Session 61 L, Economic Scenario Generators: Risk-Neutral and Real-World Considerations from an Investment Perspective

Transcription

1 Session 61 L, Economic Scenario Generators: Risk-Neutral and Real-World Considerations from an Investment Perspective Moderator: Ryan Joel Stowe, FSA, MAAA Presenter: Jinsung Yoo, FSA, Ph.D.

2 Session 61: Risk Neutral and Real World Considerations from an Investment Perspective Jinsung Yoo, FSA Ph.D. AXA US

3 Polling Questions: Do we live in a Risk Neutral World? 1. Do you expect that over the long term, a basket of risky assets (equities, credit bonds) purchased today will have a higher return than a basket of treasuries? (1 yes / 2 no) 1. Yes 2. No 87% 13% 1 2 2

4 Polling Questions: Do we live in a Risk Neutral World? 2. Does your investment department believe it? (1 yes / 2 no) 1. Yes 2. No 74% 26% 1 2 3

5 Polling Questions: Do we live in a Risk Neutral World? 3. How long a holding period would you need to feel confident that will outperform treasury bonds? 1. 3 yrs 49% I don t believe equities will outperform treasuries 10% 13% 26% 3%

6 Session Description This session will review key pricing and modeling considerations in the context of economic scenario generators. Risk neutral and real world scenarios are used for different purposes and have different characteristics that can affect risk measurement and pricing outcomes. How these scenarios are created, the management actions assumed and the investment strategies are integral components of the final outcome. At the conclusion of the session, attendees will be able to interpret and explain differences between risk neutral and real world scenarios. In addition, attendees will be able to identify key characteristics of each type of scenario and how they should (and should not) be used in modeling processes. 5

7 Guidance from the AAA Life Capital Adequacy Subcommittee Probability Measure In general, there are two probability measures for simulating investment returns. The Q measure, or risk neutral distribution, is a convenient framework used for pricing securities and is predicated on the concept of replication under a no arbitrage environment. Under the Q measure, risk is hedged (hence, securities are expected to earn the risk free rate) and derivatives (options) can be priced using their expected discounted cashflows. The Q measure is crucial to option pricing, but equally important is the fact that it tells us almost nothing about the true probability distribution. The Q measure is relevant only to pricing ( fair market value determination) and replication (a fundamental concept in hedging); any attempt to project values ( true outcomes ) for a risky portfolio must be based on an appropriate (and unfortunately subjective) real world probability model. This is the so called physical measure, or P measure. The real world model should be used for all cash flow projections, consistent with the risk preferences of the market. This is the basis for the valuation of required capital and is the focus of the remainder of this appendix. However, the risk neutral measure is relevant if the company s risk management strategy involves the purchase or sale of derivatives or other financial instruments in the capital markets. Recommended Approach for Setting Regulatory Risk Based Capital Requirements for Variable Annuities and Similar Products Presented by the American Academy of Actuaries Life Capital Adequacy Subcommittee to the National Association of Insurance Commissioners Capital Adequacy Task Force Boston, MA June

8 Polling Question Real World Do you use stochastic RW scenarios in 1. Pricing 2. Valuation 3. ALM studies 4. Strategic Asset Allocation 5. Multiple Projects 6. Not at all Real World Stochastic Pricing Valuation ALM SAA Multiple N/A 7

9 Polling Question Real World Do you use stochastic RW scenarios in 1. Pricing 2. Valuation 3. ALM studies 4. Strategic Asset Allocation 5. Multiple Projects 6. Not at all 0% 0% 0% 0% 0% 0% Countdown

10 Polling Question Risk Neutral Do you use Risk Neutral scenarios in 1. Pricing 2. Valuation 3. ALM studies 4. Strategic Asset Allocation 5. Multiple Projects 6. Not at all Risk Neutral Scenarios Pricing Valuation ALM SAA Multiple N/A 9

11 Polling Question Risk Neutral Do you use Risk Neutral scenarios in 1. Pricing 2. Valuation 3. ALM studies 4. Strategic Asset Allocation 5. Multiple Projects 6. Not at all 0% 0% 0% 0% 0% 0% Countdown

