Recent Developments in Longevity Risk Modelling with o ge ty s - - PowerPoint PPT Presentation

Recent Developments in Longevity Risk Modelling with

• ge

ty s

• de

g t Application to Longevity Risk Management Management

Sam Cox & Hal Pedersen (Presenter) (University of Manitoba) (University of Manitoba)

Longevity and Pension Funds

February 4, 2011

Axa, Paris

A Glance at the Data B f I t t d ki t lit

• Before I started working on mortality

modelling for risk management I assumed that changes in mortality were pretty that changes in mortality were pretty simple.

• If mortality was improving then the actuary

y p g y

• ught to simply use a bit more

conservative table. S f ll t ld d i

• Some of my colleagues told me during

informal discussions that they had their

• wn way of handling improving mortality
• wn way of handling improving mortality.
• Some of these colleagues even wanted to

talk about it.

A Glance at the Data O f lt th t t ki th ti f

• One person felt that taking the ratio of

q_x+1/q_x and then scaling this down by 95% or some other percentage and 95% or some other percentage and reconstructing the mortality rates after the scaling has served him well in the past.

• Actuaries need real solutions to real

business problems and most actuaries are inclined to construct their own solution if inclined to construct their own solution if

• ne they are satisfied with is not available.
• Evidently if one scales the mortality in this

Evidently, if one scales the mortality in this fashion and starts at age 0 then one

• btains a sequence of curves such as the

f ll i following.

A Glance at the Data

Mortalit Impro ements (Slope Scaled) [Initial C r e Ill strati e Gompert ]

0.2000

Mortality Improvements (Slope Scaled) [Initial Curve Illustrative Gompertz]

0.1500 q_x 0.1000 q_x new q_x_1 new q_x_2 new q_x_3 0.0500 new q_x_4 0.0000 20 40 60 80 100 120 140 60 80 00 Age x

A Glance at the Data

0.80000

m_x for US Male Mortality

0.60000 0.70000 0.40000 0.50000 1950 1960 1970 1980 0.20000 0.30000 1990 2000 0.00000 0.10000 65 68 71 74 77 80 83 86 89 92 95 98 101 104 107

A Glance at the Data

• When I first looked at mortality data in detail I

could not help but be struck by the rate at hi h it t b h i which it appears to be changing even over a period of one decade.

US Male Population Mortality US Male Population Mortality q_x 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 65 0.02098 0.02023 0.01963 0.01932 0.01881 0.01852 0.01784 0.01759 0.01717 0.01703 70 0.03142 0.03032 0.03001 0.02925 0.02930 0.02838 0.02686 0.02691 0.02568 0.02495 75 0.04755 0.04760 0.04655 0.04572 0.04506 0.04456 0.04208 0.04133 0.04001 0.03944 75 0.04755 0.04760 0.04655 0.04572 0.04506 0.04456 0.04208 0.04133 0.04001 0.03944 80 0.07524 0.07170 0.07250 0.07007 0.07016 0.06885 0.06624 0.06552 0.06396 0.06213

• As you know, the data is even more striking

for younger ages.

• Even if there is year to year noise in the data

the trend is strong.

A Glance at the Data If h t d t l t d f

• If such trends are extrapolated for

extended periods of time, one will have what is surely unrealistic life tables what is surely unrealistic life tables.

• Of course, these trends change over time

and the duration of particular trends seen p in the historical data also vary.

• The following French male mortality data

ill t t t f th i ti i illustrates aspects of the variations in trends and their duration.

• Apparently different rates of improvement
• Apparently, different rates of improvement
• ccur and for various durations.

