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Abstract

This Netspar Panel Paper discusses the pricing of contracts in an

incomplete market seng. For life insurance companies and

pension funds, it is always the case in pracce that not all of the

risks in their books can be hedged. Hence, the standard

Black-Scholes methodology cannot be applied in this situaon. The

paper discusses and compares several methods that have beenproposed in the literature in recent years: the Cost-of-Capital

method (the current industry standard), Good Deal Bound pricing,

and pricing under Model Ambiguity. Although each of these

methods has a very different economic starng point, we show

that all three converge for small me-steps to the same limit. This

convergence provides a basis for comparing the different

parameters used by the three methods. From this comparison weconclude that the current cost-of-capital of % used by the

industry and CEIOPS is too low, since it is not in line with the values

implied by the Good Deal Bound and Model Ambiguity methods. A

cost-of-capital of % is needed to bring the method in line with

the other two methods.


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. Management Summary & Policy Recommendaons

Life insurance companies and pension funds have liabilies on

their books with very long-dated maturies. The valuaon and

risk-management of these very long-dated contracts is therefore

an important problem in pracce.

The standard theory (based on replicang the cash flows) fails

because there are simply no financial contracts that last this long.

In well-developed economies (such as the euro-zone countries and

the US) government bonds have maturies up to years.On the other hand, regulators in many countries (especially in

Europe under the Solvency II project) are insisng that insurance

companies (and in The Netherlands, also pension funds) value

their liabilies on a market-consistent basis. Hence, to value

these long-dated cash flows in a market-consistent way, one is

forced to extend the term-structure of interest rates, which can be

observed from financial markets, beyond the maturity of thelongest dated instrument that can be observed in the market. In

the current economic circumstances, with low long-term interest

rates, pension funds are reporng low funding levels as a

consequence of these valuaon rules. A related issue is how to

select financial instruments that give the best possible investment

strategy (or hedge) for these very long-dated cash flows. In many

cases this involves striking a balance between seeking assets with ahigher return, at the expense of accepng a higher mismatch risk

between the liabilies and the assets.

From a scienfic point of view, the problem of pricing these very

long-dated contracts boils down to the valuaon of contracts in an

incomplete markets seng. This means trying to price contracts

where not all of the risks can be traded (and hedged) in financial

markets. In the past ten years significant progress has been made


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regarding this subject. This Panel Paper discusses and compares

several methods that have been proposed in the literature: the

Cost-of-Capital method (the current industry standard), Good DealBound pricing, and pricing under Model Ambiguity. We show that

each of these three methods converges for small me-steps to the

same limit. This convergence provides a basis for comparing the

different parameters used by the three methods.

The results presented in this paper allow us to provide the

following policy recommendaons:

The Cost-of-Capital method proposed by the insurance

industry and CEIOPS (i.e. the market-consistent price of an

insurance contract, which is determined by the market

value of the replicang porolio, plus a mark-up for the

unhedgeable risks: the risk margin; see EIOPA () for

further details) has qualitavely the right properes, but

lacks a solid theorecal foundaon. A pricing method with arigourous theorecal foundaon can be obtained by using

the pricing methods put forward in this paper.

In parcular, the formulas for calculang market-consistent

prices for mul-year productsas put forward by EIOPA in

QISlack a theorecal basis, and should be seen as a

coarse approximaon at best. The main problem is that theproposed QIS-methodology is not me-consistent. We

recommend that CEIOPS adopts a me-consistent pricing

method based on backward-inducon calculaon

techniques.

The me-consistent pricing method proposed in this paper

calculates prices under an actuarially prudent model,


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where (for each me-step) the best-esmate mean is

adjusted by k mes the standard deviaon of the

unhedgeable risk of the whole porolio. The Good DealBound approach implies k> 0.25, the Model Ambiguity

approach implies k 0.30, and the Cost-of-Capitalapproach implies k = 0.15. The values k for the first two

approaches are in line with each other, but the value

implied by the Cost-of-Capital method seems too low. A

value ofk = 0.30 is needed to bring the Cost-of-Capital

method in line with the other two methods, whichcorresponds to a cost-of-capital of % (instead of the %

currently proposed by the industry and CEIOPS).

Regulators are parcularly vulnerable to model risk. When

the regulator puts forward a very explicitly specified

standard model (as is currently happening under

Solvency II), then compeve market forces will ensure thatmost of the risk accumulates at the weakest point of the

regulators model. To guard against this model risk, we

propose that the regulator adopts a robust approach to

model risk. This can be achieved by pung forward several

alternave models, and the industry should then calculate

Solvency Capital on the basis of the worst outcome under

the different models.


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. Introducon

Life insurance companies and pension funds have liabilies on

their books with very long-dated maturies. Most people start

saving for their pension from age , and people are expected to

live to age , with the oldest people living to age . Hence,

pension funds and life insurance companies are facing contractual

obligaons that can easily last yearsand somemes even or

yearsinto the future. The valuaon and risk-management of

these very long-dated contracts is therefore an important problem.To give a feel for the size of the problem: for life-insurance and

pension products, a poron of roughly % of the net present

value of the cash flows is located in the tail of + years.

The standard theory (based on replicang the cash flows) fails

because there are simply no financial contracts which last this

long. In well-developed economies (such as countries in the

euro-zone and the US), the longest government bonds havematuries up to years. In developing countries (such as Eastern

Europe, and Lan America and Asia), government bonds are issued

with much shorter maturies (typically only up to ten years, and

somemes even much shorter).

On the other hand, regulators in many countries (especially in

Europe under the Solvency II project) are insisng that insurance

companies (and in The Netherlands, also pension funds) valuetheir liabilies on a market-consistent basis. Hence, to value

these long-dated cash flows in a market-consistent way, one is

forced to extend the term-structure of interest rates, which can be

observed from financial markets, beyond the maturity of longest

dated instrument that can be observed in the market. In the

current economic circumstances, with low long-term interest rates,

pension funds are reporng low funding levels as a consequence of


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these valuaon rules. A related issue is how to select financial

instruments that give the best possible investment strategy (or

hedge) for these very long-dated cash flows. In many cases thisinvolves striking a balance between seeking assets with a higher

return, at the expense of accepng a higher mismatch risk

between the liabilies and the assets.

Pricing calculaons serve mulple purposes. One of these is

price-seng, which involves the calculaon of the amount of

money for which a contract can be sold to a customer. A second

purpose has to do with pricing calculaons used as a basis forcorporate policy. This involves determining what the profit margin

is for each contract sold. Alternavely, by determining for which

price the profit is equal to zero, an instuon can find the

minimum price at which a product sll can be sold profitably.

These types of calculaons are typically made when new products

are being introduced by the instuon. Third, pricing calculaons

are done for reporng and capital adequacy purposes. In this case,one uses the pricing calculaons to (re)calculate the value of all

assets and liabilies in the balance sheet based on current

economic circumstances. As a result, one can then determine the

surplus (or the coverage rao) of assets versus liabilies. In

pracce, different calculaon methods are oen applied for the

different pricing purposes. Ideally, one should use the same

calculaon methodology for all applicaons in order to ensure

internal consistency.

From a scienfic point of view, the problem of pricing very

long-dated contracts boils down to the valuaon of contracts in an

incomplete markets seng. This means that we are trying to price

contracts where not all of the risks can be traded (and hedged) in

financial markets. In the past ten years significant progress has


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been made regarding this subject. Several approaches have been

invesgated with the common goal of trying to idenfy a pricing

measure (or pricing kernel) that prices traded risks consistentlywith prices observed in the market and that also includes an

extension for non-traded risks. The big problem is how to

construct such an extension in a sensible way.

