FINA 3220 Actuarial Model I Group Project Report

ACTUARIAL ELITE PLAN

LI Yuxuan (Steven) TO Chun Yuen (Jerry) ZHANG Xinyi (Cindy) ZHANG Xuanshen (Zachary) ZHANG Yanan (Victoria) ZHAO Shulun (Ellen)

CONTENTS
.Executive Summary Product Design and Features …………………………………………………………

1 1.1. Customer Analysis …………………………………………………………………

1 1.2. Coverage …………………………………………………………………………… 1 1.3. Premium ……………………………………………………………………………

1 1.4. Benefits …………………………………………………………………………… 2 1.4.1. Annuity ……………………………………………………………………… 2 1.4.2. Death Benefit ………………………………………………………………… 2 1.4.3. Optional Supplementary Benefits …………………………………………… 2 1.5. Special case (of Prof. Ma)

………………………………………………………… 3 Assumptions …………………………………………………………………………… 3 2.1. Mortality Rate ……………………………………………………………………… 3 2.2. Interest Rate ……………………………………………………………………… 4 2.3. Investment Rate …………………………………………………………………… 4 2.4. Operational Cost …………………………………………………………………… 5 Methodology …………………………………………………………………………… 5 3.1. Basic Premium Pricing: Equivalence Principle …………………………………… 5 3.2. Percentile Premium Principle under Normal Approximation ……………………… 6 3.3. Simulation …………………………………………………………………………… 7 Product Management ………………………………………………………………… 7 4.1. Projected Cash Flow ………………………………………………………………… 7 4.2. Risk Management …………………………………………………………………… 8 4.2.1. Method I: Purchasing Reinsurance ………………………………………… 8 4.2.2. Method II: Conducting Thorough Underwriting

…………………………… 8 4.3. Value at Risk (VaR) ………………………………………………………………… 8 Limitations …………………………………………………………………………… 8 5.1. Size of Customer Base ……………………………………………………………… 9 5.2. Dividend Accumulation …………………………………………………………… 9 5.3. Lapse Rate ……………………………………………………………………… 9 Appendix ……………………………………………………………………………… 10

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Executive Summary
The “Golden Years Income Plan (10 years)” provided on Moodle is a proposal for the retirement-planning insurance product. However, after evaluating the design and features of the proposal, we find it neither suitable for nor attractive to our target customers, which are a group of university students who would work as actuaries in the future. Therefore, we revise the design and features of the proposal, and come up with a new insurance product called “Actuarial Elite Plan”. We first examine the characteristics of our target customers, in order to better understand their real needs. The objective of product redesign and revision is to better match product features with customer needs at different ages. Based on our knowledge and understanding, we use several assumptions to generate the inputs of the product actuarial models. After plugging in various inputs, we utilize the Excel to construct the basic actuarial models. The models will help us arrive at a proper level of premiums that we should charge, which is calculated under both Equivalence Principle and Percentile Premium Principle. The feasibility and sustainability are critical for the success and profitability of our product. However, several limitations exist in the project, and some of which are beyond the research scopes and our knowledge. Therefore, this report mainly serves to illustrate the essential features of the revised insurance product and how we construct it. The impact of limitations should be examined thoroughly during the real product launch process.

1. Product Design and Features 1.1. Customer Analysis (target customer, salary pattern, job, mortality) “Actuarial Elite Plan” is a tailor-made insurance product for university students who are now 20 years old and major in actuarial science. Most of them plan to pursue future career as actuaries. Being confident with the demand for actuary professionals, we forecast an average-or-above salary level of our target customers. Moreover, we assume that our target customers will, on average, experience the same mortality rates with the Hong Kong population as a whole. Data for both male and female are adopted in our assumption, which is provided by The Census and Statistical Department of Hong Kong Government. As required, we also include a special customer, Prof. Alfred Ma, to our customer base. By doing so, we assume his salary level sufficiently high, his current age is 30, and his mortality rate follows the statistics as mentioned before. Please refer to part 1.5. “Special Case (of Prof. Ma)” for the details of this product design. 1.2. Coverage The death benefit of the basic plan covers ages from 20 to 85. We set the maximum age at 85 years old, considering that, on average, life expectancy of total population is around 83 years. Moreover, there is ample evidence that the mortality rates at high ages have different patterns, and some researches12 show that the mortality rates increase at a diminishing rate with age over 85. Therefore, for simplicity, we set the maximum age at 85 years. 1.3. Premium “Actuarial Elite Plan” is designed with a premium payment period of 20 years. In order to better cater for customers with different salary patterns, we provide two plans for different customers: the Two-stage Payment Plan for the rest of the students and the Level Payment Plan for Prof. Alfred Ma. (Please refer to part 1.5. “Special Case (of Prof. Ma)”). The choice of the plan is based on our customers’ decisions at the time of application, and no change of plan is accepted once the policy is issued. 20-year Two-stage Premium Payment Plan (for the Rest of Students) Under the Two-state Payment Plan, the premium amount is guaranteed to remain level at 1,080 HKD (Male) / 1,074 HKD (Female) per month within the first 10 years of the premium-paying period and then doubles to 2,160 HKD (Male) / 2,148 HKD (Female) per month in the second half.