12 Risk Neutral Scenarios Overview (1) Every asset class has same expected return No reward for taking on extra risk This comes from the assumptions that All assets can be hedged No arbitrage opportunities Investors are indifferent to risk (hence risk neutral ) No additional returns for taking on additional risk Additional spread on risky bonds will be offset (on average) by additional credit losses Traditional Capital Market Theory (CAPM) does not apply 11

13 Risk Neutral Scenarios Overview (2) Scenarios are calibrated to match observable asset prices on an expected value basis cashflows over all scenarios, discounted at the risk free rates within each scenario, should average out to current market price Typical calibration targets include swaptions (interest) and call or put volatility (equities) May target specific points, or full surface: Swaptions: Tenor and Maturity Puts or Calls: Maturity and strike price in the moneyness 12

14 Risk Neutral Scenarios Overview (2a) sample targets Implied Vol Targets swaptions Implied Vol Targets puts Mkt USD SPX Implied Vol Targets Puts maturity 60% 70% 80% 90% 100% 110% 120% 130% 140% 1Y 31.83% 28.64% 25.46% 22.41% 19.46% 16.72% 14.63% 13.41% 13.32% 2Y 30.11% 27.72% 25.40% 23.22% 21.18% 19.28% 17.63% 16.29% 15.29% 3Y 29.95% 27.93% 25.96% 24.12% 22.41% 20.82% 19.43% 18.27% 17.26% 4Y 29.74% 27.93% 26.19% 24.57% 23.07% 21.71% 20.50% 19.47% 18.56% 5Y 29.81% 28.14% 26.55% 25.08% 23.74% 22.52% 21.43% 20.50% 19.67% 7Y 30.01% 28.55% 27.16% 25.91% 24.77% 23.74% 22.83% 22.03% 21.30% 10Y 31.02% 29.76% 28.59% 27.52% 26.55% 25.68% 24.89% 24.18% 23.53% 15Y 33.45% 32.42% 31.44% 30.55% 29.71% 28.95% 28.23% 27.58% 26.93% 13

15 Risk Neutral Scenarios Overview (3) No individual scenario needs to make sense : different asset classes may perform better within a scenario in any particular scenario, risky bonds could can have large returns and zero defaults. Equities returns may or may not correlate to interest rates and other asset classes Calibration considers averages across all scenarios. Martingale test: the present value of $1 invested now in any asset class for any time period is $1 (before taxes, expenses) On average, interest rates follow the current forward rates Certaintly Equivalent scenario can be built just knowing the current yield curve The equivalent in a way of using the level scenario in a real world simulation 14

16 Risk Neutral Scenario Deflators and the Martingale Deflator: Discount factor applied to the cash flows from a particular scenario at a particular duration. Is equal to the product of annual discount factors based on the risk free cash (or 1 year) rate Martingale: The test that the present value of accumulated funds at each time t = starting MV Present value calculated as sum of the product across all scenarios at time t of deflators x accumulated values / number of scenario 15

17 Deflator and Martingale Example: Cash cash rate year scenario % 0.1% 3.6% 5.4% 3.7% 2 4.3% 4.2% 6.5% 5.1% 4.5% 3 2.5% 3.3% 6.9% 3.1% 1.4% 4 3.3% 6.6% 6.0% 1.3% 1.8% 5 3.0% 1.6% 3.9% 0.1% 2.8% accumulated cash % 106.8% 110.6% 116.6% 120.9% % 108.6% 115.6% 121.5% 127.0% % 105.9% 113.3% 116.8% 118.4% % 110.2% 116.9% 118.4% 120.5% % 104.6% 108.8% 108.8% 111.8% deflators (=v base on cash rates) % 93.6% 90.4% 85.7% 82.7% % 92.1% 86.5% 82.3% 78.7% % 94.4% 88.3% 85.6% 84.4% % 90.7% 85.6% 84.5% 83.0% % 95.6% 91.9% 91.9% 89.4% deflated % 100.0% 100.0% 100.0% 100.0% % 100.0% 100.0% 100.0% 100.0% % 100.0% 100.0% 100.0% 100.0% % 100.0% 100.0% 100.0% 100.0% % 100.0% 100.0% 100.0% 100.0% Martinga 100.0% 100.0% 100.0% 100.0% 100.0% Risk free cash rates will pass the martingale test by definition 16

