State-Price Deflators and Risk-Neutral valuation of life insurance liabilities

Transcription

1 Association of African Young Economists Association des Jeunes Economistes Africains Issue: 11 / Year: October 2014 State-Price Deflators and Risk-Neutral valuation of life insurance liabilities Bell F. Ouelega The views and interpretations in this paper are those of the author(s) and do not necessarily represent those of the AAYE. The Association of African Young Economist (AAYE) is a not-for-profit, non-partisan organization with the vision of a future for Africa where policy decisions emanate from rigorous scientific evidence by Africans and fellows for the welfare of Africans and partners. The AAYE RP WPS is double blind peer reviewed and indexed at IDEAS REPEC, SSRN and ACADEMIA. Head Quarter: Bastos, PO BOX: 959, Yaounde Cameroon; Site web: info@aaye.org; Tel.: Declaration receipt number: 005/RDA/F35/SAAJP.

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3 State-Price Deflators and Risk-Neutral valuation of life insurance liabilities Bell F. Ouelega * Abstract: Worldwide life insurance regulations are converging towards stochastic valuation of liabilities. Some regulatory framework requires the actuary to estimate the market consistent value of the liabilities. Often, a risk neutral ESG is used to project and discount future liabilities cash flows. Life insurer s liabilities cash flows are impacted by policyholders dynamics: lapses, dynamic lapses, and surrenders. Such dynamics are related to economic variables for which applying risk-neutrality is challenging, to say the least. An alternative approach is to use a real world ESG with deflators. The purpose of this paper is to contribute to the financial economics literature on state-price deflators. In this report, we compare the calculations of the market value of a call option under the risk neutral valuation (the Q-measure) and the real world valuation with the deflators (The P-measure). We also look at the Market Value of Liabilities, and the Expected Present Value of Future Profits (PVFP) under the risk neutral valuation and the real world valuation with the deflators for a profit-sharing Single Premium Deferred Annuity (SPDA)s subject to a participation rate, a spread on the index earning, and a minimum guarantee rate. The models built for this paper point to the following conclusions Firstly, State- Price Deflators and risk neutral valuation result in equivalent valuation for publicly traded securities; and secondly, State-Price Deflators and risk neutral valuation result in equivalent valuation for non-traded cash flows such as deferred annuities allowing for participation rate in an equity index, a cap and a spread. Keywords: Deflators, ESG, Monte-Carlo, Risk-neutral, Martingale. JEL Classification: G22; C63; G12. Correct citation: Ouelega, B. F. (2014), State-Price Deflators and Risk-Neutral valuation of life insurance liabilities, AAYE PR Working Paper Series, N 11, Association of African Young Economists, October * FSA CERA MAAA. Aviva USA, Director Quantitative Research. bell.ouelega@avivausa.com; Tel. :

4 METHODOLOGY Description of instruments and products priced in this analysis Simple options We have priced call options and put options of various maturity and strike. The call option provides the holder the right to buy the underlying asset at a predetermined price (the strike price) on a specified date (the maturity). The put option provides the holder the right to sell the underlying asset at a predetermined price (the strike price) on a specified date (the maturity). Profit-sharing simple SPDA The SPDA credits an annual interest based upon the appreciation of the portfolio of assets (invested in the equity, for the continuous runs; and invested in a combination of equity and the risk-free bond for the discrete models). The crediting rate is subjected to a participation rate, a spread and a minimum guaranteed rate. In the discrete case, we looked at a three years contract (a three-step binomial lattice), and in the continuous case, we looked at a 10 years contract. The Economic environment We are in a world where equity returns are assumed to follow a Geometric Brownian Motion (GBM) process. Thus, at any future time t, the process for the terminal stock price S(t) in the Q-measure is such that: ( ( )) ( ( )) (( ) ) Where r is the annual risk-free rate and is the annual volatility of the equity s return. The process for S(t) in the P-measure (the real world measure ) will take into account the Market Price of Risk (MPR) of the underlying stock. It is: ( ( )) ( ( )) (( ) ) Where is the drift (r + MPR) of the stock price. We have assumed that interest rates are constant throughout the term of the contracts. We are essentially in the Black-Scholes- Risk Neutral world (the Q-measure), and in the Black-Scholes-Real World with the Black-Scholes Deflators. The Deflators Essentially, the deflators are stochastic discount rates used in the P-measure to maintain the arbitrage-free, market-consistency property. 2