A Glance at the Data

0.60

French Male Mortality Data (1816 ‐ 2007)

0.40 0.50 0.30 20 40 60 0.10 0.20 60 80 100 0.00 1816 1824 1832 1840 1848 1856 1864 1872 1880 1888 1896 1904 1912 1920 1928 1936 1944 1952 1960 1968 1976 1984 1992 2000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2

A Glance at the Data

0.12

French Male Mortality Data (1816 ‐ 2007)

0.08 0.10 0.06 20 40 0.02 0.04 60 0.00 1816 1824 1832 1840 1848 1856 1864 1872 1880 1888 1896 1904 1912 1920 1928 1936 1944 1952 1960 1968 1976 1984 1992 2000

A Glance at the Data Th h i t lit t t b

• The changes in mortality rates seem to be

volatile when measured annually.

• If we average the mortality rates over non
• If we average the mortality rates over non-
• verlapping trailing windows such as 5

years then the qualitative nature of the y q data might be easier to follow.

• If we look at ratio of this time series to a

b i d th f base period, then one measure of mortality improvement is obtained.

• The following is US male mortality post
• The following is US male mortality post

WWII.

A Glance at the Data

US M l M t lit R d ti F t 5 T ili

1.20

US Male Mortality ‐ Reduction Factors on 5‐year Trailing Averages on 1947 Base Rates (1952, 1957, ..., 2007)

0.80 1.00 10 20 0.60 20 30 40 50 60 0.20 0.40 60 70 80 90 0.00 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 2007 100

Lee-Carter Model

• The Lee-Carter model remains the

benchmark model for mortality modelling.

• The parameters a and b are age dependent

parameters and k is a time dependent process the reflects the temporal variation in mortality.

• Estimated over an age range and a time

range [t_0,t_N].

Lee-Carter Model I d t th d l f th i k

• In order to use the model for the risk-

management simulation of mortality, one must adopt a stochastic process for k that must adopt a stochastic process for k that permits simulation going forward.

• The determination of such a stochastic

process is an involved issue and one that is not fully settled in the literature, particularly in the case of the multi factor particularly in the case of the multi-factor version of the model.

• However for the moment let us suppose

However, for the moment, let us suppose that such a stochastic process is adopted.

• We then simulate mortality as:

y

Lee-Carter Model

• Traditionally the series k is treated as a
• Traditionally, the series k is treated as a

random walk beyond the last data point.

Lee-Carter Model

• Traditionally the series k is treated as a
• Traditionally, the series k is treated as a

random walk beyond the last data point.

Lee-Carter Model

0 12

Lee‐Carter Mortality Projected Using Random Walk Drift Only (US Both Sexes Grouped)

0.1 0.12 0.06 0.08 2007 2017 2027 0.04 2027 2037 2047 2057 0.02 50‐54 55‐59 60‐64 65‐69 70‐74 75‐79 80‐84 85‐89

Lee-Carter Model Th t ti t f thi US

• The parameter estimates for this US

grouped both gender data 1933-2007 are shown below shown below.

a 0.6 0.8 b

• 5

0.2 0.4 10 20 30

• 10

10 20 30 8 k 4 6 20 40 60 80 2

Lee-Carter Model A h f d t t US l 65 85

• A change of data to US male ages 65-85

from 1950 has the parameters shown below

b

below.

• 3
• 2

a 0.3 0.35 b

• 5
• 4

0.2 0.25 10 20 30 5 10 20 30 4 k 1 2 3 20 40 60 1

Lee-Carter Model

• The time series for k is not a relatively

The time series for k is not a relatively straight line anymore but it can be approximated as such in the latter part of th i (W l i th d t t ) the series. (Was also in other data set.)

• Some authors have suggested using a

multi factor version of Lee Carter multi-factor version of Lee-Carter.

• For most data sets, the first principal

component captures the majority of the component captures the majority of the variation and the next few principal components capture virtually all of the i ti variation.

• However, for a multi-factor model one

must be able to model the k’s to simulate must be able to model the k s to simulate for risk management.