18 Deflators and Martingale: Equity (1) equity returns first simulation % 11.3% 17.9% 13.5% 18.0% 2 0.8% 9.0% 21.6% 9.7% 24.5% % 13.8% 13.7% 14.9% 1.6% % 24.6% 7.4% 23.5% 19.4% 5 9.8% 21.9% 24.8% 20.2% 12.0% accumulated equity % 106.0% 125.0% 141.9% 116.4% % 91.7% 111.6% 100.7% 125.4% % 103.1% 117.2% 99.8% 98.2% % 57.0% 61.2% 75.5% 90.2% % 70.5% 88.0% 105.7% 93.0% martingale 92.6% 80.0% 89.3% 90.2% 87.2% (SumProduct(accumulated equity,deflator) / 5) Does not pass the martingale: average PV is not close to 100% adjustment f 107.9% 125.0% 112.0% 110.9% 114.7% (factor to apply to accumulated equity to force the martingale) 17

19 Deflators and Martingale: Equity (2) accumulated equity Adjusted accumulated % 132.5% 140.1% 157.3% 133.5% % 114.7% 125.0% 111.7% 143.8% % 128.8% 131.4% 110.6% 112.6% % 71.2% 68.5% 83.8% 103.5% % 88.1% 98.6% 117.2% 106.7% adjusted equity returns % 28.9% 5.7% 12.3% 15.1% 2 8.9% 5.3% 9.0% 10.7% 28.8% % 0.2% 2.0% 15.8% 1.8% % 12.7% 3.7% 22.2% 23.6% 5 2.6% 9.5% 11.9% 18.9% 9.0% deflated accumulated value % 124.0% 126.6% 134.9% 110.4% % 105.6% 108.1% 91.9% 113.2% % 121.6% 116.0% 94.7% 95.1% % 64.6% 58.6% 70.8% 85.9% % 84.2% 90.6% 107.7% 95.4% average 100.0% 100.0% 100.0% 100.0% 100.0% Martingale looks good Assets other than risk free cash may not pass the martingale on the first try may need to recalibrate or otherwise adjust 18

20 Risk Neutral Calibration Targets Risk Free Rates Replicate the starting yield curve: avg of all deflators at year n should equal 1/(n year spot) from starting curve Interest rate volatility: Expected value of a swaption (avg discounted cashflows over all scenarios) should match current swaption price (for selected tenors / maturities) 19

21 Refresher: Par, Spot, and Forward Rates Par: Coupon rate necessary for bond to trade at par value Spot: Rate for zero coupon bond Forward: Rate expected for zero coupon bond purchased in the future In the Risk Neutral world, you expect the same wealth accumulation after n years no matter how you invest the forward rate can be derived as the difference between spots of different maturities, e.g. E.g. invest in a five year zero coupon bond OR invest in a 4 year, reinvest for 1 year Same expected wealth after 5 years Year 4 forward rate = (5 yr spot) 5 / (4 year spot) 4 20

22 Certainty Equivalent scenario derived from Yearend 2014 Treasury curve par curve time % 0.67% 1.10% 1.38% 1.65% 1.81% 1.97% 2.04% 2.10% 2.17% spotcurve time o 0.25% 0.67% 1.11% 1.39% 1.67% 1.84% 2.00% 2.07% 2.14% 2.21% forward curve time % 1.09% 1.98% 2.23% 2.81% 2.67% 3.02% 2.55% 2.70% 2.86% forward curve time % 1.98% 2.23% 2.81% 2.67% 3.02% 2.55% 2.70% 2.86% forward curve time % 2.23% 2.81% 2.67% 3.02% 2.55% 2.70% 2.86% forward curve time % 2.81% 2.67% 3.02% 2.55% 2.70% 2.86% forward curve time % 2.67% 3.02% 2.55% 2.70% 2.86% forward curve time % 3.02% 2.55% 2.70% 2.86% spot curve time % 1.54% 1.77% 2.03% 2.16% 2.30% 2.34% 2.38% 2.43% spot curve time % 2.11% 2.34% 2.42% 2.54% 2.55% 2.57% 2.60% spot curve time % 2.52% 2.57% 2.68% 2.66% 2.67% 2.69% spot curve time % 2.74% 2.83% 2.76% 2.75% 2.77% spot curve time % 2.85% 2.75% 2.74% 2.76% Par curve % 1.53% 1.76% 2.02% 2.14% Par curve % 2.11% 2.34% 2.42% 2.53% Par curve % 2.52% 2.57% 2.68% 2.65% Par curve % 2.74% 2.83% 2.77% 2.75% Par curve % 2.84% 2.75% 2.74% 2.76% 21