5 The deflators allow the modeler to use realistic scenarios in projecting future cash flows. In particular, assets expected returns are greater than the risk-free rate. For instance, when valuing the call option in the P-measure with the state-price deflators, one will follow the following steps: o Simulate N possible ending stock price using the P-measure process for the stock price. You assume that the stock s return is the Real World return (r+mpr). That gives you N possible states of nature. o For every simulated stock price ( ( )), compute the payoff of the option ( ( ( ) )). o Compute the specific state-deflator for each of the N possible states of nature. o Discount each simulated payoff of the call by the specific state-deflator. o Take the average of all the discounted payoffs. In mathematical terms, the value of the call option is simply: ( ) ( ( ) ) Where: ( ) ( ) ( ) ( ) (( ) ) And ε follows a normal distribution of mean 0 and variance 1. The deflators are scenario and time specific. The deflators must also satisfy two Martingale Conditions: (i) The deflators shall be able to price the underlying asset: ( ) ( ) (ii) The deflators shall be able to price the zero-coupon bond. 3

6 In a more general setting, we will be given a real world ESG (Economic Scenario Generator) comprising of equity returns, yield curve, real estate, inflation, credit risk. In such setting, the deflators are computed as: Where: ( ) ( ( ) ) ( ) ( ) The above formula provides satisfactory deflators (meet the two Martingale conditions) in the examples that we have considered. In Appendix 1, we provide a mathematical proof for the deflator s formula. In the simple Black-Scholes-economy that we have considered, the formula for the deflator boils down to: { ( ( ) ) ( ) } Notice that the formula involves the Sharpe Ratio for the equity risk. In the financial economics literature, this is known as the Black-Scholes-Deflators. RESULTS Call Option For a wide variety of tradable securities, risk-neutral valuation and real-world with deflator s valuation are consistent. In fact, we calculated the value of a simple call option under three methods: (i) (ii) (iii) Black-Scholes Closed form: The first method is the closed form solution, or the Black-Scholes solution. In fact, it is not surprising that this method is consistent with the risk-neutral valuation approach. Q-Measure Valuation: Then we value the option using a Risk-Neutral Monte Carlo simulation and finally, P-Measure with deflators: We priced the option using a Real World simulation with the deflators. 4

7 Details about the equations, methodology and results are found in Appendix 2. We provide the price of an at-the-money option on a stock with the following characteristics: Then we allow the time to maturity (T) of the option to be 1 year and 10 years. Result: With large sample sizes, RN and RW approaches provide results that are closed enough, even when the maturity of the option increases. Profit-sharing SPDA Then, we performed discrete-time valuations of some simple SPDAs. First, we looked at a one-period model Example 1: Simple SPDA, One period binomial tree model We consider a simple economic environment characterized as follows: Where the risky equity is allowed to either go up or down (the real world process) at real world probabilities (p and 1-p) depicted in figure 1. This gives rise to two possible states of nature at time 1 (the equity up scenario and the equity down scenario). The risk-free rate is assumed to be r=0.05. We represent the two scenarios with their real-world probability occurrence in table 1. Let us assume that an insurance company operating in this one-period economic environment has issued the following liability depicted in table 2. Assume the insurance company adopts the investment strategy in table 3. We are going to calculate the Present value of Future Profits (PVFP) under the P- measure (The Real world measure with the deflators) and the Q-measure (The Risk- Neutral Measure with risk-free rate discounting): First, we need the risk-neutral probabilities to be used in the Q-measure (q, and 1-q). We solve ( ) And get: 5

8 For the real world calculation (the P-measure), we need to find the deflators: In Appendix 1, we provide and proof the general formula for deflators. Thus for Scenario 1 (Equity Up, and bond level), we get: Likewise, for scenario 2 (Equity down, and bond level), we get: Based on the liability profile and the asset mix, we can now project the future cash flows under both scenarios, as depicted in table 4. Thus, the Real world PVFP is calculated as: ( )( )( ) ( )( )( ) ( )( )( ) For the Risk-Neutral calculation, we perform the following valuation instead: ( )( ) We end up with the same value: $ ( )( ) ( )( ) Thus, both the P-measure and the Q-measure provide the same PVFP for this SPDA in this simple one-period economic environment. The MV of assets under the P-measure and the Q-measure is the same, it is calculated as: ( ) ( )( )( ) ( )( )( ) ( )( )( ) Thus: ( ) So what is the difference between the two paradigms? The expected asset s return in the RW (the P-measure) is: (0.6)*(10.525%)+(0.4)*(0.775%)=6.62% 6