Lee-Carter Model

• Three factor model with initial fit to 2007
• Three-factor model with initial fit to 2007

(US both sexes grouped from 1950).

a 0.5 b

• 5

0 5 10 20 30

• 10

10 20 30

• 1
• 0.5

2 4 k 1 1.005 Explaining the Variance 0 98 0.99 0.995 20 40 60

• 2

10 20 30 0.985

Lee-Carter Model

• It appears that it might be reasonable to
• It appears that it might be reasonable to

use random walks with drift to model all three factors. three factors.

• There are data sets for which some of the

factors are non-linear and this is a f potential roadblock to using a multi-factor Lee-Carter.

• Instead of fitting a multi factor Lee Carter
• Instead of fitting a multi-factor Lee-Carter

and then dealing with the dynamics of the k time series on an ad hoc basis, perhaps , p p we can force a linear structure on a the model?

Lee-Carter Model

• Borrowing an idea from the term structure
• Borrowing an idea from the term structure
• f interest rates, we look at the structure
• f the change in the logarithms of the
• f the change in the logarithms of the

mortality rates. Fitting to the log differences is in contrast to the usual estimation of Lee Carter which fits to the estimation of Lee-Carter which fits to the logs of mortality rates.

• We assume that the change in the logs of
• We assume that the change in the logs of

mortality rates can be captured with a linear drift.

• Performing principal components yield the

following estimates for US both sexes grouped from 1950 grouped from 1950.

Lee-Carter Model

• There are no k processes for this model
• There are no k processes for this model

as a specific structure is directly imposed. 3 factors gets 75% and 6 get 90%. g g

0 01 D 0 02 0.04 factors

• 0.02
• 0.01

0.02 10 20 30

• 0.03

10 20 30

• 0.02

1 Explaining the Variance 0 6 0.8 10 20 30 0.4 0.6

Lee-Carter Model

• The analytic specification for this model is
• The analytic specification for this model is

as follows.

• In this form market prices of risk for the
• In this form, market prices of risk for the

various mortality factors can be imposed.

• Of course whether this really makes

Of course, whether this really makes sense and to what they would be calibrated is an open question.

• The model does produce reasonable

looking mortality curves.

Lee-Carter Model

• The following shows the 2007 curve
• The following shows the 2007 curve

together with simulated values at 10 year intervals for age group 0 to 80-84. g g p

0.06 0.07 data1 data2 0.04 0.05 data3 data4 data5 data6 0.02 0.03 0.01 0.02 2 4 6 8 10 12 14 16 18

Lee-Carter Model

• The model does capture the empirical
• The model does capture the empirical

correlation of the changes in mortality across the curve.

• The approach is naïve and most likely has

a number of drawbacks relative to a multi- f t L C t factor Lee-Carter.

• In simulation it does not send mortality

rates to 0 as rapidly as the usual Lee- rates to 0 as rapidly as the usual Lee- Carter.

• For data that is not grouped there is more

For data that is not grouped there is more jaggedness in the simulated mortality curves than is seen in a multi-factor Lee- Carter Carter.

Our Mortality Stochastic Model

µ(x, t) = ˆ µ(x, t) · exp(V (x, t) − G(x, t) + H(x, t)),

◮ ˆ

µ(x, t): the long-term trend estimated from the Renshaw et

• al. (1996) model.

◮ Diffusion Process V (x, t):

V (x, t) = σ(x)Zt − 1 2σ2(x)t.

◮ Permanent Longevity Jump G(x, t) ◮ Temporary Adverse Mortality Jump H(x, t)

Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Permanent Longevity Jump G(x, t)

G(x, t) = K(x, t) + D(x, t).

◮ Jump reduction component K(x, t): one-time permanent

jump K(x, t) =

∞

• s=1

ysAs(x)1{t≥ηs},

◮ ηs: time of jump reduction event s ◮ ys > 0: maximum mortality improvement of all ages in jump

reduction event s

◮ As(x) ∈ [0, 1] distributes the effects of event s across different

ages x.

Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Permanent Longevity Jump G(x, t) (Con’t)

G(x, t) = K(x, t) + D(x, t).