23 Certainty Equivalent scenario derived from Yearend 2014 Treasury curve Spot and par very close at time 0 due to low rates / coupons Forwards are a bit jagged, maybe due to my linear interpolation of 4, 8, and 9 year par rates Yield curve goes up by year, especially at the short end in the current environment, this is a more aggressive assumptions than holding rates level 22

24 RN issues calibration (1) Only one RN curve (if pricing on the swap curves, then treasuries won t be RN) RN only applies to time 0 How to make a buy and hold portfolio RN Difficult to calibrate with low / negative forwards CRA, LP, VA more market consistent than the market? Number of scenarios required to pass a martingale, tamp down random noise 23

25 RN issues calibration (2) Calibration with no observable targets For example, to calibrate 40 years worth of 30 year scenarios, need 70 years worth of forward rates at time 0 If trying to calibrate to a swaption surface, there are many points at with no market or a very thin market Calibration with no observable asset types May look at history as opposed to something immediately observable on the market, e.g. what is the correlation between interest rates and equities? Credit spreads and equities? Implied vols are reproduced on an expected value basis. Shocks take you out of RN framework 24

26 RN issues modeling and using results Spread business looks bad with no spread Assets with high expenses look bad CAPM leftmost point. Not useful for strategic asset allocation Modeling management and policyholder behavior Do you consider corporate spread in your crediting? What are your competitor constraints in a risk neutral world? Extreme conditions, exploding rates or negative rates Results highly sensitive to starting conditions; No RTM Only Average is meaningful percentile and CTE results not meaningful Used for valuation models (EV) change in EV as the starting curve is shocked can be used as a measure of duration 25

27 RN: example of 2 different calibrations (2FBK, LMM+) Given a set of targets, there is more than one way to calibrate Choice of calibration tools / models can make a substantial difference in the resulting scenarios Example: comparison of scenario sets generated by the 2 factor Black Karasinski (2fBK) model and * LMM+ model Both run within the B&H ESG 2fBK models short rates as a mean reverting process, with the mean reversion target also being a mean reverting process LMM+ is a model proprietary to B&H which applies an offset to the LMM model, limits the explosive upside seen in LMM models by allowing negative rates Graphs in this exhibit are based on a small subset (100 scenarios) of 2 sets generated explicitely to compare the two models 26

28 10 yr Par rates 2FBK 27

29 10 yr Par rates LMM+ 28

30 10 yr Par rates average rates compared 29

31 Polling Question From the prior slides, which set of interest scenarios would be better suited to model your business? 1: 2fBK 2: LMM+ 3: The New York 7 are looking pretty good right now! Scenario Choice 2fBK LMM+ No preference 30

32 Polling Question From the prior slides, which set of interest scenarios would be better suited to model your business? 1. 2fBK 2. LMM+ 3. The New York 7 are looking pretty good right now! 0% 0% 0% Countdown

33 Deflators compared 32

34 And yet The two scenario sets reproduce similar yield curves. They also (not illustrated) produce fairly similar swaption volatilities at key points (5, 10, 15 yr maturities of the 10 year tenor.) 33

35 Real World Highlights Can reflect a realistic relationship between asset classes e.g. risk/return tradeoff Can be deterministic (e.g. NY7, reverse stress scenarios) or stochastic (e.g. AG43) Calibration of stochastic sets is generally tied to observations of history e.g. hypothesize a parametric form of the variable being modeled, look at the actual history of the variable, and use a maximum likelihood estimator Normal / lognormal / regime switching Can model implied volatilities, or fx volatilities, or any other desired parameter AAA / NAIC requirements on scenarios include wealth accumulation percentiles Can define reasonableness as a scenario by scenario test, as an average, percentile, or CTE Possible use of RTM to smooth results 34

36 Issues with Real World History is not all that extensive, especially when considering tail events Since 1928, S&P 500 lost > 33.3% of its value in 3 years, grew by > 33.3 in 7 years How to avoid model arbitrage if a model with risky assets gets you a more favorable result, can end up backing fixed liabilities with (say) private equities Asset based capital charges Use x th percentile results or CTEs instead of averages allow volatility to hurt you Use discount rates consistent with the risk How well do any of these approaches work when you need to take a risk to clear some hurdle, say a minimum guaranteed rate? Calibrations and targets will be sensitive to the historical period included in your analysis 35