This panel paper first discusses the Cost-of-Capital (CoC) method

proposed by the insurance industry. This method has become the

de facto industry standard, which has also been adopted by the

European Union for the Quantave Impact Studies (QIS) in theSolvency II process. The idea behind the CoC method is that the

insurance company has to hold a buffer for the non-hedgeable

risks on top of the replicang porolio. Hence, pricing consists of a

best-esmate term plus a mark-up for the non-hedgeable risks.

We discuss how to construct a me-consistent extension of the

CoC methodology, and we derive an equaon for how to calculate

CoC prices.A second approach discussed in this paper, is the Good Deal

Bound (GDB) method. The GDB approach looks at the risk/return

trade-off of non-hedgeable risks. This risk/return trade-off for the

non-hedgeable risks is then compared to the risk/return trade-off

that we can observe for traded assets (where it is called the market

price of risk). The GDB method then calculates prices for

non-hedgeable assets by making sure that the risk/return trade-off

for any asset does not exceed a given upper bound. This upper

bound is put on the prices, under the assumpon that economic

agents will exploit trading opportunies that are too good

(i.e. have a risk/return trade-off that is too high).

The third approach discussed in this paper is based on model

ambiguity and robustness. Although this methodology has been


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widely used in engineering for decades, it has aracted aenon

in economics only in recent years. (See, for example, the book

Robustness by Hansen and Sargent ().) The fundamentalpremise in the robustness approach is that we are uncertain about

the correct specificaon of our model. Therefore, when we try to

make decisions (like pricing and hedging a liability) we explicitly

want to take the model-uncertainty into account. This can be

implemented mathemacally as follows. First, specify a set of

alternave models to the current base model. Then assume that

we are playing against a malevolent mother nature that tries topick the worst possible model out of the set of alternave models

(given the decisions we have commied to). Since we are,

however, aware of this, we try therefore to make decisions that are

as resilient as possible given the worst-case acons of mother

nature.

This Panel Paper shows that each of these three approaches

converges for small me-steps in the limit to the same pricingequaon. This is illustrated with several examples. Unfortunately,

most of the academic literature discussed in this paper is wrien in

rather abstract mathemacal language, making the results very

difficult to access for non-technical readers. One of the

contribuons that this Panel Paper hopes to make is to present the

results in a more intuive way.

The remainder of this paper is organised as follows. Secon

briefly recalls the results of how to calculate prices in a complete

market seng. Secon then analyses the other extreme, an

incomplete market seng when we only have risks that are not

traded in a market. In this seng we derive our main results about

the mathemacal equivalence of the three pricing methods under

consideraon. Secon considers the case in which we have both


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types of risks (traded and non-traded), and we show how the

results from the previous secon generalise in this case. Finally,

Secon shows some applicaons of the pricing methods we havedeveloped.


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. Pricing in Complete Markets

This secon provides an overview of the theory of pricing payoffs

in complete markets. In a complete market, every risk driver can be

traded in a marketand every risk can thus be hedged. In the case

of complete markets, every payoff can be priced explicitly using

arbitrage-free pricing.

. Binomial Tree

To illustrate the main ideas, we use a simple mathemacal seng.

We have a risk driver Wx(t), which is a Brownian Moon. We alsoassume an asset price process x(t), which is given by the diffusion

equaon

dx = m(t, x) dt + (t, x) dWx, (.)

where m(t, x) and (t, x) denote the dri and diffusion of the

return process x(t). We assume that x(t) can be traded in a

market.

For example, if we model a stock price S(t) as x(t), and we setm(t, x) = x and (t, x) = x, then we recover the famous

Black and Scholes () model.

We also assume a riskless asset B, which earns the risk-free

interest rate r. The value of the riskless asset is given by

dB = rB dt. (.)

We wish to consider a discresaon scheme for the returnprocess x(t) for the me period [t, t +t] in the form of a

binomial tree:

x(t+t) = x(t)+mt+

{+t with prob. 1

2

t with prob. 12 ,(.)

where we have suppressed the dependence ofm(t, x) and

(t, x) on (t, x) for ease of notaon.


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. Pricing by Replicaon

Suppose we have a derivave f(t, x) that has a payoff that

depends on x. Suppose that we know the price of the derivave atme t +t for any value ofx(t +t); i.e. we know the

funcon f(

t +t, x(t +t))

. The queson is: how do we

determine the value for fone me-step earlier at me t?

The answer to this queson was developed by Fisher Black,

Myron Scholes and Robert Merton in the early s. They used

the noon ofpricing by replicaon, which won Scholes and Merton

the Nobel Prize in economics in . The idea works as follows.Suppose we buy a porolio ofDunits of the risky asset x and an

amount B invested in the risk-free asset. Then at me t this

porolio has value (Dx(t) + B). At me t+tthe porolio has

two possible values (using the binomial discresaon (.))

{Dx+ + e

rtB with prob. 12

Dx + ert

B with prob.12 ,

(.)

where x is shorthand notaon for x := x(t) + mtt.

Given the binomial discresaon for x(t +t), the derivave

f() has two possible values at me t +t: either

f+ := f(t +t, x+) or f := f(t +t, x). If we want to

match the values of our porolio (Dx(t) + B) with the value of

our derivave fat me t +

t, we have to solve the followingsystem of equaons:{Dx+ + e

rtB = f+Dx + e

rtB = f.(.)

The soluon is given by D = f+fx+x

and B = ert fx+f+xx+x

.

We have now explicitly constructed the replicang porolio for

the derivave f(). The brilliant insight of Black, Scholes and


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Merton was that the price of the derivave f(t, x) at me t must

be equal to the price (Dx(t) + B) of the replicang porolio. If

this would not be the case, there would be an arbitrageopportunity: two different prices for two instruments that have

exactly the same value at me t +t. Therefore, we calculate

the value at me t of the derivave f(t, x) by evaluang

(Dx(t) + B):

f(t, x) = 121 rt

m(t,x)rx

(t,x)t f

++

12

1 rt +

m(t,x)rx(t,x)

t

f. (.)

The term m(t,x)rx(t,x)

measures the excess return above the risk-free

rate of the risky asset divided by the standard deviaon of the risky

asset. This rao is known as the market price of risk, which will be

denoted by

(t, x) :=m(t, x) rx

(t, x). (.)

The market price of risk is a posive quanty, as the return

m(t, x) on a risky asset is larger than the return rx on a risk-free

asset. The market price of risk will turn out to be quite crucial in

the rest of our story.

. Deflator Pricing

The binomial pricing equaon (.) admits several different

interpretaons. The first interpretaon worth nong is the

Please note that the equality is not exact, as we have omied terms of higher

order thant. On the other hand, the binomial approximaon (.) of the pro-

cess x is also not exact. However, when we consider the limit fort

0 then

all of the approximaons converge to the correct answer.


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interpretaon as a pricing operator with respect to a deflatoror

pricing kernel.

Interpret (.) as taking an expectaon, using the originalbinomial probabilies 1

2and 1

2, of the adjusted derivave values(

1 rt (t, x)t) f+ and(1 rt + (t, x)

t

)f. The adjustment factor is different

for the plus and the minus state of the world; hence, the

adjustment factor is a random variable. In fact, we can interpret

the adjustment factor as the binomial discresaon of the random

variable (t), which is given by

d = r dt (t, x) dWx. (.)