1 Coale, A. and Guo, G. (1989). Revised Regional Model Life Tables at Very Low Levels of Mortality. Population Index, 55(4): 613–643. 2 Coale, A. and Kisker, E. (1990). Defects in Data on Old-age Mortality in the United States: New Procedures for Calculating Mortality Schedules and Life Tables at the Highest Ages. Asian and Pacific Population Forum, 4(1): 1–31.

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This premium payment design fully takes into consideration of the customers’ salary patterns. Compared with the level payment plan, our customers only need to pay lower premiums in the first 10 years when they just start working in their 20’s and have a relatively lower level of salaries; and later, as their capability to pay increases with time, they will find it much easier to afford relatively higher premiums in their 30’s. With a better match of our customers’ salary patterns, this considerate design will definitely be a strong selling point. 1.4. Benefits 1.4.1. Annuity Income The income period comes 20 years after the premium-paying period, starting when the insureds are 60 years old. The insureds will receive a monthly income throughout the 25-year income period. The monthly income is designed to be 5,000 HKD per month. A common measurement tool for retirement planning is the replacement ratio, which is defined as a person’s gross income after retirement, divided by his or her gross income before retirement3. International experience and conventional wisdom indicate that most people will need around 75% replacement ratio to maintain similar standards of living in retirement, compared with that before the retirement. Currently in Hong Kong, the protection of the Mandatory Provident Fund (MPF) system cannot provide sufficient retirement protection. Therefore, our “Actuarial Elite Plan” will be a valuable choice for our customers to provide additional benefit. 1.4.2. Death Benefit In the unfortunate event that the insured dies after the policy becomes effective, a death benefit will be paid to the beneficiary. a) If the insured dies within the premium-paying period (age from 20 to 39), the beneficiary will receive 100,000 HKD. b) If the insured dies after the premium-paying period and before the income period (age from 40 to 59), the beneficiary will receive an increasing benefit proportional to the insured’s age, linearly increasing from 100,000 HKD up to 700,000 HKD. c) If the insured dies during the income period (age from 60 to 85), the beneficiary will receive 700,000 HKD. 1.4.3. Optional Supplementary Benefits (Riders) Good health and stable job status value a lot to our customers; however, our insurance product cannot guarantee a full protection of our customers. Many uncontrollable factors, such as the overall economy performance, the possibility of accidents, and the occurrence of critical illnesses, will exert extra risks on our customers. Therefore, we

3 2008 Replacement Ratio Study, Aon Consulting, http://www.aon.com/about-aon/intellectualcapital/attachments/human-capital-consulting/RRStudy070308.pdf.

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suggest provide appropriate riders and supplementary benefits to our basic plans, such as Terminal Illness Advance Payment Benefit, Unemployment Benefit, etc. The feasibility and sustainability of this suggestion is not thoroughly examined in this report. Further research and work are needed to arrive at a more concrete conclusion. Below is an illustration of the “Actuarial Elite Plan”.

1.5. Special Case (of Prof. Ma) Under the design and coverage of the basic plan, our customers are 20 years old. In order to consider the special case of Prof. Ma, we will revise the basic plan and change the insured age from 20 to 30, and we will keep the other features. We only provide “20-year Level Premium Payment Plan” for Prof. Ma, and the premium amount remains level at 2,236 HKD per month and will not increase with age throughout the 20-year premium-paying period, allowing easy financial management. Therefore, under this special case, the premium-paying period is from age 30 to 50, and the income-period from age 60 to 85. 2. Assumptions As our objective is to design the most attractive product as long as it is feasible and sustainable, we would only consider factors that will have a significant impact on the feasibility and sustainability of product. 2.1. Mortality Rate: Male and Female We adopt Hong Kong Life Table for Males / Females, 2011 from Hong Kong Census and Statistics Department as raw data to calculate annual mortality rate. It is the latest official data from the most thorough population survey in the past five years, and also the best estimation for the future.

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We assign month as the unit of payment and benefit periods. Based on the assumption of Uniform Distribution of Death (UDD), we can calculate the monthly mortality rate. 2.2. Interest Rate According to the regulation in Hong Kong, the rates of interest to be used in calculating the present value of future payments by or to an insurer shall be no greater than the rates of interest determined from a prudent assessment of the yields on existing assets attributed to the long term business. While the long-term yield assumed should not exceed the lowest among the following three. a) A prudent assessment of the yield of long-term fixed interest securities issued by the national government of the country in which currency the liabilities are denominated b) If the yield referred in (a) exceeds 6% per annum, 6% plus one quarter of the excess c) 7.5% per annum4 Historically, Hong Kong’s Government Bond Yield for 10-Year Notes averaged 4.80 % from 1996 to 2012. Although it reached a record low of 0.60% in November of 2012, we can regard it as a mere fluctuation of the yield.5 Therefore, we can safely assume the interest rate to be 4.00% per annum just for the illustration purpose. 2.3. Investment Return According to the research of asset allocation of insurance companies6, we assume that the company allocates 75% of its assets in bonds, 5% in cash or short-term debt, 8% in equities and 12% in others.