9 The expected asset s return in the RN (the Q-measure) is: (0.4333)*(10.525%)+( )*(0.775%)=5%= the risk-free rate This finding is consistent with the projection of terminal stock prices in calculating the value of the call option. In the real world measure, assets expected return reflects their riskiness and include a risk premium, whereas in the risk-neutral world, assets expected return is the same as the risk-free rate, irrespective of risk. In the next example, we extend the time period to three years; we add a spread to the product profile and adjust earnings for DTA (Deferred Tax Assets). Example 2: Profit-sharing SPDA, Three time-period binomial tree model The economic environment is characterized by: The risky equity process is depicted in the three-steps recombining binomial tree in figure 2a. The tree is recombining, thus, the Cox-Ross-Rubinstein condition holds (in the real world). Where assumption for u and d are in figure 2a. The risk-free rate and the real world probability of the up-movement are in figure 2c. Based on this, we solve for the expected real world annual return of the equity and the RN probability of the up movement as in figure 2d. The risk-neutral probability of the up-movement is calculated as in figure 2e. Now, we are in a position to build the real world ESG for our environment. Since we are dealing with a three time steps binomial tree, they are 8 possible states of nature: table 5. We compute the Real World and the Risk-Neutral World probabilities of each scenario in table 6. For instance, the P-measure probability of scenario 1 is the probability of (3 up movements of the equity): an up and up and an up move of the in the real world, to be consistent with our notation, this is: Likewise, the Q-measure probability of scenario 1 is found to be: 7

10 We are now in a position to use the equation in Appendix 1 and find the deflators for every scenario and all three time periods as in table 7a. Let us determine whether or not the deflators satisfy the two Martingale conditions: (i) (ii) The deflators shall be able to price the underlying asset: The deflators shall be able to price the zero-coupon bond. The calculations are detailed in table 7b. The liability profile issued by the insurance company operating in this environment is the following comprises a participation rate, a minimum guarantee rate and zero spread: table 8. So we initially assumed that there is no spread. Let us assume that the asset mix is as in table 9. Now we start the cash flows projection. First the investment returns, table 9a. Then the liability cash flows, table 9b. The shareholders surplus before taxes, table 9c. The cash taxes, table 9d. And the shareholders surplus adjusted for taxes and DTA, and the PVFP calculation, table 9e. Once again, both methodologies return the very same PVFP. We later introduced a spread of 1%, and this increases our PVFP to $ The details are in table 10. In conclusion, with the three-period binomial model, both the RN valuation and RW with deflators valuation provide the same value for the PVFP, the MV of liabilities and the MV of assets. Example 3: Profit-sharing SPDA, Continuous-time model We also looked at a continuous-time version of the simple SPDA. We computed the Present Value of the liabilities under both paradigms (RN valuation and RW valuation with the deflators). Product: 10 years profit sharing SPDA. Money invested in equity. There is a minimum guaranteed rate and a spread. Over the 10 years period, the product credits the annual 8

11 appreciation of the equity index, subject to a spread and a minimum guaranteed rate with a full return of premium at the end of the period. Details about the simulations are provided in table 11. The PV(Liabilities) is calculated as follows: ( ) Where, in the risk-neutral world, for every scenario i: ( ) ( ) ( ) Where, in the real-world with deflators, for every scenario i: ( ) ( ) ( ) Where: We found that both paradigms results are pretty consistent. Even when the minimum guaranteed rate varies wildly above and below the risk-free rate, the PV(Liabilities)s under both paradigms are still statistically closed. COMPARISON BETWEEN THE RISK-NEUTRAL MEASURE and THE REAL-WORLD WITH DEFLATORS MEASURE Risk measures/risk margin Under the P-measure valuation with the deflators we can build a real world distribution for the PVFP. This allows the modeler to compute relevant risk measures: 1. The mean 2. The CTE (Conditional Tail Expectation) and the VAR (Value At Risk) 3. The Economic Capital 4. And the confidence interval for all these estimates. Under the Q-measure with discounting at the risk-free rate, the only risk measure of relevance is the mean. 9