The random variable (t) is known as the deflatoror the pricing

kernel. Note that the volality of the pricing kernel is equal to

minus the market price of risk

(t, x). The minus sign indicates

than whenever Wx(t) decreases, then (t) increases. This hasthe effect of pung more weight on bad outcomes of the

process x (i.e. low values ofWx) than on good outcomes (high

values ofWx).

Using the deflator interpretaon, re-write the pricing equaon

(.) as

f(

t,

x) =

Et[(t +t)f(t +t)]

(t) ,(.)

where Et[ ] denotes the expectaon operator condional on the

informaon available at me t, in parcular the informaon that

the process x(t) at me t is equal to the value x.

. Risk-Neutral Pricing

An alternave interpretaon of the pricing equaon (.) is as a

discounted risk-neutralexpectaon. Instead of using the original


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binomial probabilies and adjusng the payoff (as was done in

Secon .), we can adjust binomial probabilies and leave the

payoff unchanged. When doing this, we must ensure that the newprobabilies are created sll sum to . The adjusted binomial

probabilies are given by 12

(1 rt (t, x)t) and

12

(1 rt + (t, x)t). However, when these two numbers

are added together we get (1 rt), which is less than . Anelegant way to adjust the weight-factors is by re-wring them as

ert12(1

(t, x)t) and ert12(1 + (t, x)

t). Now

re-write the pricing equaon (.) as

f(t, x) = EQt[

ertf(t +t)]

, (.)

where EQt [ ] denotes the condional expectaon operator with

respect to the adjusted binomial probabilies

q = 1

2(1 (t, x)

t) (.a)

1 q = 12

(1 + (t, x)

t

)(.b)

for the plus and minus state, respecvely.

Like in Secon ., the adjusted probabilies qand (1 q) putmore weight on the minus state compared to the original

binomial probabilies of 12

. In fact, the expectaon ofx(t +t)

is calculated using the adjusted binomial probabilies, we find that

EQt [x(t +t)] = x(t)e

rt. Hence, under the adjusted

probabilies, the process x(t) grows with the risk-free rate r,

which is lower than the true growth rate m(t, x) of the process

x(t).

Wrapping up this secon, we would like to stress that the pricing

formul (.) and (.) will give exactly the same outcome. They

Again, we are ignoring terms of higher order thant.


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are nothing more than different representaons of the same

binomial pricing equaon (.).

. Paral Differenal Equaon

Up unl now, we have focused extensively on the pricing of a

derivave contract for one single me-step [t, t +t]. Obviously,

we are ulmately interested in pricing contracts of the whole life

[0, T]. One method for converng a one-step pricing formula

into a whole interval pricing formula is to apply the one-step

pricing formula using a backward-inducon procedure. In otherwords, start at the end-date T and then move backward in me by

repeatedly applying the one-step pricing formula for each

me-stept. The backward-in-me nature of this algorithm

ensures that at each me t during the calculaon the valuaon

formula accounts for all of the remaining uncertainty unl the

maturity date T. Hence, use of backward-inducon makes it

possible to construct a pricing operator that is me-consistent.This subsecon considers the limit fort 0. Assume that

f(t +t, x) is sufficiently smooth in t and x, such that we can

apply for all values of(t, x) the Taylor approximaon

f

(t +t, x + h

)= f

(t +t, x

)+

fx(

t +t, x)

h +1

2 fxx(

t +t, x)

h2

+ O(h3

), (.)

where subscripts on fdenote paral derivaves. If we apply the

binomial approximaon (.) for the process x(t), this yields

f+ = f(

t +t, x + mt + t

)and

f = f(

t +t, x + mt t).If we substute the Taylor approximaon (.) into these

expressions and then substute into the one-step binomial pricing


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equaon (.), this yields

0 = f(

t +t, x) f(t, x) + rxfx(t +t, x)t +

122fxx

(t +t, x

)t rf(t +t, x)t +O(t2).

(.)

Note that due to the adjustment factors in the binomial pricing

equaon (.), the true growth rate m(t, x) has disappeared

from (.), and has been replaced by the risk-free growth rate rx

that mulplies the term fx(

t +t, x)t.

Now, divide byt and take the limit fort 0, which yields

ft + rxfx +122fxx rf = 0, (.)

where we have suppressed the dependence on (t, x) to lighten

the notaon. Equaon (.) is a paral differenal equaon (pde)

for the derivave price f(t, x). The price of any derivave on theunderlying process x(t) is a soluon to (.) with respect to a

boundary condion f(

T, x(T))

that defines the payoff as a

funcon ofx(T) at the maturity date T.

. Literature Overview

The literature on pricing in complete markets has been developed

and extended since the s. It started with the seminal papers

by Black and Scholes () and Merton (). The binomial treepricing model was developed by Cox et al. (). The connecon

to marngale measures was developed by Harrison and Kreps

() and Harrison and Pliska (). Significant generalisaons

were achieved for more general stochasc processes by Delbaen

and Schachermayer (). For an introducon to pricing

derivaves, see the textbook by Hull ().


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. Pricing non-hedgeable Risk

Secon considered the case of a complete marketone in whichthe underlying risk driver can be traded. This secon considers the

opposite case, where the underlying risk drivers cannotbe traded.

This makes it no longer possible to construct a replicang porolio,

which was the underlying basis for the pricing method in Secon .

Instead, we have to define a pricing operator to determine the

value of a payoff.

This has been the subject of study of actuaries for a long me.The basic idea for a pricing operator is to use the expected value of

the payoff minus a penalty term that depends on the risk of the

payoff. Many different pricing operators have been proposed (for

an overview, see Gerber (), Deprez and Gerber (), Young

(a) and the textbook by Kaas et al. ()). Actuaries make a

disncon between two main classes of pricing operators. One

class uses standard deviaon as a measure of risk, the other classuses variance as the measure of risk. This Panel Paper focusses on

pricing operators of the first class. Secon . provides a literature

overview of alternave pricing methods that belong to the second

class.

. Binomial Tree

Let us introduce a new risk driver Wy, which is a Brownian Moon.We also assume a process y(t), which is given by

dy = a(t,y) dt + b(t,y) dWy. (.)

Note that we assume that y(t) cannotbe traded in a market. Like

in Secon , here we also consider the binomial discresaon for


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the process y for a me-steptas

y(t+t) = y(t) + at+{

+bt with prob.1

2bt with prob. 12

. (.)

Furthermore, we want to consider a derivave g(t,y) which has a

payoff that depends on y.

. Cost-of-Capital Pricing

A pricing principle that is widely used in pracce is the

Cost-of-Capital (CoC) Principle. This was introduced by the Swissinsurance supervisor as a part of the method to calculate solvency

capitals for insurance companies (see, e.g. Keller and Luder, ).

In recent years, the CoC method has been widely adopted by the

insurance industry in Europe, and has also been prescribed as the

standard method by the European Insurance and Pensions

Supervisor for the Quantave Impact Studies (see EIOPA, ).

The CoC approach is based on the following economic reasoning.First consider the expected loss E[g(T,y)] of the insurance

claim g(T,y) as a basis for pricing. But this is not enough; the

insurance company also has to hold a capital buffer against the

unexpected loss. This buffer is calculated as a Value-at-Risk over

a me horizon (typically one year) and a probability threshold q

(usually ., or even higher). The unexpected loss is then

calculated as VaRq[g(T,y)]. The capital buffer is borrowed fromthe shareholders of the insurance company (i.e. the buffer is

subtracted from the surplus in the balance sheet). Given the very

high confidence level, in many cases the buffer can be returned to

the shareholdersalthough there is a chance that the capital

buffer is needed to cover an unexpected loss. Hence, the

shareholders require compensaon for this risk in the form of a

cost-of-capitalpremium. This cost-of-capital premium needs to be


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included in the pricing of the insurance contract. If we denote the

cost-of-capital by , then the CoC pricing equaon is given by

g(t,y) = er(Tt) (Et[g(T,y)] + VaRq,t[g(T,y)]). (.)