37 Uses of Real World Scenarios Deterministic: Test a particular, well defined concern NY7: what happens if interest rates follow particular patterns Reverse Stress scenarios: make sense out of a stochastic result by finding a single scenario that can replicate the numbers Stochastic: Reserves and Capital as defined by regulation (AG43, RBC C3) Asset Allocation use a full blown ALM model to search out an efficient frontier 36

38 Summary observations Risk Neutral: Since the scenario can reproduce the price of (some) assets observable on the market, makes sense to try to use to derive non observable prices, e.g. value of a block of business Should try to calibrate the scenarios to assets that should be related to the liabilities you are looking to value (e.g. in 2009, calibrate to in the money put options for equity based guarantees issued before 2008) Real World: Assumptions used in developing the scenario may strongly impact the results Risk Neutral view is that models should be built around data that can be derived from a snapshot of current market conditions. Real World allows the incorporation of historical data and explicit judgment 37

39 Session 61: Risk-Neutral and Real-World Considerations from an Investment Perspective Jinsung Yoo, FSA Ph.D. AXA US

40 Session Description This session will review key pricing and modeling considerations in the context of economic scenario generators. Risk neutral and real world scenarios are used for different purposes and have different characteristics that can affect risk measurement and pricing outcomes. How these scenarios are created, the management actions assumed and the investment strategies are integral components of the final outcome. At the conclusion of the session, attendees will be able to interpret and explain differences between risk neutral and real world scenarios. In addition, attendees will be able to identify key characteristics of each type of scenario and how they should (and should not) be used in modeling processes. 2

41 Risk Neutral versus Real World Risk Neutral Scenarios Real World Scenarios View Forward looking Backward looking Presumption History is irrelevent History will repeat What's oberved now is Learn from history a reflection of future events same expected return depending on riskness of security Asset return risk free rate risk rate plus risk premium CAPM: Probability Risk neutral Q measure Realistic P measure Discount risk free rate Weighted Average Cost of Capital Method Objective mechanical Subjective explicit judgment Risk metric Implied volatility Standard Deviation, beta Philsophy Hard to beat the market I can beat the market No arbitrage I can create "alfa" Pricing Correct Incorrect Risk bearing No extra return on average Requires extra return Data Sanpshot of current market Historical market data 3

42 Risk neutral or real world probability? Selected a trick question appeared on SOA exam Q1. Suppose the current IBM stock price is $100. Assume no dividend payment. After one year, there are 60% chance of stock price to increase by 10% and 40% chance of stock price to decrease by 10%. If the continuous risk free rate is 5%, what is the price of the call option on this stock with strike at 100? Show all your work using risk neutral valuation. See the answers on the next two slides. 4

43 Student Answer Sheet S 0 S 1 option payoff % 40% C = exp(-5%)[ 60%* %*0] = 5.71 His score is 3 out of 10 with a partial credit!! 5

44 Correct Answer Sheet S 0 S 1 option payoff 100 p 1-p = exp(-5%)[ p*110 + (1-p)*90] = exp(-5%)[110p - 90p +90] => 100 exp(5%) = 20p + 90 => 20p = 100 exp(5%) - 90 => 20p = 100 exp(5%) - 90 => 20p = => p = /20 = => 1 - p = C = exp(-5%)[ * *0] =

45 What did you learn from this SOA exam question? The student tried to calculate with a real world probability P- measure (subjective opinion) SOA key answer: risk neutral probability Q-measure Lesson: You have to read the question carefully!! Market option quote is based on Q-measure, otherwise there is an arbitrage opportunity 60% 40 % subjective view in Real world probability turns out to be a mispricing of the option. You should have used risk neutral probability of 74% 24%. No matter how the stock price would be realized at year 1, it should have a present value of 100 on average when you discount back with risk free rate. 7

46 Another trick question for fun Is it legal everywhere in the world for a man to marry his widow s sister? The answer will be given at the end of the session. 8

