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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
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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.
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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
().
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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
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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.
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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, (.)
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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
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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
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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 ).
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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
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= 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.
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. 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
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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 (),
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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 ().
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. 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
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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|, (.)
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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), (.)
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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
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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).
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-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
Return on Financial Market
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
Return on Financial Market
ConfInt mu0 r a*
(c) = 0.4
Figure : Confidence interval for for different values of.
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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 ..
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. 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
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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
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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
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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
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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.
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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).
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