.. Time-Consistency

The pricing method defined in equaon (.) has a methodological

problem: it is defined for a one-year horizon (i.e. t = T 1). Animportant praccal queson is, how to extend the pricing formula

to longer horizons? The approach adopted by the industry is a

simple rule-of-thumb; see Keller and Luder (). The idea is asfollows: you first make a projecon of the contract value along the

best-esmate path of the risk driver given by

Et[

g(T,y(T)) | y(t) = E0[y(t)]]

for all 0 t T. Then, atannual points (t = 1, 2, 3, ...) you approximate the Value-at-Risk

(VaR) by considering the impact of a .% shock for the risk driver

from the best-esmate path. Finally, the present value of all shocks

is added and mulplied by .

Let us consider an example. For ease of exposion assume

r = 0. Suppose there is a two-year product with a payoffebWy(2).

The best-esmate path is given by Et[ebWy(2)|Wy(t) = 0] =

e12b

2(2t) for t = 0, 1, 2. A one-year .% worst-case shock on

Wy(t) is given by an increase in value to Wy(t) + 2.58. Hence,

the Value-at-Risk in year t is approximated by applying the

one-year shock to the best-esmate path as e12b

2(2t)(e2.58b 1).Finally, the CoC price for this two-year product would be calculated

as

e12b

22 + (e12b

2

+ 1)(e2.58b 1). (.)Ifb = 50% and = 6% then we calculate a price of1.62.

A disadvantage of the best-esmate path method is that the

dynamics of the risk drivery(t) are completely ignored for the VaR


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calculaon. If we move one year ahead in me, then the risk driver

will be at the value y(1), which will differ from the best esmate

value E0[y(1)]. Hence, the CoC price of the product at me t = 1is based on a different best-esmate path than the calculaon at

t = 0. Therefore, the best-esmate path method used by the

industry is notme-consistent.

How can we obtain a me-consistent version of the CoC pricing

operator? One approach (similar to that taken for complete

markets in Secon ) to use a backward-inducon method. In fact,

Jobert and Rogers () prove that every me-consistentvaluaon operator can be obtained by backward-inducon of a

one-step pricing operator. Returning to the example, given the

payoff at T = 2, we can calculate the price at me condional

on the value ofWy(1) as

ebWy(1)+12b

2

+ (eb(Wy(1)+2.58)+12b

2

ebWy(1)+

12b

2

).

This expression can be simplified to

ebWy(1)+12b

2 (1 + (e2.58b 1)) .

Given the price at me , which is now an explicit funcon of

Wy(1), we can again calculate the CoC price at t = 0. This leads

to the formula

e12b

22(

1 + (e2.58b 1))2 . (.)If we take again b = 50% and = 6%, we find a price of1.72.

When we compare the price (.) of the best-esmate path

method with the backward-inducon price (.), then we

immediately see the effect of the best-esmate path

approximaon. In (.) one adds the terms (e2.58b 1) and


22/55

e12b

2(e2.58b 1) to the price e12b22. Whereas the

backward-inducon method explicitly takes the capital-on-capital

effect into account by mulplying the price e12b22 twice with the

factor (1 + (e2.58b 1)), the inclusion of the capital-on-capitaleffect leads to a me-consistent pricing operator.

.. Paral Differenal Equaon

As a final step in our argument, we change the length of the

me-step in the Cost-of-Capital pricing operator from one year to

t, and consider the limit fort 0. Note that whencomparing Value-at-Risk quanes at different me-scalest,

these have to be scaled back to a per annum basis; this is done by

dividing the VaR term byt. Then, consider that the

cost-of-capital behaves like an interest rate: it is the

compensaon the insurance company needs to pay to its

shareholders for borrowing the buffer capital over a certain period.

The cost-of-capital is expressed as a percentage per annum; hence,over a me-stept the insurance company has to pay a

compensaon oft pere of buffer capital. This yields a net

scaling oft/t =

t.

For a single me-stept, this yields the following expression

for the CoC price:

g(

t,y(t))

= ertEt[g

(t +t,y(t +t)

)]+

tVaRq,t[g

(t +t,y(t +t)

)]

. (.)

The scaling byt is a result of using Brownian Moon to describe the

evoluon of risk. More general stochasc process (such as Lvy processes) may

require different scaling factors, but this is beyond the scope of the current paper.


23/55

With this pricing operator for at-step we apply the

backward-inducon method to determine the me-consistent CoC

price for a payoffg(T,y) at me T, and take the limitt 0.Note that for smallt, the variance at me t of the process

g(t +t,y) is given by b2g2yt. Furthermore, in a diffusion

seng, for smallt, all risks are very close to a normal

distribuon. Hence, the VaR at me t is closely approximated by k

mes the standard deviaon kb|gy|t, where the constant k is

given by the inverse cumulave normal distribuon of the VaR

confidence level q(i.e. k = 1(q)). Given that g(t +t,y) issufficiently smooth to be twice connuously differenable in y, we

can then (similar to the manipulaons in Secon .) substute

the Taylor approximaon of the funcon g(t +t,y) for the

binomial approximaon (.) into (.), divide bytand take the

limit fort 0, which yields the following paral differenalequaon (pde) for the price operator g(t,y):

gt + agy +12

b2gyy + kb|gy| rg = 0. (.)A comparison of the pricing equaon (.) and the complete

market pricing equaon (.) reveals two important differences.

First, note the addional term kb|gy|. This is the penalty termthat the Cost-of-Capital method adds for wring the

non-hedgeable claim g(

T,y

). Second, it seems that we have not

changed the dri term afor the process y in the pricing equaon.

But this is not enrely true. In fact, whenever the payoffg(T,y)

is monotonous in y(T), then the sign ofgy is unique, and the two

terms depending on gy can be added together to obtain(a kb)gy. Therefore, the CoC price g(t,y) can be

For a more elaborate derivaon, see Bayraktar and Young () or Delong

().


24/55

represented with respect to the risk-adjusted process y:

dy =(

a(t,y) kb(t,y)) dt + b(t,y) dWy, (.)where the sign ofk is determined by the sign ofgy. This allowsfor a very nice interpretaon of the pricing equaon (.). When

pricing an non-hedgeable claim g(T,y), we adjust the

best-esmate dri of the process y in a conservave direcon.

In other words, the dri is adjusted upwards or downwards by

kb(t,y) depending on the sign ofgy. Making the price more

conservave by adjusng the dri is a me-honoured actuarialpracce known as prudence.

Revising the example from Secon .., recall there was a

payoff ofey(2) with r = 0, a = 0 and b = 0.50. This payoff is

monotonically increasing in y and this claim can be priced by

adjusng the dri ofy upward to kb. Hence, we calculate a price

at me 0 ofe(kb+12b

2)2. If we take = 6%, k = 2.58 and

b = 0.50, then the price at me 0 is 1.50.

This secon concludes by nong that that Cost-of-Capital

method suffers from a weakness: there is relavely lile economic

jusficaon for choosing the correct values of and k. The report

CRO-Forum () considers a wide variety of arguments that lead

to a wide range of possible parameter values. In the end, the

CRO-Forum recommends seng = 0.06 and

k = 1(0.995) = 2.58, leading to a total factor ofk = 0.15.