Asset Allication of Insurance Company

12% 8% 5%

Bonds Cash and Short-tem Debt Equities
75%

Others

4 Section 59, Cap 41, Insurance Companies (Determination of Long Term Liabilities) Regulation http://www.legislation.gov.hk/blis_pdf.nsf/6799165D2FEE3FA94825755E0033E532/43F71AE9B9B8BF63482 575EE0032DCAF/$FILE/CAP_41E_e_b5.pdf 5 Hong Kong Government Bond 10Y, http://www.tradingeconomics.com/hong-kong/government-bond-yield 6 https://www2.blackrock.com/webcore/litService/search/getDocument.seam?contentId=1111147085&Source= SEARCH&Venue=PUB_INS. Here we utilize an American case for illustration.

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The average yield to maturity of 364-day Exchange Fund Bills issued by Hong Kong Monetary Authority from 1996 to 2012 is approximately 3.00%. And Hong Kong’s Government Bond Yield for 10 Year Notes averaged 4.80% from 1996 to 2012. From 1996 to 2011, the average return of Hang Sang Index is around 4.90%. Assuming the average return from others is 6.00%. According to WACC, the investment return of the company assets can be approximated by 75%×4.80% + 5%×3.00% + 8%×4.90% + 12%×6.00% = 4.86% per annum. We can safely assume that the average investment return of the company is 4.00%. 2.4. Operational Costs For simplicity, we only consider the expenses that affect the individual insurance policy significantly. The initial cost consists of 2,000 HKD and 50% of the first premium per policy. The renewal cost is 3% of each subsequent premium, i.e. premium paid from the 2nd policy month to the 240th policy month. And the fix cost is 10 HKD per policy month throughout the policy period. The termination cost is too small relative to other costs; hence we ignore it during the calculating process. 3. Methodology 3.1. Basic Premium Pricing: Equivalence Principle Having designed the product structure and correctly extracted the life contingency factors from the Hong Lifetable for each group of customers, we are able to obtain the gross premium. We firstly set E ������!! = 0 for each individual (i=1, 2 … 50), and get three sorts of premium: P1 for 20-year-old male, P2 for 20-year-old female, and P3 for 30-year-old male. Since
!" !

������ ������������������ ������

! ������!! ! ������!!

= 0 ������������������ ∀ ������ ������������������������������������������ ������
!!!

������!! = ������ ������ = 0
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= 0 ������������������ ∀ ������ ������������������������������������ ������������������������������������ ������
!!!

������!! = ������ ������ = 0

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Then we can know {������������ | ������ ������������ = ������} ⊊ {������������ ������ ������ = ������ . We can conclude that these P1, P2, P3 ������������ are more supportive prices than that under the portfolio equivalence principle for our company. We have calculated the gross premium as the following, Premium Male Female Prof. Alfred Ma Annual 10,797 10,739 21,816 Monthly 900 895 1,818

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Knowing only the expected loss for the group is not sufficient to determine the sustainability and profitability of the product. We need to investigate the probability distribution of loss. 3.2. Percentile Premium Principle under Normal Approximation Since ������!! s are independent but not identically distributed, by the Lyapunov Central Limit Theorem, we know that approximately, ������������ − ������(������������ )
������ ������������������(������������ ) ������ ������ !

~������(������, ������)

Then we set a margin percentage δ such that,
! ! ! ������! = ������! 1 + ������ , ������! = ������! 1 + ������ , ������! = ������! 1 + ������ , and ! Pr ������! < 0 = 95%

Using the Excel “Solver” function, we obtain ������ = 2.83% Simulation to Verify the Normal Assumption Now we are going to check if the loss (L) of our product follows a normal distribution. ! Since we know the distribution of each random variable ������!! , we can use simulation to get sample distributions to check the fitness of the normal approximation. We choose Inverse Transform Method to simulate. See next part for details. Then we ! figure out that the sample distribution of ������! is a significant positive kurtosis, so normal approximation may not be accurate enough. Also, by q-q plot, we can easily draw the conclusion. One of the representative simulation results is as follows.

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3.3. Simulation We sort the present value of loss in each year in an ascending order and paired with the associated with the probability of having that amount of loss, we are able to get the probability mass function of the loss (L), ie. P(L< l) for any l. As a result, we can also get the c.d.f. of the loss (L). (see Excel worksheet ”male”, ”female”, ”Alfred” Column AB and AC. One of the individual c.d.f. plots is presented as follows: Then we are going to generate a sample distribution of 50 policyholders (34 male students, 15 female students and Prof. Alfred Ma) by using inverse function transformation. The mechanism is as follows L = F-1 (U) where U~U(0,1) and F-1 is the inverse c.d.f. function. U is generated by the embedded random number generator in Excel.
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0.8 0.6 0.4 0.2 0

Since the c.d.f. of L is discrete, if we obtain -­‐150000 -­‐100000 -­‐50000 0 50000 100000 a U such that L1< F-1 (U)