in later chapters that knowing the formulae for European call and putoptions is surprisingly helpful for more complicated benefits

A European put option is a derivative security based on an underlying assetwith (random) value at If is the maturity date of the option and

guaranteed minimum maturity benefit (GMMB), where is the guarantee,

is the maturity date, and is the segregated fund value at , so the

)

In fact, all of the financial guarantees that were described in Chapter 1can be viewed as derivative securities, based on some underlying asset In thesegregated fund or variable-annuity (VA) contract, the underlying security

is the separate fund value Similarly to derivative securities in the bankingworld, financial guarantees in equity-linked insurance can be analyzed usingthe framework developed by Black, Scholes, and Merton

First, we give a very simplified example of option pricing, using a binomialmodel for stock returns, to illustrate the ideas of replication and no arbitragepricing

Suppose we have a liability that depends on the value of a risky asset.The risky asset value at any future point is uncertain, but it can be modeled

by some random process, which we do not need to specify

The no-arbitrage assumption (or law of one price) states that two tical cash flows must have the same value Replication is the process offinding a portfolio that exactly replicates the option payoff—that is, themarket value of the replicating portfolio at maturity exactly matchesthe option payoff at maturity, whatever the outcome for the risky as-set So, if it is possible to construct a replicating portfolio, then the price

iden-t

T

T T

A REVIEW OF OPTION PRICING THEORY

K ⱖ F or nothing if K< F This structure is identical to the standard

payoff under the guarantee is (K– F

is the strike price, then the put option pays at time T, either (K – F ) if

t

K S

a

t

a bS P

a bS

of that portfolio at any time must equal the price of the option at time ,because there can only be one price for the same cash flows

For example, suppose an insurer has a liability to pay in one month

an amount exactly equal to the price of one unit of the risky asset at thattime The amount of that liability at maturity is uncertain The insurermight take the expected value of the risky asset price in one month, usingsome realistic probability distribution, and discount the expected value

at some rate That method of calculation would be the traditional actuarialapproach The beautiful insight of no-arbitrage pricing says that such acalculation is essentially worthless in terms of a market valuation of theliability If the insurer buys one unit of the risky asset now, it will haveenough to precisely meet the liability due in one month If the liability isvalued at any amount lower or higher than the current price of one unit

of the risky asset, then an arbitrage opportunity exists that would quickly

be exploited and therefore eliminated So, the replicating portfolio is oneunit of risky asset, and the valuation is the price of one unit of risky asset.Replication and valuation are inextricably linked

To see how the theory is applied to a more complicated contingentliability, such as an option, we use a simple binomial model in which twoassets are traded:

A risk-free asset that earns a risk-free force of interest of 05 per timeunit, so an investment of 100 at time 0 will pay 100 at 1

A risky asset (or a stock) that pays 110 if the market goes upover one time unit, and 85 if the market goes down No otheroutcomes are possible in this simple model Assume that the time 0price of the risky asset is 100

Suppose we sell a put option on the stock The option gives the buyerthe right to sell the stock at a fixed price of, say, 100 at time 1 Thisright will be exercised if the stock price goes down, because in that case thepurchaser receives 100 under the contract compared with 85 in the market

If the stock price goes up, the purchaser can sell the asset in the market for

110 and, therefore, has no incentive to exercise the option and sell for only

15 if the market goesdown (since they have to buy the stock at but end up with an asset worthonly ) and 0 if the market goes up

Now assume the option seller buys a mixed portfolio of the risk-freeasset and the risky asset; the portfolio has units of the risk-free asset andunits of the risky asset, so its value at 0 is and at

1 its value is

if the market goes up

if the market goes down

r u d

d

d

r

u d

The situation is illustrated in Figure 7.1

Now, we can make the portfolio exactly match the option liability bysolving the two equations for and :

ⴱ ⴱ

ⴱ

ⴱ

ⴱ ⴱ

A very interesting feature of the result is that we never needed to know

or specify the probability that the stock rises or falls We have not used theexpected value of the payoff anywhere in this argument

In general, this binomial setup for the put option gives a price:

looks like a probability and the portfolio value looks like an expectedpresent value, because if we treat as the probability that the market falls

term discountsthe expected payoff to the time zero value at the risk-free force of interest

So, even though we have not used expectation anywhere, and even though

is not the true probability that the market falls, we can use the language

of probability to express the option as an expectation under this artificialprobability distribution

This illustrates the third concept of option valuation: the

Using the artificial probabilities for the down market

1 is

So under this artificial probability distribution, the expected value of

at 1 is the same as if the stock earned the risk-free rate of interest

) is known as the In financial economics literature, it isalso commonly known as the (measure is just used to meanprobability distribution) The real probability distribution for the stock

r u d

r d r u

Based on our results, we know that S < S e < S (since any other

ordering breaches the no-arbitrage assumption) so that 0< p < 1 Now

and (1 – p ) is the probability that the market rises, (C (1 – p )+C p )

is the expected payoff at t = 1 under the option, and thee

and (1– p ) for the up market, the expected value of the risky asset at time

This is why the probability distribution p and (1 –p

Q only

price (which we have not needed here) is known as “nature’s measure,” the

“true measure,” or the “subjective measure,” but is always shortened in thefinance literature to the

The difference between the and probability distributions is veryimportant, and is the source of much misunderstanding In particular, thetheory does assume that equities earn the risk-free rate of interest

on average, even though the -measure might give this impression The-measure is a device for a simple formulation for the price of an option

as an expected value, even though we are not using expectation to value

it but replication The -measure is therefore crucial to pricing, but also,crucially, is relevant to pricing and replication Any attempt to projectthe true distribution of outcomes for an equity-type fund or portfolio must

be based on an appropriate -measure Say we wanted to predict howfrequently the option in the binomial example above ends up in-the-money,which is the probability that the stock ends up in the “down” state, the-measure “down” probability is quite irrelevant to this frequency, andcan give us no useful information

The derivation of the risk-neutral measure from the market model, ingeneral, does require some information about the underlying -measure:The risk-neutral measure must be to the -measure Equiv-alence means (loosely) that the two measures have the same nullspace—or in simple terms, that all outcomes that are feasible under the-measure are also feasible under the -measure, and vice versa.The expected return on the risky asset using the -measure must beequal to the return on the risk-free asset

These two requirements are sufficient in the binomial example todetermine the risk-neutral probabilities The first requires that the onlypossible outcomes under the -measure are , the probability of moving

to the “up” state, and , the probability of moving to the “down” state.Clearly, under the first requirement,

The second requirement states that

(7 10)These equations together give the probability distribution in equation 7.8.Now we extend the binomial model above to two periods to illustrate the

of a liability

u d

The Risky Asset

The Option Liability

100 – 72.25

100 – 93.50 0

Two-period binomial model

Replication and No-Arbitrage Pricing

We keep the same structure so that, over each time period, the price

of the risky asset rises by 10 percent or falls by 15 percent, and we make

no assumptions about the relative probabilities of these events The stockworth 100 at 0 then follows one of the paths in the top diagram ofFigure 7.2

Now consider a put option that matures after two time units The strikeprice is 100, giving a liability at the end of the period of 0 if the stockhas risen in both time units, 6.50 if it has risen once and fallen once, and27.75 if the stock price fell in both time units We can replicate the optionpayoff in this model by working backwards through the various paths Theidea is to break the two-period model down into two one-period models

At time 1 we know if we are in the up state or the down state If we are inthe up state, then we need a portfolio

(7 11)which will exactly meet the liabilities after the next time step, that is:

,