. Good Deal Bound Pricing

A very different approach on pricing in incomplete markets was

introduced by Cochrane and Sa-Requejo (). It is based on the

following idea. Suppose you are offered the opportunity to enter

the following loery: with a probability of 12

you get a payoff of

1000, or 1. The inial price of the loery is 2. In terms of the


25/55

theory developed in Secon , this is not an arbitrage opportunity.

However, it does represent a very good deal. We get something

with an expected value of500.50 for a price of2. There is,however, risk involved: the expected value is12

1000 + 12

1 = 500.50, and the standard deviaon is12 (1000 500.50)2 + 1

2 (1 500.50)2 = 499.50. But

you have to be extremely risk-averse to bring the price you are

willing to pay from 500 down to below 2, in order to not

parcipate in this loery.

The tools developed in Secon can be used to calculate the

price for this loery by adjusng the probabilies of the outcomes.

In this example we have to solve for the adjusted probability qthe

equaon 2 = q 1000 + (1 q) 1, which leads to q = 1/999and (1 q) = 998/999. Comparing the rao between theadjusted probabilies qand the original probabilies 1

2, like in

equaon (.), we see that the rao is extremely largealmost afactor 2000 in this example. Hence, extremely good deals imply

very large probability raos. Another way of looking at this is to

look at the factor () that was introduced in (.), which is the

volality of the deflator (). Secon . established that the

deflator volality () can also be interpreted as the market price

of risk, if it was in a complete market. Hence, for extremely good

deals the market price of risk (t,y) is very large.This brings us to the idea ofGood Deal Bounds. In an incomplete

market seng, we cannot trade in the underlying risk driver y.

Hence, we cannot calibrate the marngale measure to the prices

of traded assets. On the other hand, it is unrealisc to assume that

agents in the economy will leave extremely good deals

For ease of exposion we ignore the effect of discounng in this example.


26/55

unexploited. Cochrane and Sa-Requejo () introduced the

idea of pung an upper bound on the deflator volality () to

disnguish normal deals from extremely good deals.Furthermore, Cochrane and Sa-Requejo () proposed using

market prices of risk that we can observe for traded risks as a

benchmark for non-traded risks.

Suppose that we put an upper bound on the deflator volality.

This makes it possible to search for the upper and lower bounds on

the price for a derivave g(T,y) by considering all pricing

deflators with a volality less than or equal to . These upper andlower bounds for the price represent the ask and bid prices for

an agent with good deal bound .

This may all sound quite complicated, but it allows us to exploit

the structure between deflators, risk-neutral probabilies, and the

dri of the risk driver already explored in Secon . Let us return

to the binomial discresaon (.) of the process y. By pung a

bound on the deflator volality, we infer from equaon (.)that we are considering adjusted probabilies in the range

q12

(1

t

), 12

(1 +

t

). (.)

But, changing the binomial probabilies is equivalent to changing

the dri of the process y. Hence, alternavely, we can also say

that we consider specificaons for the stochasc process y where

the adjusted dri a(t,y) is somewhere in the range

a(t,y) [a(t,y) b(t,y), a(t,y) + b(t,y)] . (.)Using the derivaon from Secon ., we infer that any price of a

derivave g(t,y) that falls within the Good Deal Bounds is

described by the paral differenal equaon (pde)

gt + a

gy + 12b2gyy rg = 0, (.)


27/55

where a is taken from the interval (.).

When seeking the highest and lowest prices that are on the

edge of the good deal bound interval, we have to find the dri a

that minimises or maximises the price g(t,y) for each me-step.

For example, when we want to maximise the price g(t,y), then

we should either put a at the upper bound a(t,y) + b(t,y)

whenever gy(t,y) > 0 or put a at the lower bound

a(t,y) b(t,y) whenever gy(t,y) < 0. Therefore, we canrepresent the good deal bound price g(t,y) with respect to the

risk-adjusted process y:

dy =(

a(t,y) b(t,y)) dt + b(t,y) dWy, (.)where the sign of is determined the sign ofgy.

Note that the structure of the risk-adjusted process (.) is

exactly the same as the structure of the Cost-of-Capital pricing

process (.), provided we take = k.

On the other hand, given that we have the interpretaon of as

an upper bound for the deflator volality, which in traded markets

is equal to the market price of risk, this informaon can be used to

get more guidance on seng . Considering equity markets , then

we can calculate the market price of risk. For typical equity markets

(see e.g. Dimson et al. ()) we see an excess return above the

risk-free rate of around 4%, and a volality of around 16%,

leading to a market price of risk of approximately 4/16 = 0.25.

From this calculaon we infer that the upper bound should be

larger than 0.25. Note that in this light the value k = 0.15

implied by the Cost-of-Capital method seems to be on the low side.

. Model Ambiguity & Robustness

This subsecon introduces a third perspecve for pricing contracts

in incomplete markets. This is the noon ofModel Ambiguity. This


28/55

means that we explicitly take into consideraon that the

mathemacal models we use to describe the world are not exact,

but may be misspecified.Model ambiguity can be illustrated as follows. Suppose we try to

esmate the expected return of invesng in an equity index (say,

the Standard & Poors (S&P) index). Historical observaons can

then be used to esmate the expected return, but the esmate of

the expected return will then be subject to esmaon error. It

turns out that since equity returns are relavely volale, it is very

difficult to obtain an accurate esmate for the expected return.Assume that the volality of the S&P index is around %, and that

we use years of data. Then the standard error for the esmate

of the expected return is 16%/

25 = 3.2%. Suppose that the

esmate of the expected return is 8%; then the % confidence

interval for this esmate is

[8%

1.96

3.2%, 8% + 1.96

3.2%] = [1.7%, 14%]. Even if

we would use years of historical data, our % confidenceinterval is sll [4.9%, 11%]. Using more years of historical data

will give us a more accurate answer only, if the data-generang

process has remained the same during the enre period. It is

highly quesonable whether the economy of years ago is

representave of todays economy. Thus, it is clearly very difficult

to obtain an accurate esmate of something as simple as the

expected return of an equity index. The same observaon is also

true for the expected increase in human longevity: actuaries have

been constantly revising their projecons about forecasts of

human longevity in the last years.

Suppose we accept the impossibility of accurately knowing

the correct model specificaon. How can we deal with this model

ambiguity? One approach is to assume that economic agents are


29/55

concerned about making bad decisions based on misspecified

models. To deal with this problem, assume that agents resort to

robust opmisaon methods. This means that agents try to maketheir decisions in such a way that they explicitly incorporate the

fact that the true model of Mother Nature may deviate from the

mathemacal model used by the agent for decision making. The

noon of model ambiguity and robust opmisaon in economics

has been made popular in recent years by Hansen and Sargent

(see, for example, their book Robustness, Hansen and Sargent

()).How can the noon of robust opmisaon be implemented in

our seng? We start by making some strong simplifying

assumpons. First, assume that the true model for the process

y(t) is of the form (.). The only uncertainty that we have

concerns the correct specificaon of the dri term a(t,y). Hence,

we assume that we know the correct specificaon of the diffusion

term b(t,y). These are, of course, very strong assumponsindeed, but given the difficules in esmang even a simple

parameter as the expected return, this seems like a good starng

point.

Second, assume that the agent is able to specify a confidence

interval of reasonable values for the dri a(t,y). We will

return later to the queson of how to specify a confidence interval

of reasonable values. In the one-dimensional case being

considered in this secon, a confidence interval for the dri has

the form a(t,y) [aL, aH]. Another way of represenng aconfidence interval is to say we have a point esmate a(t,y) that

is located in the centre of the confidence interval, and a width of

For a more elaborate jusficaon of considering only uncertainty in the dri,

see Hansen and Sargent (, Chapter ).


