Session 31, Statistical Techniques for Fund Mapping and Other Applications. Moderator: Douglas L. Robbins, FSA, MAAA. Presenter:

Transcription

1 Session 31, Statistical Techniques for Fund Mapping and Other Applications Moderator: Douglas L. Robbins, FSA, MAAA Presenter: Douglas L. Robbins, FSA, MAAA

2 Statistical Techniques for Fund Mapping and Other Applications Douglas L. Robbins August 29, 2016

3 Topics Fund Mapping: What, Why, How? Where do Linear Models Fit? What Can Go Wrong? Other Statistical Thoughts 2

4 Topics Fund Mapping: What, Why, How? 3

5 Definitions Fund A subaccount within a Variable Product Held within the Separate Account Has an objective (growth, income, etc) Many managed outside the ins co Index An indicator or measure of the performance of a certain broad asset class; may be real (e.g., S&P500) or notional (e.g., Large Cap) 4

6 How are the Two Related? One of them truly exists within typical variable products The other, at least for VA / VL, exists only w/i models What does that mean? Why does it matter? 5

7 Fund Mapping Goals By its nature, every fund probably has different risk & expected return characteristics Conservative funds tend to be low on both Aggressive funds have the opposite tendency But there s more to it, as asset blends all have their own tendencies Fund mapping is an attempt at capturing these tendencies 6

8 What will this Affect? For recent business, conservatism will generally reduce rider costs bonds especially so (more on that below) For seasoned business, it s less clear: For an AB or DB may still be true For an IB or WB, think about the possible futures for Cash and/or Bonds Bonds can end up being a worst case once a contract is deeply in the money 7

9 How Does One Map Funds? There are really three approaches, which may be complementary Find a published info source for each fund, which indicates long-run goals Find a source that gives you current mix Use a linear model of some sort to analyze return data and come up with a mapping It is probably a good idea to try to do more than one of these 8

10 One Recent Actual Case Available information on a given fund Morningstar allocation by asset classes Reasonable given what one might expect Had a subset of unmappable classes Fund and Index Returns by month for 20 years Amenable to a combined look, using a true statistical approach Morningstar becomes null hypothesis Linear model used to accept or reject 9

11 Topics Where do Linear Models Fit? 10

12 Definitions What does Linear Model even mean? At the end of the day, it means you are mapping predictors to predictees using linear factors One way to think about this involves matrices (something from WAY back) 11

13 What do you mean? 12

14 A Couple Important Points The matrix of predictors and the vector of coefficients can have n dimensions There is no scientific reason you need a column of constants (i.e., the 1s ), or a least squares goal Then why is it the most common textbook example? Because it s elegant, intuitive, & easy to visualize And also, this

15 Here is Some Data Does it look linearly related? What will happen if I model w/o a constant? 14

16 Here is Some Data Does it look linearly related? What will happen if I model w/o a constant? 15

17 With the Constant, This Happens: Bingo!! 16

18 Anything Else we should Know? 17

19 Anything Else we should Know? Yeah: For one, that isn t the only plausible answer Here s another, within 95% confidence limits 18

20 When that can Haunt You When does this matter a little? When would it matter a lot? Should this issue concern me, vis-à-vis Fund Mapping? What else, related to Fund Mapping? Using a zero constant is a practical necessity Multiple X variables are a key feature 19

21 As an Aside.... Does this data set look like a good candidate for linear modeling? Why? 20

22 Here is my Linear-Modeling Result 21

23 Here is my Linear-Modeling Result In Linear Models, only coefficients must be linear Data can be anything, which is part of what makes the technique so powerful 22

24 A couple more asides: Does this data look linear? 23

25 A couple more asides: Does this data look linear? It is, in fact; but what is the equation? 24

26 A couple more asides: Does this data look linear? It is, in fact; but what is the equation? That s right: y = 15 (plus e ) Including X is not useful 25

27 A couple more asides: Does this data look linear? 26

28 A couple more asides: Does this data look linear? It is, and this time there is a relationship 27

29 A couple more asides: Does this data look linear? It is, and this time there is a relationship But the picture looks like: x = 6 (or very nearly) And for that reason, a regression is hopeless 28

30 Summary - A Fund-Mapping Linear Model: Examines a fund s returns ( Y ) Has no constant (row of 1s) Has dimensions ( Xs ) equal to the number of indices being mapped to Coefficients more or less constrained: Must all be positive Must sum to 100% 29

31 Topics What Can Go Wrong? 30

32 Cash is Typically Tough to Map 31

33 Cash is Typically Tough to Map Recall this data from earlier: Use about 0.2% instead of 6 for X above, and it s about what you get 32

34 Most Funds won t map well to all 5 Indices Some will have a trivial or essentially a 0 coefficient relationship to the fund s returns If your process doesn t sort these out, future predictability will be hurt Additionally, due to constraints, and since Cash has minimal impact on final return, it will tend to rise or fall very fluidly; result? May need to estimate Cash outside of your Linear Model (as in my Morningstar example 33 can t reject null hypothesis, really)

35 Possibly the Biggest Potential Pitfall 34

36 Possibly the Biggest Potential Pitfall Unlikely to affect many of you today 35

37 Possibly the Biggest Potential Pitfall Unlikely to affect many of you today Take a look at this data (it is a 60/40 fund) 36

38 Possibly the Biggest Potential Pitfall 60%X and 40%Y -> indices, blue dots -> returns Your Linear Model is doing something analogous to an Algebra problem that you probably recall 37

39 Possibly the Biggest Potential Pitfall It s something like the following: X + Y = 20 X Y = 14 You have 2 equations and two unknowns, so you quickly solve 38