47 Volatility: Risk Metric Although both use the same notation s for volatility, they have different meanings. In Real world scenarios, it is a statistical metric standard deviation In Risk neutral scenarios, it is an implied volatility derived from Black-Scholes formula so that theoretical option price equals to the market price of the market traded option 9

48 Historical volatility versus Implied volatility Historical Volatility Implied Volatility present Past 30 days Next 30 days Historical stock price movement Current option price maturing in 30 days Statistical standard deviation Value maikng BS price equals market price A measure of fluctuations of stock prices Expected fluctuations of the underling during the past 30 days stock price over the next 30 days Up/down movement equally treated Stock Up => low IV, Stock Down => high IV Variation metric Fear index, uncertainty index 10

49 Example will make the concept clear I am going to show you how we generate S&P 500 equity price index return scenarios in two different ways. Real world scenarios Risk neutral scenarios Both will follow lognormal process for stock prices 11

50 Equity Scenario Generation Real world scenarios Step 1: Collect Historical data (S&P 500 price ) Step 2: Calculate m and s for lognormal stock price model Step 3: Using the above parameters, generate 1000 paths Risk neutral scenarios (simple setting) Step 1: Collect current market(s&p 500 put option implied volatility & risk free rate) Step 2: Using the above information, generate 1000 paths 12

51 Illustrative Graph In general, RN scenarios produce more conservative results due to lower mean and higher volatility Equity Price Real Wrold Risk Neutral 13

52 RN: example of 2 different calibrations (2FBK, 2FHW) Given a set of targets, there is more than one way to calibrate Choice of models can make a substantial difference in the resulting scenarios and financial impact Example: comparison of scenario sets generated by two different interest rate models 2FBK (Black-Karasinsky) model: lognormal model using short rates as a mean reverting process with the mean reversion target also being a mean reversion process 2FHW (Hull-White) model: normal model which is very simple and was popular in the 90 s 14

53 RN: Characteristics of BK2F and HW2F Lognormal model BK2F: No negative rates yet rates can be explosive Normal model HW2F: rates not explosive yet rates can be negative Cash Total Returns % 95.00% 75.00% Hull & White BK2F 55.00% 35.00% 15.00% -5.00% % 15

54 RN issues Hard to communicate with senior management Looks too theoretical and artificial, unusual individual paths Difficult to calibrate with low / negative forwards Hard to justify sensitivity cases to be more aligned with market consistency Calibration with no observable targets For example, swap spot curve or swaption volatility data beyond 30 years Yet still have to project at least up to 50 years Calibration with no observable investment instruments the correlation between different equity indexes The correlation between interest rates and equities The correlation between credit spreads and equities 16

55 Real World Highlights Actuaries still feel comfortable with real world scenarios since they look realistic Can reflect a realistic relationship between asset classes for example, can assume that classes with higher volatility must also offer higher expected return although still subjective Can be deterministic (e.g. NY7, reverse stress scenarios) or stochastic (e.g. AG43) AAA / NAIC requirements on scenarios include wealth accumulation percentiles Can define reasonableness as a scenario-by-scenario test, as an average, as a percentile, or as a CTE 17

56 Issues with Real World History is not all that extensive, especially when considering tail events 1 in 200 years of event scenario (not enough data except Great Britain) Under a situation of having to sell a block of business, buyers would not agree to your asking price based on real world scenarios Calibrations and targets will be sensitive to the historical period included in you analysis (arbitrary and subjective) Discount rates 18

57 Uses of Real World Scenarios Deterministic: Test a particular, well defined concern NY7: what happens if interest rates follow particular patterns Reverse Stress scenarios: make sense out of a stochastic result by finding a single scenario that can replicate the numbers Stochastic: Reserves and Capital as defined by regulation (AG43, RBC C3) Asset Allocation use a full blown ALM model to search out an efficient frontier 19

58 Summary observations Risk Neutral: Since the scenario can reproduce the price of assets observable on the market, it makes sense to try to use to derive non-observable prices, e.g. value of a block of business Real World: Assumptions used in developing the scenario may strongly impact the results Risk Neutral view is that models should be built around data that can be derived from a snapshot of current market conditions. Real World allows the incorporation of historical data and explicit judgment 20

59 Another trick question for fun Is it legal every where in the world for a man to marry his widow s sister? It doesn t matter It would be impossible! Since he has a widow, he must be dead himself 21