◮ Trend reduction component D(x, t): steeper

downward-sloping mortality curve D(x, t) =

∞

• i=1

ζi(t − υi)Fi(x) exp(−ξi(t − υi))1{t≥υi},

◮ υi: time of trend reduction event i ◮ ζi > 0: maximum annual excess mortality improvement of all

ages in trend reduction event i

◮ Fi(x) ∈ [0, 1] distributes age effects. ◮ (t − υi) accumulates this mortality improvement effect. ◮ ξi > 0 specifies the length of trend reduction event i. Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Temporary Adverse Mortality Jump H(x, t)

H(x, t) =

∞

• j=1

bjBj(x) exp(−κj(t − τj))1{t≥τj}, where

◮ τj: time of adverse mortality event j ◮ bj > 0: maximum mortality deterioration among all ages in

jump event j.

◮ Bj(x) ∈ [0, 1] distributes mortality jump impact across ages. ◮ exp(−κj(t − τj)) where κj > 0 models transitory nature of

mortality jumps.

Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Parsimonious Model

µ(x, t) = ˆ µ(x) · exp

• σ(x)Zt − 1

2σ2(x)t

• · exp

  −

M(t)

• i=1

ζ(t − υi)F(x) exp(−ξ(t − υi))1{t≥υi}    · exp   

N(t)

• i=1

bB(x) exp(−κ(t − τj))1{t≥τj}    , where

◮ we exclude the longevity jump reduction component K(x, t); ◮ we assume constant mortality and longevity jump effects for

age x.

Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Parameter Estimates for Different Ages

Figure: The graph on the top presents the volatility σ(x) of log

µ(x, t + 1) µ(x, t) where year t = 1901, 1902, · · · , 2005. The function F(x) (left graph at the bottom) shows the normalized annual excess percentage decrease in µ(x, t) of different ages in the 1970’s. The right graph at the bottom illustrates the function B(x) that distributes the impact of the 1918 worldwide flu on µ(x, t) across ages. The x-axis of all three graphs represents the age. Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Simulated Sample Paths of Force of Mortality: Ages 30, 40, 50 and 60

20 40 60 80 100 0.01 0.02 0.03 0.04

Figure: Simulated sample paths of the force of mortality. The top curve is the simulated force of mortality for

age 60, the one just below it is for age 50, then 40 and the bottom curve is for age 30. The vertical axis stands for force of mortality and the horizontal axis represents time. Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Actual and Simulated Sample Paths of Force of Mortality: Ages 35 and 75

Our model: µ(x, t) = ˆ µ(x, t) · exp(V (x, t) − D(x, t) + H(x, t))

Figure: Actual (linked dotted line) and sample path (solid line) of force

• f mortality for the US male at age 35 (left) and male at age 75 (right).

The vertical axis stands for force of mortality and the horizontal axis represents year.

Samuel H. Coxa, Yijia Linb and Hal Petersena Mortality Risk Modeling: Applications to Insurance Securitization

Lee-Carter Model

• The Cox-Lin-Pedersen model is

complicated, overparameterised and must be calibrated with a fair amount of user judgement judgement.

• A simple idea that produces useful results

is to alter a factor model by introducing a is to alter a factor model by introducing a simple but flexible drift dynamic.

• The drift of the random walk process is

p allowed to change value.

• The drift is permitted to take on values

f 0% t 100% f it hi t i l from 0% to over 100% of its historical values, representing periods of no mortality improvement or accelerated mortality improvement or accelerated mortality improvement.

Lee-Carter Model E h i ibl l f t lit

• Each permissible value of mortality

improvement is tagged with a duration range range.

• Lastly, some migration dynamic must be

specified to direct where mortality p y improvements evolve to when they leave a given value. A M k h i ld b d

• A Markov chain could be used.
• In the following, we use durations

corresponding to draws from a simple corresponding to draws from a simple collection of uniform random variables.

Lee-Carter Model

• The effect of the stochastic drift is shown
• The effect of the stochastic drift is shown

in the following. Improvements are 100%, 130% and 160% of historical.