30/55

the confidence interval given by 2mes the standard deviaon

b(t,y) of the process y(t). This leads to the representaon

a(t,y) [a(t,y) b(t,y), a(t,y) + b(t,y)]. (.)Note that this confidence interval is of exactly the same form as

the good deal bound equaon (.). However, the parameter

now has the interpretaon as being the width of the confidence

interval for a(t,y).

In this setup we can achieve a robust noon of pricing a

derivave g(T,y) by calculang the expectaon ofg(T,y)

under the worst model specificaon for the non-hedgeable

process y. In parcular, for an insurance company that has wrien

the claim g(T,y) payable at me T, the worst model

specificaon is that choice for the dri a(t,y) in the

interval (.) that maximises the value of the expectaon. Using

exactly the same argumentaon as in Secon ., we find that the

robust price g(t,y) can be represented by taking the expectaon

with respect to the worst case process y:

dy =(

a(t,y) b(t,y)) dt + b(t,y) dWy, (.)where the sign of is determined the sign ofgy.

Given our interpretaon ofas the width of the confidence

interval, how can we determine ? In other words: how can weestablish an interval of reasonable values for a(t,y)? We can

offer two (closely related) arguments. The first argument is that

historical data can be used to esmate the parameter a. The

confidence interval from the parameter esmate the becomes the

measure for the interval of reasonable values for a. Using a %

confidence interval based on years of historical data yields

= 1.96/25 = 0.39. For years of historical data we obtain


31/55

= 1.96/

50 = 0.28. The second argument is to consider the

queson: which alternave model specificaons are stascally

indisnguishable from our current model, given the available data?The answer (in the case of uncertainty in the mean) is given by all

values for the dri athat are in the % confidence interval (.).

Comparing the values for of . or . to the lower bound

of . found for reveals these values are nicely in agreement

with each other. Furthermore, we arrive (once again) at the

conclusion that the value k = 0.15 used by the insurance

industry is on the low side.

. A New Value of the Cost-of-Capital

To summarise our discussion on the Cost-of-Capital: the

conclusions drawn both in Secon . and in the previous secon

indicate that the CoC parameter k = 0.15 currently used by the

insurance industry seems too low.

A value ofk = 0.30 seems much more appropriate when thisis compared to the values implied by the Good Deal Bound and the

Model Ambiguity methods. Seng k = 0.30 and using

k = 1(0.995) = 2.58 implies a cost-of-capital parameter

= 12%, which is basically doubling the current value of6%

proposed by the insurance industry.

Seng = 0.30 in the Model Ambiguity method, this

corresponds to using (1.96/0.30)2

= 43 years of historical datato esmate the mean of the process y(t). This also seems a

reasonable tradeoff between using as much historical data as

possible, without going back so far in me that it becomes hard to

believe that the data is sll representave for todays economy.


32/55

. Alternave Approaches

This secon presents various alternave approaches to the theory

developed in this paper so far.

.. Variance Pricing & Ulity Indifference Pricing

A large body of literature focuses on ulity indifference pricing.

The roots can be traced back to Hodges and Neuberger (). The

idea is that the assumpon is made that the behaviour of agents

can be described by a ulity funcon, then a ulity indifference

price for accepng an (non-hedgeable) claim can be found. Forexponenal ulity funcons, quite explicit results can be found

(see Zariphopoulou (); Young and Zariphopoulou ();

Musiela and Zariphopoulou (); Hugonnier et al. (); Hu

et al. (); Musiela and Zariphopoulou (b); Henderson

(, ) and Henderson and Hobson ()). Also for power

ulity funcons, paral results are known, see Hobson ();

Monoyios (). For a general overview, see the bookIndifference Pricing by Carmona ().

A big disadvantage of ulity-based pricing is that it depends on

the specificaon of the ulity funcon at a specific horizon T. This

introduces an arficial dependency in the pricing on the horizon T.

Aempts to resolve this issue were proposed by Henderson and

Hobson () and Musiela and Zariphopoulou (, a).

.. Strong Time-Consistency

Our derivaons have used condional expectaons that are

sequenally evaluated using backward-inducon arguments. This

leads to the pricing pdes we have found in equaon (.). Use of

backward-inducon techniques allows us to construct pricing

methods that are strongly me-consistent; see Hardy and Wirch

() and Jobert and Rogers (). However, the concept of


33/55

strong me-consistency for pricing methods is not uncontroversial.

See Roorda et al. (); Roorda and Schumacher () for a

discussion.

.. Bayesian Approach

Finally, note that as an alternave to robust opmisaon, it is

possible to use Bayesian methods. In the Bayesian approach the

uncertainty about the model specificaon is specified in the form

of prior and posterior probability distribuons on the parameter

space. Porolio opmisaon and pricing is then carried out byaveraging over the parameter space (i.e. averaging over the

different alternave model specificaons). For a discussion and

examples, see Lutgens (); Lutgens and Schotman ().

. Literature Overview

As menoned in the text, the Cost-of-Capital (CoC) approach was

originally proposed by the insurance industry (see CRO-Forum()), based on ideas put forward by the Swiss insurance

supervisor in the so-called Swiss Solvency Test (SST) (see Keller and

Luder ()). For a crical discussion on the risk measure implied

by the SST see Filipovic and Vogelpoth (). The CoC method

was adopted by the European Union as the standard method for

the calculaons in the Quantave Impact Studies of the

Solvency II process; see CEIOPS (); EIOPA ().Good Deal Bound pricing has been introduced by Cochrane and

Sa-Requejo (). Their basic ideas were extended by ern and

Hodges (), Becherer (), Bjrk and Slinko ()

andKlppel and Schweizer (b). The connecons between

Good Deal Bound pricing and the numraire porolio (which we

have called the stochasc discount factor, see eq. (.)) have been

explored by Becherer (, ), Karatzas and Kardaras (),


34/55

Christensen and Larsen () and Delong ().

Jaschke and Kchler () highlighted the connecons

between Good Deal Bound pricing and the rich theory of coherentrisk measures. Coherent risk measures were introduced by Artzner

et al. (, ). Later, this has been extended to the more

general class of convex risk measures by Fllmer and Schied ()

and Cheridito et al. (). The connecon between convex risk

measures and pricing can be found in Cvitani and Karatzas (),

Carr et al. (), Frielli and Rosazza Gianin (), Detlefsen and

Scandolo (), Klppel and Schweizer (a), Stadje (),Delbaen et al. (), and the papers by Cherny (, ,

). In the actuarial literature, see Denuit et al. () and

Goovaerts and Laeven ().

Model Ambiguity and robustness were made popular in

economics by Hansen and Scheinkman (), Hansen and Sargent

(), Cage et al. (), Cont (), Hansen et al. () and

Hansen and Sargent (). However, ideas for robustness instascs are much older, and date at least back to Huber ()

(for a new edion, see Huber and Ronche, ). Also, several

authors have applied robustness ideas to porolio opmisaon;

see Kirch (), Goldfarb and Iyengar (), Maenhout (),

Coleman et al. (), Gundel and Weber (), Rogers (),

Fllmer et al. (), Iyengar and Ma () and Kerkhof et al.

(). Another branch of literature studies the noon of model

ambiguity on decisions that economic agents make; see Duffie and

Epstein (a), Duffie and Epstein (b), Chen and Epstein

(), Maccheroni et al. (), Epstein and Schneider () and

Riedel ().


