# Analyzing the Surrender Rate of Limited Reimbursement Long-Term Individual Health Insurance

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1 Analyzing the Surrender Rate of Limited Reimbursement Long-Term Individual Health Insurance Chih-Hua Chiao, Shing-Her Juang, Pin-Hsun Chen, Chen-Ting Chueh Department of Financial Engineering and Actuarial Mathematics, Soochow University October 12 15, 2009

2 Introduction Pricing model Numerical example Surrender rate Outline Rational surrender behavior Impact factors Conclusions Discussion

3 Introduction 1 Long term individual health insurance in Taiwan Facts Noncancelable level premium during premium paying period Ageing population, longer life expectancy, advanced medical technology, increasing medical expenditure Then... How to relieve the risk of this kind of product? Taiwan regulates the limit of reimbursement for the total amount in claims throughout the policy period.

4 Introduction 2 Limited reimbursement long term individual health insurance total reimbursement limit Daily hospitalization benefit Appointed multiple For example, NT\$1 million limit is specified using NT\$1, , 000. Cumulative claims exceed the limit, contract will terminate automatically. Risk control more efficiently. Death benefits

5 Introduction 3 Consider A long term individual health insurance with a limited reimbursement cap NT\$1 million The cumulative claim payments of NT\$ 300,000 during first 5 policy years NT\$ 700,000 for the remainder of the life of policy

6 Introduction 4 future total premium payments and future coverage Future total premium payments > future coverage, = Future total premium payments future coverage, =

7 Introduction 5 A rational insured may decide to surrender because the accumulated future premium may be more than the amount of benefit payments received in the future. Lack of suitable surrender rate assumption for the limited reimbursement long term individual health insurance, however. Three topics in this talk analytical solution of single (level) premium rational surrender rate sensitivity analysis

8 Premium model without surrender rate 1 Notations L : reimbursement limit T : future lifetime of the insured aged x, 0 t ω x, with limiting age ω B i : the total claim amount paid at the end of the ith year, i = 1, 2,, ω x. Assume that B i are independent random variables with same probability function.

9 Premium model without surrender rate 2 T : the time of the cumulative claims payments exceeding the limit L, i.e., { } τ T = min τ : B i > L i=1 K : the insured s curtate future lifetime, K = T δ : assumed interest rate

10 Premium model without surrender rate 3 K and T K T The contract of the insured has expired at T. Current benefits amount of the T th year, K < T T 1 L i=1 B i The upper limit L is not reached yet. Death benefits ( K ) L B i + B K+1 i=1

11 Premium model without surrender rate 4 Present value of total claim amount Y B = = [ ] T 1 T 1 B i e δi + L B i e δt if K T i=1 [ K B i e δi + L i=1 min(k,t 1) i=1 i=1 ] K B i e δ(k+1) if K < T i=1 B i e δi + L min(k,t 1) i=1 B i e δ(min(k,t 1)+1)

12 Premium model without surrender rate 5 Actuarial present value E[Y B ] = ω x k=0 { k B 1 (τ)p (T = τ) + τ=1 τ=k+1 B 2 (k, τ)p (T = τ) } P (K = k) where B 1 (τ) = µ(τ)a τ + [L τµ(τ)] e δτ, B 2 (k, τ) = µ(τ)a k + [L kµ(τ)] e δ(k+1), and µ(τ) = E [B i T = τ], τ = 1, 2,, ω x

13 Premium model without surrender rate 6 µ(τ) : given T, annual average claim amount T K E[Y B ] is associated with B 1 (τ). Recall that B 1 (τ) = µ(τ)a τ + [L τµ(τ)] e δτ Present value of annuity of τ years with annual average claim amount µ(τ). Present value of average account balance [L τµ(τ)] when contract is invalid. Intuitively, B 1 (τ) is present value of total reimbursement for τ years.

14 Premium model without surrender rate 7 T > K E[Y B ] is associated with B 2 (k, τ). Recall that B 2 (k, τ) = µ(τ)a k + [L kµ(τ)] e δ(k+1) Present value of annuity of k years with annual average claim amount µ(τ). Present value of the average death benefits [L kµ(τ)]. Intuitively, B 2 (k, τ) is present value of total reimbursement for k years.

15 Premium model without surrender rate 8 As to distribution of K in E[Y B ], or P (K = k), k = 0, 1,, ω x, one can refer to Life Table, say 2002TSO in Taiwan. P (T = τ) can be determined by using renewal process (refer to Ross, 2002). Sample distribution of T Empirical distribution of T

16 Premium model without surrender rate 9 E[Y B ] life annuity immediate with uncertain benefit amount and annuity period. No medical occurrence, zero annuity amount The annuity amount is uncertain. It is related to the severity and frequency of medical requirement. The period of paying benefit is uncertain. It is related to the mean residual lifetime and L.

17 Premium model without surrender rate 10 Level premium At the beginning of year Agreed to contribute h periods Premium contribution periods minimum of h, K and T. For NT\$1 premium paid at the beginning of year, the present value will be Y P = ä K+1, if K = min{k, T, h} ä T, if τ = min{k, T, h} ä h, if h = min{k, T, h}

18 Premium model without surrender rate 11 APV three parts E[Y P ] = E[Y P (K)] + E[Y P (T )] + E[Y P (h)] where E[Y P (K)] = E[Y P (T )] = h 1 ä k+1 P (K = k)p (T k + 1) k=0 h 1 ä k+1 P (K k + 1)P (T = k + 1) k=0 E[Y P (h)] = ä h P (K h)p (T h + 1)

19 Premium model without surrender rate 12 Illustration of E[Y P ]

20 Premium model without surrender rate 13 AREA (I): ä 1 [P (K = 0)P (T 1)] AREA (II): ä 1 [P (K 1)P (T = 1)] AREA (III): ä 2 [P (K = 1)P (T 2) + P (K 2)P (T = 2)] AREA (IV): ä h 1 [P (K = h 2)P (T h 1)+P (K h 1)P (T = h 1)] E[Y P (h)] denotes that h periods of premium have been fully paid.