12 Implementation It is quite complex to produce a real world ESG with deflators. However, there are actuarial software that can operate in either paradigm. Comfort of valuation Under the Risk-neutral valuation, scenarios are generated under the assumption that the expected return from all assets is the risk-free rate, irrespective of the asset s risk or volatility. As a result, the probability measure (the q-measure) attached to every scenario (risk-neutral scenario) reflects how likely it is to occur in the future, assuming zero risk premium for all assets. In short, the so-built ESG (Risk neutral ESG) cannot be used for answering real world business questions. Because, in reality investors require compensation for risk, and will not invest in a more risky asset unless there is compensation for doing so. The Real-World with the deflators valuation uses a realistic probability distribution for all assets (or economic variables). This is an estimate of the real world probability of the scenario occurring in the future. True, the calibration of such ESG is difficult. But once this is done, the ESG can be used for both arbitrage-free, market consistent valuation as well as answering real-world business problems requiring realistic projections. In fact, with a real world ESG, management and internal stakeholders gain comfort with the underlying risks of the business. Policyholders Dynamics When using a risk neutral ESG, economic variables are generated under the a ssumption that on average, instruments earn the risk free rate. Such assumption impacts policyholders dynamics. In fact, policyholders lapsation depend not only market performance, it also depends on some economic variables for which finding a riskneutral equivalent is problematic: for instance, unemployment rate, and inflation rate. As a result, a risk neutral ESG valuation might mis-represents life insurance liabilities because of the difficulty to formalize risk-neutral policyholder dynamics. This is not the case when using a real world ESG. IASB level II and FASB 157 Both accounting standard emphasize an unbiased, probability-weighted average value of future cash flows (Current Exit Value, under IASB and Fair Value under FASB 157). Under the P-measure and the Q-measure this value can be estimated for a wide range of life insurance contracts. Both accounting standards diverge in how risk margins are computed. Under IASB, risk margins are to be calculated in one of three ways: (1) A confidence interval (2) A conditional Tail expectation or 10

13 (3) A cost-capital approach. Under the P-measure, methods (1) and (2) are directly applicable. The P-measure is the appropriate measure to understand the tail-behavior of assets and liabilities, not the Q- measure. With the Q-measure, it is unclear how that will be achieved, since the mean is the only relevant metric. A potential solution might be to shock the scenario set and rerun the simulation. CONCLUSION Summary of Results In this paper: 1. We obtained the same value for traditional call and put options using the following methods: (i) (ii) (iii) The Closed-form Black-Scholes-Merton equation The Risk-Neutral Monte Carlo Simulation approach The Real World with Deflators Monte Carlo Simulation 2. When computing the PVFP (Present Value of Future Profits) for a simple profitsharing SPDA contract (The premium is invested in a risk-free government bond and a risky security with real world dynamics captured by a recombining Binomial lattice) using a three-period binomial lattice on a one hand, and a continuous GBM process on the other, under the Risk-Neutral measure (the Q- measure) and the Real World with the Deflators Measure (The P-measure), the value of the PVFP under both paradigms is the same. These results do not change even after allowing for a participation rate on the equity appreciation, a spread and a minimum guaranteed rate features. Possible divergence between both valuation paradigms may arise because policyholders dynamics might be tied to economic variables for which risk-neutrality might be difficult or impossible to achieve or obtain. Even though this paper did not address this issue specifically, the ground work has been laid out and depending on a specific insurer lapses assumption; one can test whether or not a real world ESG is really necessary to obtain market consistent liabilities. REFERENCES The Staples Inn Actuarial Society report (2001), Modern Valuation Techniques, February Franck de Jong (2004), Deflators: An introduction. Milliman Report (2007): Deflators demystified, April

14 International Actuarial Association (2010), Stochastic Modeling: theory and reality from an Actuarial Perspective TABLES Figure 1: one period Binomial Tree Table 1: Two scenarios with real world probabilities occurrence Table 2: Profit-sharing SPDA Liability issued by the life insurance company Table 3: Life insurance investment strategy (mix of asset) Table 4: real World cash Flows Projections 12

15 Figure 2a: Three-steps recombining binomial tree for the equity process Figure 2b: assumption for u and d Figure 2c: The real world probability of the up movement and the risk-free rate Figure 2d: The Equity Risk Premium 13

16 Figure 2e: The risk-neutral probability of the up move Table 5: the Real World ESG 14

17 Table 6: Risk-neutral and Real World probabilities of each scenario in the ESG in table 5 Table 7a: deflators for the Real World ESG Table 7b: deflators satisfying the two Martingales conditions. 15

18 For condition 2, the results were also satisfactory. Table 8: SPDA product offered by the life insurance company Table 9: The life insurance company investment strategy 16

