Understanding the Death Benefit Switch Option in Universal Life Policies

Transcription

1 1 Understanding the Death Benefit Switch Option in Universal Life Policies Nadine Gatzert, University of Erlangen-Nürnberg Gudrun Hoermann, Munich

2 2 Motivation Universal life policies are the most popular insurance contract design in the U.S. - Lifelong policies with flexible premium payments (frequency, amount) as long as cash value remains positive - Death benefit either - level: fixed face amount, or - increasing: pays available cash value in addition to fixed amount - Embed the option to switch from one death benefit scheme to the other: death benefit switch option - Switch from increasing to level without costs / evidence of insurability (unlike switch from level to increasing)

3 3 Motivation Option of no concern for insurers? - Death benefit is fixed at the current level, does not affect net amount at risk - But, crucial: dependence on premium payment behavior after switch (not prescribed by insurer) Combination of two options, can be very valuable Has not been investigated to date

4 4 Aim Enhance understanding of this feature - Develop model framework of increasing universal life policies - Incorporate switch probabilities and stochastic interest rates - Investigate effects of adverse exercise behavior depending on health status of insureds - Consider mortality heterogeneity using a frailty factor - Assume different switch probabilities for different health status - Account for modified premium payment behavior after switch

5 5 Aim Based on this model: quantify net present value of option from the insurer s perspective - Conduct simulation analysis - Sensitivity analysis with respect to frailty distribution - Derive policy recommendations for life insurers

6 6 The model of an increasing universal life policy Pool of increasing lifelong universal life policies - Cash value (policy reserve) V t - Increasing death benefit Y = Y + V, t = 1, K, T - One-year table probability of death at age x+t - Constant annual interest rate i - Constant annual premiums B (equivalence principle) T 1 T 1 t h ( 1 ) ( 1 ) B + i = Y q + i h= 0 h= 0 x+ h t h 1 t t q, t = 0, K, T 1 x+ t

7 7 Death benefit switch option Death benefit at time t given exercise of the switch option at time τ ( τ ) Yt, t = 1, K, τ Yt = Yτ, t = τ + 1, K, T - Switch before peak of cash value: original premiums too high, need to be reduced - Switch near, at, after peak of cash value: higher premiums needed due to higher death benefit

8 8 Premium payment scenarios Cannot analyze death benefit switch option alone: we need to make assumptions about premium payment Consider two viable scenarios after switch: - Minimum constant premium (level premium scenario) - Minimum flexible premium (risk premium scenario) Ensure positive cash value throughout the contract term Any other constant or flexible premiums need to exceed these premium amounts B, ( τ ) t = 0, K, τ 1 Bt = ( τ ) t = τ, K, T 1 B,

9 9 Model framework: Premium scenario Minimum constant premium (level premium scenario) T τ 1 T τ 1 ( τ ) t ( t+ 1) t x+ τ ( 1+ ) + τ = τ t x+ τ x+ τ+ t( 1+ ) t= 0 t= 0 ( τ ) B B p i V Y p q i T τ 1 ( t+ 1) Yτ t p x q + τ x+ τ+ t( 1+ i) Vτ t= 0 = max,0 T τ 1 t t p x+ τ ( 1+ i) t= 0

10 10 Model framework: Premium scenario Minimum flexible premium (risk premium scenario) B ( τ ) t B, t = 0, K, τ 1 = ( τ ) 1 ( τ ) max{ 0, q x+ tyt+ 1 ( 1 + i) Vt }, t = τ, K, T 1

11 11 Contract valuation Valuation depends on mortality Consider mortality heterogeneous insureds - Individual frailty factor d specifies individual's state of health d q x, d q x < 1 qx = 1, x= min x% { 0, K, ω} : d q x% 1 for x { 0, K, ω} and qω : = 1 for d < 1 0, otherwise d 1 d <1 Insureds with average or below-average life expectancy Insureds with above-average life expectancy 1 α 1 Γ β f(,, )( d) = ( d γ) e, for d γ, γ, α, β > 0. αβγ α Γ α β ( ) d γ

12 12 Contract valuation Switch probabilities s(t,d) ( ) ( ) = P ( τ ) = ( ) 1 ( ν ) Short-rate process: Vasicek model k - Affine term structure, zero bond price formula: h 1 F k k s h s τ h= 1 ν = 1 ( ) ( ) Q dr ( t) = κ θ r ( t) dt + σdw t t t rudu ( ) 1 e 2 ( ) 0 t P κ 0, t e σ σ σ κ =Ε = exp θ r t θ ( 1 e ) κ 2κ 2κ 4κ