21 Premium model without surrender rate 14 According to the equivalence principle, the annual level premium P satisfies An illustrative example E[Y B ] = P E[Y P ] Daily benefit: NT\$1,000 Appointed multiple: 1,000 Limit L : 1, 000 1, 000 Assumed interest rate δ : 2% B i : exponential distribution with mean θ = 20, 000 Life Table: 2002TSO Premium paying period h : 20 years

22 Premium model without surrender rate 15 Level premium at each issue age Level Premium (NTD) 65,000 60,000 55,000 50,000 45,000 40,000 35,000 Male Female Issue Age

23 Premium model without surrender rate 16 Level premium at issue age 35 for various assumed interest rates Level Premium (NTD) 45,000 40,000 35,000 30,000 25,000 Male Female Assumed Interest Rate

24 Premium model without surrender rate 17 Level premium at issue age 35 for various multiples Level Premium (NTD) 70,000 60,000 50,000 40,000 30,000 Male Female Multiplier

25 Rational surrender behavior A rational insured may decide to surrender if the accumulated future premium exceeds the amount of aggregate benefit payments received in the future. After the premium paying period, the insured has paid all premiums and he cannot get the surrender value back. So we assume that the insured will not surrender the policy after the premium paying period.

26 Given Surrender rate model 1 reimbursement limit L, premium paying period h, level premium GP (loading β) payable at the beginning of the year Claim amount in the ith year, B i iid assumed Surrender condition L t B i < (h t) GP, t = 1, 2,, h 1 (1) i=1

27 Let s rewrite (1) Surrender rate model 2 (L hgp ) + tgp t i=1 B i } {{ } U t < 0, t = 1, 2,, h 1 (2) {U t, t = 1, 2,, h 1} in (2) can be viewed as a surplus process with Initial surplus (L hgp ), Earned premiums tgp, and Total claims paid t i=1 B i

28 Surrender rate model 3 Refer to Klugman, Panjer and Willmot (2008) Loss Models: From Data to Decisions, a discrete time ruin model is used to establish surrender rate model. U 0 = L hgp U t = U t 1 + GP B t W t = GP B t 0, if Ut 1 < 0 Wt = W t, if Ut 1 0 U t = U t 1 + W t

29 Surrender rate model 4 Probability of surrendering at the end of tth year L [h (t 1)GP ] { u 0 f Wt (w t )dw t } f U t 1 (u)du

30 Numerical analysis 1 Assume that Issue age: 35 Gender: male Limit of total reimbursement: NT\$1,000,000 Amount of claim each year follows exponential distribution with mean θ = 20, 000, 25, 000, 30, 000, respectively. Loading: β = 5%, 10%

31 Numerical analysis 2 Surrender rate for each policy year at 5% loading rate and various expected values of annual claim Surrender Rate theta=20000 theta=25000 theta= Policy Year

32 Numerical analysis 3 Surrender rate for each policy year at 10% loading rate and various expected values of annual claim Surrender Rate theta=20000 theta=25000 theta= Policy Year

33 Numerical analysis 4 Both Figures show that Decreae surrender rate with policy year Variable surrender rate assumption for various medical behavior should be considered in premium pricing. The higher the expected value of annual claim amount, the higher the surrender rate in the first year.

34 Numerical analysis 5 Surrender Rate theta=20000 theta=25000 theta= Policy Year 10% loading rate: Suppose θ = 30, 000, the surrender rate in the first year is as high as 17%, and falls to 5.8% in the second year, and below 1% in the fourth year.

35 Numerical analysis 6 Surrender Rate theta=20000 theta=25000 theta= Policy Year 5% loading rate: Surrender rates at loading rate of 5% are very low, below 5%; that is, the smaller the loading rate, the cheaper the premium; then under the same coverage condition, consistent with visual trend of surrender rate

36 Numerical analysis 7 To facilitate application of surrender rate in premium calculation, the multiplier of surrender rate of each policy year at various reimbursement multiple against the surrender rate of the first policy year are referred to the following Table.

37 Numerical analysis 8 Ex: Surrender rate at the 1st policy year: 10% Surrender rate at the 2nd policy year: 10% =3.331% Surrender rate at the 3rd policy year: 10% =1.605% policy year limit multiple

38 Numerical analysis 9 policy year limit multiple

39 Numerical analysis 10 1,000 multiple, surrender rate decreases more quickly in the first 5 years, and significantly slows down decreasing afterwards. 2,000 multiple, surrender rate decreases more slowly in the first 5 years, and more quickly after 5 years.

40 Conclusions 1 Pricing without surrender rate Analytical approach Both the probability of using up total reimbursement limit and probability of death of the insured are taken into consideration in process of premium calculation. More realistically. No specified calim distribution necessary

41 Conclusions 2 Surrender rate based on the rational surrender behavior Impact factors on surrender rate: Loading rate, annual reimbursement, assumed interest rate and total reimbursement limit multiple The surrender rate in initial policy years is heightened significantly. Providing more accurate assumption in profit analysis.

42 Discussion How about the critical illness insurance? Premium waived after getting ill No recovered, then substandard status Almost impossible to get insurance converage. To hold contract until expiration

43 Thank you for your attention!

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