19 Table 9a: The investment returns Table 9b: Liability Cash Flows Table 9c: The shareholders surplus before taxes Table 9d: The cash taxes 17

20 Table 9e: the shareholders surplus adjusted for taxes and DTA, and the PVFP calculation (PN PVFP is the Risk Neutral PVFP) Table 10: RW and RN PVFP calculation with a spread of 1%. Everything else in the same as in table 8, and table 9. 18

21 Table 11: Real World Simulation versus Risk neutral World simulation basis Appendix 1: Mathematical Proof for solving the Deflator for a real world ESG We show that the deflators that will make real-world valuations preserve marketconsistency can be obtained using the following equation: Where: ( ) ( ( ) ) ( ) ( ) We use the proof by mathematical induction approach on the number of simulation n. Let a security V with the following characteristics: Under the Q-measure, the price of this security satisfies: 19

22 ( ( ) ) This price is essentially a market consistent valuation of the security. The r isk neutral probabilities are artificial devices to ensure that when discounting the cash flows in the risk neutral world, the expected value is equal the market value of the security ( ). Under the P-measure, we use the stochastic discount factors (deflators) to discount the cash flows under each scenario, and we assign the real world probabilities (of scenario) to get the following value for the security: ( ) To preserve the market consistency property, the deflators must be determined so that: ( ( ) ) ( ) Initial point: Let us assume that we are only generating one scenario. Thus n=1; for all t; we have: Thus: ( ( ) ) ( ) ( ) ( ( ) ) In fact, under one scenario, the probabilities equate to one, and the deflator is simply the discount rate. So, we have proved when running only one scenario (the initial point), the deflator satisfies our equation. Induction: Let us assume that the property holds under n scenarios, to finish the proof by mathematical induction, we need to establish that when simulating n+1 scenarios, in order to preserve market consistency, the (n+1)th deflator shall be: ( ) ( ( ) ) 20

23 Let us assume that we have simulated n+1 economic scenarios for pricing the security V, the risk-neutral value of V is: ( ( ) ) Thus ( ( ) ) ( ( ) ) Under the hypothesis that the deflators for the first n scenarios satisfy the equation, we have that: ( ) ( ( ) ) In order for the deflator for scenario n+1 to satisfy market consistency for the security, we need to have: ( ) This is the same as: ( ) ( ) Let us equate both expressions ( ): ( ) ( ( ) ) ( ) ( ) We get rid of the term: ( ) Thus: ( ( ) ) ( ) And: ( ) Thus, if the equation holds for n scenario, to ensure market consistency when simulating n + 1 scenarios, the equality still holds. 21

24 In conclusion, irrespective of the number of scenarios, the equation holds and we can use it to calculate the vector of deflators ( as t varies from 1 until the end of the projection period (50 years)) for every scenario i: ( ) ( ( ) ) Appendix 2: Valuation for the Call Option The option is written on a stock S, the strike price is denoted K, the risk free rate of interest is r, the volatility of the stock return is, the drift of the stock is.the payoff of the option at maturity (time T) is: ( ) Closed Form Solution: The Black-Scholes Risk Neutral Valuation of the Call option ( ) ( ) The closed form solution is based on the RN process for the terminal stock price. Thus, if we were to simulate this process in the RN world, and calculate the payoff of the derivative under each simulated value, we shall be able to get the same valuation as the closed form solution. ( ( ) ) Where: ( ) Real World with Deflators Valuation of the Call option The third approach consists of simulating the real probability distribution function of the terminal stock price (that is allowing for the market price of risk). Each simulated value can be interpreted as a state of nature. 22

25 Each state of nature is associated with a specific deflator ( derivative is: ). The value of the ( ( ) ) Where: ( ) Are these three methodologies equivalent? Here are the results: With large sample sizes, RN and RW approaches provide results that are closed enough, even when the maturity of the option increases. 23

26 Association of African Young Economists Association des Jeunes Economistes Africains The Association of African Young Economist (AAYE) is a not-for-profit, non-partisan organization with the vision of a future for Africa where policy decisions emanate from rigorous scientific evidence by Africans and fellows for the welfare of Africans and partners. The AAYE RP WPS is double blind peer reviewed and indexed at IDEAS REPEC, SSRN and ACADEMIA. Head Quarter: Bastos, PO BOX: 959, Yaounde CMR; Site web: info@aaye.org; Tel.: Declaration receipt number: 005/RDA/F35/SAAJP.