1500 2000 stochastic drift + Single-Factor Lee-Carter 500 1000 4 5 6 7 8 9 10 11 12 x 10

• 3

2000 Single-Factor Lee-Carter 500 1000 1500 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 500

S-Forwards

• The majority of transactions involving the
• The majority of transactions involving the

hedging of longevity risk are indemnity swaps. swaps.

• The LLMA has proposed standardised

index-based instruments such as S- forwards.

• There are, of course, a number of

complicated practical issues involved in complicated practical issues involved in using S-forward contracts in place of indemnity swaps. y p

• We are interested in illustrating the use of
• ur model to price an S-Forward.

S-Forwards

• We assume that we are interested in
• We assume that we are interested in

determining the forward price on a 20- year S-forward on US male lives aged 65. year S forward on US male lives aged 65.

• We apply the single-factor Lee-Carter

model with stochastic drift. We retain the f bias toward improvement from our previous illustration.

• In practice the possibility of below trend
• In practice, the possibility of below trend

improvement will, of course, reduce the forward price. p

• We use simple indifference pricing with

exponential utility to determine the forward i price.

S-Forwards

• We quote directly from the LLMA technical
• We quote directly from the LLMA technical

note to remind ourselves of the nature of the contract. the contract.

• Consider an example in which a longevity

protection buyer pays the fixed rate and receives p p the floating rate on an S-forward contract with maturity T.

• Th fi

d t f th t ti i th f d

• The fixed rate for the transaction is the forward

survival rate agreed between the parties for the index at time t = 0 corresponding to the p g forward time period T and is denoted by p_forward(0:T).

S-Forwards

• Th

li d i l (fl i ) f h

• The realised survival rate (floating rate) for the

index at maturity T is denoted by p_realized(0:T).

• Th

t t th t t k l t t it T

• The net payment that takes place at maturity T,

the Net Payoff Amount (NPA) is given by: NPA(T) = N ti l NPA(T) = Notional x [ p_realized (0:T) − p_forward (0:T)]

• If NPA(T) is a positive number, this cashflow is

an amount receivable by the longevity protection buyer. If NPA(T) is a negative number, this is an amount payable by the longevity protection buyer.

S-Forwards

• It is seen that the fixed rate payer (protection

It is seen that the fixed rate payer (protection buyer) has a long longevity position, since the fixed rate payer benefits if realized survival rates i b i (i li f ll rise above expectations (i.e., mortality falls below expectations).

• By contrast a floating rate payer (protection
• By contrast a floating rate payer (protection

seller) would receive the fixed rate and pay the floating rate (realized survival rate), benefiting if g ( ), g the realised survival rates fall below expectations (i.e., mortality rises above expectations) expectations).

• In our example the notional is 100mm and

the contract initiation is 2007. the contract initiation is 2007.

S-Forwards

• The 20 year S forward price for the model
• The 20-year S-forward price for the model

with stochastic drift is 0.4924.

• The mean value of 20S65 for the
• The mean value of 20S65 for the

simulation for the model with stochastic drift was 0.4900.

• The 20-year S-forward price for the model

without stochastic drift is 0.4808. Th l f 20S6 f h

• The mean value of 20S65 for the

simulation for the model without stochastic drift was 0 4786 drift was 0.4786.

• The histograms for the 20S65 processes

are shown below.

S-Forwards

20S65 stochastic drift 2000 3000 20S65 stochastic drift 1000 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 3000 20S65 no drift 2000 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 1000

S-Forwards

• The LLMA document gives some
• The LLMA document gives some

suggestions on simple pricing models using assumed risk premia applied to using assumed risk premia applied to mortality projections.

• Presumably, these contracts could be

ff quoted as percentage reductions off projections from a standard model such as a single-factor Lee-Carter as a single-factor Lee-Carter.

• In the end, if these contracts trade in a

liquid market we will have some q interesting data to analyse.