13 13 Contract valuation Net present value (NPV) of the increasing policy T 1 t T 1 t+ 1 Q r( u) du r( u) du 0 0 NPV ( d ) B 1 { ( ) } e Q =Ε Yt 1 1 K x t + { K( x) t} e Ε = t= 0 t= 0 T 1 T 1 ( ) ( ) = B p P 0, t Y p q P 0, t+ 1. t x t+ 1 t x x+ t t= 0 t= 0 NPV of the increasing policy with death benefit switch option ( ) ( ) T 1 t T 1 t+ 1 ( ) ( ) ( ) ( ) r u du Q r u du t { ( ) } t 1 1 K x t + { K( x) = t} τ Q τ τ =Ε Ε t= 0 t= 0 NPV d B e Y e T 1 T k 1 T 1 T k 1 ( k) ( k) = Bt s ( k) ( 1 s( h) ) t pxp( 0, t) Yt+ 1 s( k) ( 1 s( h) ) t pxqx+ tp( 0, t + 1 ). t= 0 k= 1 h= 1 t= 0 k= 1 h= 1

14 14 Contract valuation Expectation NPV = E Q ( NPV ( D) ) ( ) ( τ) ( τ = E Q ) ( ) NPV NPV D Value of the death benefit switch option: Difference between value of policy with switch option and value of policy without the switch option ( τ ) Opt NPV = NPV NPV

15 15 Numerical examples: Input paramters Policy face value Y = $ Age at inception: x = 45 years Actuarial minimum interest rate i = 3.5% U.S CSO male ultimate composite mortality table Frailty distribution D Γ( 2.0;0.25;0.5 ). Constant annual premium B = $5,937 (calibrated) NPV for increasing policy (without switch) from the insurer s perspective is NPV = $2,866

16 16 Value of the death benefit switch option by constant switch probability (insurer perspective) $2'000 "risk premium" "level premium" $1'000 NPV Opt $0 -$1' % 5% 7.50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% -$2'000 -$3'000 Switch probability Negative for level premium scenario Positive but decreasing for risk premium scenario, due to high risk premiums, but: turns negative in case of policy lapse High s implies more negative values (early switch, long lifetime)

17 17 Value of the death benefit switch option by switch probability and health status: Level premium Switch probability if d<1 (above-average life expectancy) 0% 2.5% 5% 7.5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 7.5% 5% 2.5% 0% $2'000 $1'500 $1'000 $500 $0 -$500 -$1'000 -$1'500 -$2'000 -$2'500 -$3'000 -$3'500 -$4'000 NPV Opt Switch probability if d>=1 (average or below-average life expectancy)

18 18 Value of the death benefit switch option by switch probability and health status: Risk premium $2'000 $1'500 $1'000 $500 NPV Opt $0 -$500 -$1'000 0% 2.5% 5% 7.5% 10% 20% 30% 40% Switch probability if d<1 (above-average life expectancy) 50% 60% 70% 80% 90% 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 7.5% 5% 2.5% 0% Switch probability if d>=1 (average or below-average life expectancy)

19 19 Switch option value for specific exercise scenarios depending on health status s=100% at t=41 (peak) s=10% t=25 to t=41 s=10% to s=100%* (linear) t=25 to t=41 s=10% t=5 to t=15 All Below LE Above LE All Below LE Above LE All Below LE Above LE All Below LE Above LE Level premium Risk premium Risk premium (lapse) d 1 d <1 Insureds with average or below-average life expectancy Insureds with above-average life expectancy Lapse scenario: lapse as soon as risk premium exceeds 10% of new level death benefit

20 20 Results Strong adverse effects can be observed depending on premium payment method and health status Scenarios that are intuitively rational pose greatest threat to insurers, namely - If insureds with above-average life expectancy switch early and thus save risk premiums by making level payments - If impaired insureds set out premium payments after switch, being aware of possibly not surviving until high risk premiums have to be paid

21 21 Policy implications for life insurers Analysis allowed identification of four key factors of relevance for the switch option value: - insureds life expectancy - premium payment method after switch - switch probabilities (time of switch) - lapsation Combination of these factors can make switch option either valuable or risky for insurer Problem: Option can be valuable when exercised early as well as late during the contract term

22 22 Policy implications for life insurers Switch as an alternative to surrender Impose new evidence of insurability (tradeoff: costs, penalizes healthy insureds) Prescription of premium payments after switch, combined with charges for group that causes adverse effects Restrict switch exercises to predefined time ranges In summary: death benefit switch option can pose a threat to insurers in case of adverse exercise behavior with respect to insureds health status Careful monitoring is crucial

23 23 Thank you very much for your attention! Understanding the Death Benefit Switch Option in Universal Life Policies Nadine Gatzert, University of Erlangen-Nürnberg Gudrun Hoermann, Munich

24 24 Deterministic results: Level premium $20'000 $0 -$20'000 N P V O p t -$40' $60'000 -$80'000 Time of death Switch exercise time

25 25 Deterministic results: Risk premium $100'000 $80'000 $60'000 $40'000 $20'000 $0 -$20'000 -$40' Time of death Switch exercise time