35/55

. Combining Hedgeable and Non-Hedgeable Risk

This secon pushes our analysis one step further. We invesgate

an environment in which we have both a financial risk process

x(t) that can traded and hedged in a market, and also an

non-hedgeable insurance risk process y(t). Basically, this secon

seeks to combine the results from Secons and .

. Model Ambiguity and Hedging

The process for the financial risk x(t) is given in equaon (.) and

the process for the non-hedgeable risk y(t) is given in (.).Similar to the setup in Secon , an agent is considered that is

uncertain about the true value of the dri parameters m and aof

the financial and the insurance processes, respecvely. Assume

that the agent faces no uncertainty about the diffusion coefficients

, band the correlaon parameter between the the Brownian

Moons Wx and Wy.

To help us describe the uncertainty set, we introduce somefurther notaon. The vector of dri rates , and the covariance

matrix are defined as follows

:=

m

a

, :=

2 b

b b2

. (.)

We now make the assumpon that the joint uncertainty in the dri

rates is described by the following set:

K := {0 + | 1 k2}. (.)

The specificaon of the uncertainty in this form is movated (like

in Secon .) by the fact that the economic agent can use

econometric esmaon techniques to esmate the dri rates. The

esmaon leads to the point esmate 0. However, there is


36/55

uncertainty surrounding this esmate. This uncertainty typically is

proporonal to the covariance matrix. In other words, the agent

assumes that the true values of the dri parameters liesomewhere within the confidence interval given by the set K. Inthe one-dimensional case we considered in Secon ., the

uncertainty setK for the dri rate asimplifies toa [a0 kb, a0 + kb].

We want to invesgate what price the agent will aribute to a

derivave that depends both on financial and insurance risk and

has payoffg(t +t, x,y) at me t +t. We furthermoreassume (like in Secon .) that the agent can hedge the financial

risk by invesng an amount Dt in the risky asset x at me t, but

cannot trade in the insurance asset y. Hence, the robust raonal

agent solves the following opmisaon problem for each me-step

[t, t +t]:

maxDt mint ert

E[t]t[

g(

t +t, xt+t,yt+t) Dtxt+t]

s.t. 1 k2,(.)

where E[t][ ] denotes taking the expectaon using the dri term

0 + t for the processes x and y.

This maxmin opmisaon problem can be interpreted as a

two-player game. First, the agent chooses an amount Dt to invest

in x, and then Mother Nature chooses the worst possible

perturbaon t of the dri 0. But the agent is aware of the bad

intenons of Mother Nature, and therefore chooses the amount

Dt that maximises the minimal outcome of Mother Nature.

The opmal soluon for Dt is given by

D

t := gx + b

gy

+

k2 2b1

2

|gy|, (.)


37/55

where is the market price of risk defined in equaon (.).

Several interesng things about this soluon are noteworthy.

First, the soluon Dt is only well-defined for 2 < k2. Thiscorresponds exactly to the condion encountered in Secon .:

that the good deal bound should be larger than the market price

of risk for the financial market. Stated differently, if the agent is

confident that even in the worst case a posive excess return can

be made by invesng in the financial market (i.e. when 2 > k2),

then the agent will try to invest a massive amount in the financial

market and has a confident expectaon of geng very rich.The second interesng thing to note is that the opmal hedge

posion consists of two parts: the hedge porolio(fx + b fy)and a speculave porolio that is determined by the product of

the residual non-hedgeable risk b

1 2/ fy and the marketconfidence factor /

k2 2. The market confidence factor

shoots to infinity if approaches k. For small values of, the

market confidence factor is approximately equal to /k, and thespeculave investment is then approximately proporonal to the

market price of risk scaled down by a factor ofk.

. Agents Valuaon

If we substute for each me-step [t, t +t] the opmal

soluons for (Dt, t) into (.), and then take the limit for

t 0 (as we did in Secon .), we find a paral differenalequaon for the price g(t, x,y):

gt+ rgx+ agy +

122gxx+bgxy+

12

b2gyy rg = 0, (.)

where the dri term a for the insurance process is given by

a

= a0 b b(1 2)(k2 2), (.)


38/55

where the sign of the last term depends on the sign ofgy.

Although the expressions for a may look a bit complicated, some

very nice interpretaons can be given for the expressions.The first interpretaon for a is an economic interpretaon.

Recall that the formula for the opmal hedge given in

equaon (.) consists of two parts: a hedge porolio and a

speculave porolio that depends on . Suppose that the agent

would only choose the hedge porolio Dt := (fx + b/fy).Substugn Dt into (.) yields an expression very similar to (.),

except that the dri term for the insurance process would be givenby a = a0 b kb

1 2. Therefore, by including the

speculave porolio into the opmal hedge, the agent can finance

part of uncertainty in a by exploing the expected excess return

on equies. This then results in the opmal dri term a, where

the residual non-hedgeable insurance risk kb

1 2 is shrunkby an addional factor

k2

2.

Note also the downward adjustment of the dri of thenon-traded assety by the termb. This downward adjustmentcompensates exactly the excess return ofy due to the correlaon

with the traded asset x. This effect makes sense if we think in

terms of the Capital Asset Pricing Model (CAPM). In the CAPM, the

expected return of any asset is given by the formula

E[dy(t)] = (r + (m r))dt, where is given by the formula := b/2. Combining the expression for with the definionof the market price of risk given in (.) yields

E[dy(t)] =(

r + b)

dt. Hence, the dri term a in the agents

valuaon adjusts the dri in two steps: the first stepbcorrects the dri ofy for the excess return injected by the

correlaon with the traded asset, and the second term

b(1 2)(k2

2) adjusts the dri ofy for the


39/55

non-hedgeable risks.

The second interpretaon of equaon (.) is a geometric

interpretaon. Recall that the uncertainty setK is an ellipsoidcentred around 0. Because the financial component x of the risk

vector is perfectly replicated, this means that the uncertainty

regarding the mean of the financial risk is eliminated, and is

replaced by the risk-free return r. The uncertainty for the mean of

the insurance process y is now confined to the intersecon of

uncertainty set

Kand the line m = r. The intersecon of a line

and an ellipsoid has two soluons: exactly the two soluons givenin equaon (.).

This geometric interpretaon is illustrated in Figure , using the

following parameters: m0 = {4%, 7%, 10%}, r = 4%, a0 =0, = 0.15, b = 1, = 0.75 and k = 2/

25 = 0.4. The three

different values ofm0 lead to = {0,0.2,0.4}, and these threecases are illustrated in the sub-figures (a), (b) and (c). The

uncertainty setK is given by the interior of the ellipse, and thepoint esmate 0 is given by the point in the centre. The line

m = r is the vercal doed line, and the intersecon with the

ellipse gives the two soluons for a. Figure (a) illustrates the

case = 0. In this case, a = a0 kb

1 2, which is thenaive confidence interval for aequal to the point esmate a0plus/minus k mes the non-hedgeable insurance risk b1

2.

The other sub-figures illustrate that for larger values of, the

ellipse moves to the right. This is a reflecon of the fact that higher

values of correspond to higher point esmates ofm0. When the

Note that this geometric interpretaon is equivalent to the result found in

Barrieu and El Karoui (), where the pricing measure is characterised as the

inf-convoluon of the set of test measures (in our case, the ellipsoid) and the

set of marngale measures (in our case, the line m = r).


40/55

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

iftofInsuranceProcess

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-4% 0% 4% 8% 12% 16%

DriftofInsuranceProcess

Return on Financial Market

ConfInt mu0 r a*

(a) = 0

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

iftofInsuranceProcess

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-4% 0% 4% 8% 12% 16%

DriftofInsuranceProcess


ConfInt mu0 r a*

(b) = 0.2

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

iftofIn

suranceProcess

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-4% 0% 4% 8% 12% 16%

DriftofIn

suranceProcess


ConfInt mu0 r a*

(c) = 0.4

Figure : Confidence interval for for different values of.