40 Possibly the Biggest Potential Pitfall It s something like the following: X + Y = 20 X Y = 14 You have 2 equations and two unknowns, so you quickly solve X = 17, so Y = 3 39

41 Possibly the Biggest Potential Pitfall It s something like the following: X + Y = 20 X Y = 14 You have 2 equations and two unknowns, so you quickly solve X = 17, so Y = 3 Yay! 40

42 Possibly the Biggest Potential Pitfall In this case, your solution is a solution set The blue dots represent this continuum of 60%X + 40%Y s in a 3 rd dimension 41

43 Possibly the Biggest Potential Pitfall Anyway, then your teacher did this: X + Y = 20 3X + 3Y = 60 42

44 Possibly the Biggest Potential Pitfall Anyway, then your teacher did this: X + Y = 20 3X + 3Y = 60 And you threw your hands up in the air, and said, But... the answer could be anything!! 43

45 Possibly the Biggest Potential Pitfall That is analogous to this same data as before But the way they are ordered, the Linear Model now throws up its hands, just like you did!! 44

46 Possibly the Biggest Potential Pitfall In the first case, a regression s output will almost always be very accurate 45

47 Possibly the Biggest Potential Pitfall In the first case, a regression s output will almost always be very accurate In the second, it will likely look roughly like: 46

48 Possibly the Biggest Potential Pitfall Unfortunately, a differing Beta doesn t help 47

49 Possibly the Biggest Potential Pitfall Unfortunately, a differing Beta doesn t help If any of the indices are close to 100% correlated, a Linear Model will fail 48

50 Possibly the Biggest Potential Pitfall Why could this be quite important? 49

51 Possibly the Biggest Potential Pitfall Why could this be quite important? What do you think this data represents? 50

52 Possibly the Biggest Potential Pitfall Why could this be quite important? What do you think this data represents? Correct! It s a portion of the market 51

53 Possibly the Biggest Potential Pitfall Analysis of a balanced fund here will likely fail 52

54 Possibly the Biggest Potential Pitfall Analysis of a balanced fund here will likely fail Your model will probably detect a relationship overall, but your coefficients may not even be close 53

55 Summary of 3 Potential Pitfalls Lack of variation in X data e.g., Cash Inclusion of variables that aren t really statistically significant Intercorrelation between X variables 54

56 Topics Other Statistical Thoughts 55

57 Something to Consider.... In my experience, actuaries tend to do stochastically only at the economy What if assumptions are not independent? Some risk factors, like economy and lapse, may be prima facie dependent Others might seem independent, but not be 56

58 Something to Consider.... In my experience, actuaries tend to do stochastically only at the economy What if assumptions are not independent? Some risk factors, like economy and lapse, may be prima facie dependent Others might seem independent, but not be A couple simple examples to try to make my meaning clear: 57

59 The Ice Cream / Oreo Paradox Consider a proposed relationship between outside temperature, and how much ice cream you want (Sm, Md, Lg) I would claim this is prima facie dependent The hotter it is, the more ice cream you want! Now, what if you are considering your desire for ice cream, alongside your desire for Oreo cookies? Independent? Let s assume so; but what if

60 What If you know your spouse will kill you if you go large on both, in a single day? How effectively will you analyze the Risk Margin related to a given snack, if you hold the other snack at expected? 59

61 What If you know your spouse will kill you if you go large on both, in a single day? How effectively will you analyze the Risk Margin related to a given snack, if you hold the other snack at expected? You won t you will die! 60

62 What If you know your spouse will kill you if you go large on both, in a single day? How effectively will you analyze the Risk Margin related to a given snack, if you hold the other snack at expected? You won t you will die! Moral: Two independent events can become dependent in terms of tail risk, especially when PV ing future losses 61

63 Thoughts on Loss Distribution, and: How large a block is large enough? Concept: Velocity of Diversification, Case Study How many of you are on the Life Side? Have your PD guys developed ROP Term? If so, have you thanked them? There is one stochastic / statistical sense in which ROP Term is far superior to ordinary Term 62

64 Ordinary 10-Year Term Example A little corny: no i, no tax or v /RBC 63

65 75% ROP Term Example Same Basis also no SNFL 64

66 How has ROP improved my Risk situation? P(Any 1 policy sold having a positive profit margin) is the same 72% Standard Deviation of the profit margin of a single policy is massively better For straight Term it was 1493% For ROP it is now only 335% Those are not typos 65

67 How has ROP improved my Risk situation? This, by the way, alludes to why Vegas works for both sides; and also part of table limit theory E.g., Roulette black: Expected return = -5.3%; sigma ~= 100% Bettor says, Maybe I ll win! House says, I win in the long run! 66

68 How has ROP improved my Risk situation? But how about betting on 00? Expected = -5.3%; sigma ~= 576%! Bettor says, Maybe I ll win! House says, I win in the long run, but you re not going to bankrupt me in 1 spin! It takes the house 33 times as long to become sure (-ish) of winning 67

69 How has ROP improved my Risk situation? By designing ROP Term, PD folks have done something similar In my version, you must sell almost 34,000 straight Term policies to even get sigma = mu, for total profit For the ROP term, it s only ~1,700 ROP Term diversifies ~20x as fast 68

70 How has ROP improved my Risk situation? Corollary: Any experience study estimates a time series, so.... Early future is always more solid than distant future If mortality in years 7+ ends up 0.5% higher than estimated: Term profit margin = -38.6% ROP Term margin = 0.5% 69

71 Moral In at least some important ways, we cheat ourselves if the only assumption we think of stochastically is the economy Thinking stochastically doesn t have to mean modeling things that way However, it may mean more than simple sensitivity testing 70