41/55

ellipse moves to the right, the points a move down (due to the

correlaon term

b), and the points move closer together (due

to the factork2 2). Figure (c) illustrates the largest allowedvalue for . For larger values of, the ellipse no longer intersects

the line m = r, and the opmisaon (.) no longer has a finite

soluon.

. On the choice ofk

This secon concludes by elaborang on the choice ofk. In the

one-dimensional case treated in Secon . we took

k = 1.96/43 = 0.30. This choice was movated byconsidering the % confidence interval for the esmate of the

mean of the process y(t) using years of historical data.

In the two-dimensional case we considered in this secon, we

have to modify this argument slightly. If we used years of data

to esmate the vector = (m, a), then the confidence interval

for is given by the setK defined in equaon (.). The set Kdescribes a % confidence interval if we take k2 equal to the %crical value of a 2-distribuon with degrees of freedom,

divided by the number of observaons. This value is given by

k2 = 5.99/43 = 0.139, which leads to the value k = 0.37.

For the one-dimensional case, we would take the % crical

value of a 2-distribuon with degree of freedom, divided by ,

which leads to k =

3.84/43 = 0.30, which is the same valuethat was derived in Secon ..


42/55

. Applicaons

This secon uses several examples to illustrate the concepts

developed in this paper.

. Pricing Long-Dated Cash Flows

As menoned in the introducon, the pricing of very long-dated

cash flows is an important problem for insurance companies and

pension funds. This secon illustrates the applicaon of the pricing

approach outlined in this paper to this problem.

We start by assuming that the interest rates are stochasc, andcan be described by a Vasicek () model. In this model we take

the instantaneous short rate r(t) and model it as

dr = a( r)dt + dWr. (.)

In this equaon, is the long-term average of the interest rates,

and ais the speed of mean reversion.When pricing cash flows, we have to disnguish two cases. For

maturies up to years, there are bonds traded in financial

markets, and the market is complete. This means that a porolio

of cash flows is priced at the same price as the value of a

replicang porolio of bonds that matches the cash-flow paern.

This price can also be calculated by discounng the cash flows at

the zero-curve implied by the market.For cash flows beyond years, there are no bonds available,

which leads us in an incomplete market situaon. This calls for

applicaon of the methods outlined in Secon to determine the

value of the cash flows.

For pricing short cash flows (i.e. cash flows promised to

policyholders), we have to adjust the interest rates downwards.

Applying the methodologies of Secon to the interest rate


43/55

2.00%

3.00%

4.00%

5.00%

6.00%

Extrapolated zero-curve

0.00%

.

0 10 20 30 40 50 60

Zero-up Zero-dn

Figure : Extrapolated term-structure of interest rates

dynamics (.), we get

dr = a( k

a r) dt + dWr, (.)

where we set k = 0.30. This formula implies that a zero rate with

maturity T > 30 has to be adjusted up or down (for long or short

cash flows, respecvely) by the formula

kaT

(T 30)

1 ea(T30)

a

. (.)

Assume that the long-term nominal interest rate is % (for

example, composed of % inflaon plus % real interest rates). If it

is also assumed that a = 0.05 and = 0.01, then it is possible to

explicitly calculate the extension of the zero-curve beyond the

-year point using formula (.). This is illustrated in Figure . The


44/55


45/55

80

85

90

95

Life expectancy at birth for Dutch males

70

75

1950 1970 1990 2010 2030 2050

Figure : Life Expectancy at Birth for Dutch Males

forecasng of human mortality so difficult.Based on almost years of data, an average increase in life

expectancy of . months per years is esmated, and a standard

deviaon of . months per year. The Dutch Actuarial Associaon

has recently published new life tables, idenfying a trend of .

months increase in life expectancy for Dutch males per year, which

is very close to the number found here.

However, when we want to price a porolio of contracts wherewe worry about people living longer (as is typical for life insurance

and pension porolios). Using the methods outlined in Secon ,

we should (for pricing purposes) therefore adjust the trend

upwards (in the more conservave direcon) by . standard

deviaon, leading to a prudent trend of .+.*. = .

months per year increase in life expectancy. The prudent up and

down trends are also illustrated in Figure for the period from


46/55

unl .

For further examples, see Wang et al. (), Young and

Zariphopoulou (), Young (b), Milevsky et al. (),Bayraktar and Young (), Milevsky and Young (), Bayraktar

and Young (), Hri et al. (), Ludkovski and Young (),

Young () and Bayraktar et al. ().

. Pricing Non-Hedgeable Equity Risk

The final example examines the pricing of equity risk. A typical

market-consistent seng assumes that equies can be freelytraded in financial markets, and that equity risk is fully hedgeable.

This means that equity risk is priced using the risk-neutral methods

outlined in Secon .

However, in the case of very large pension funds this

assumpon is quesonable. If very large pension funds would buy

or sell very large equity posions, they would move the market

prices. Hence, large pension funds are not price takers, but canmove the market. In parcular, in mes of crisis, large pension

funds could push the market down even further if they would try

to sell equies in response to the drop in prices. Fortunately, this

has not happened during the last two crises, thanks to adequate

relaxaon of the underfunding rules by the Dutch Central Bank.

If we take the fact that large pension funds cannot trade without

moving the market to the extreme that large pension funds cannottrade at all, then we have again an incomplete market situaon,

and equity exposures should be priced using the methods of

Secon .

Another example of equity incompleteness is the case of

insurance companies that give profit-sharing to their policyholders

based on the performance of their own investment porolio. In

the tradional approach, such as that of Grosen and Jorgensen


47/55

6 0 0 %

8 0 0 %

1 0 0 0 %

1 2 0 0 %

1 4 0 0 %

1 6 0 0 %

Cumulative Return S&P Index

0 %

2 0 0 %

4 0 0 %

6 0 0 %

1 9 5 0 1 9 7 0 1 9 9 0 2 0 1 0 2 0 3 0 2 0 5 0

Figure : Cumulave Return in Standard and Poors Stock Index

(), such profit-sharing opons are priced with the risk-neutralmethods of Secon . However, there is a problem: risk-neutral

pricing is based on the cost of the replicang porolio of the

contract. But in the case of profit-sharing on your own porolio, it

is impossible to hedge: at the moment you start buying

instruments to hedge your own profit-sharing opons, you start

changing the composion of your asset porolio, which then starts

changing the nature of the profit-sharing. One approach to solvethis problem was suggested by Kleinow (), where one tries to

find the investment porolio with profit-sharing that is

self-hedging. The soluon to this approach is that the only

porolios that are self-hedging are those that have no

investment risk (and thus have perfectly predictable

profit-sharing). Although mathemacally correct, this outcome

does not reflect the behaviour of insurance companies in reality.


48/55

An alternave approach would be to assume that these

profit-sharing opons are non-hedgeable and should be valued

using the methods of Secon .What kind of pricing do we get for the non-hedgeable equity

approach? Figure plots the cumulave return of the S&P equity

index between and . The average return over this period

is .% with a standard deviaon of .%. When we need to

price an non-hedgeable equity posion as an asset or as a wrien

put-opon, we adjust the return down by . standard deviaon

to .%. This higher level of conservasm reflects the addionalrisk associated with holding an non-hedgeable posion.

For further examples, see Davis (, ) and De Jong

(b).


49/55

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