Behavioral Finance Driven Investment Strategies

Transcription

1 Behavioral Finance Driven Investment Strategies Prof. Dr. Rudi Zagst, Technical University of Munich joint work with L. Brummer, M. Escobar, A. Lichtenstern, M. Wahl 1

2 Behavioral Finance Driven Investment Strategies Introduction: Portfolio insurance with changing interest rate 1 0,95 0,9 VV tt x Floor value 0,85 Cushion F tt x 0,8 0,75 0, Time r=3% r=1% 2

3 Behavioral Finance Driven Investment Strategies Introduction: Underfunding in public and private pensions Source: Bloomberg, July/August

4 Behavioral Finance Driven Investment Strategies Overview Expected Utility Dynamic Investment Strategies Behavioral Finance Asset Liability Management 4

5 Behavioral Finance Driven Investment Strategies Expected Utility The determination of the value of an item must not be based on its price, but rather on the utility it yields there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount. Daniel Bernoulli (* ) 5

6 Behavioral Finance Driven Investment Strategies Expected Utility X: Wealth (the return of a portfolio) at (terminal) time T. U: Utility function under which X is evaluated. Goal: [ ( X) ] max E u J.v. Neumann (* ) Arrow-Pratt measure of absolute risk aversion for a wealth of x (J.W. Pratt (1966), K.J. Arrow (1971)): A ( x) Relative risk aversion: u := u ( x) ( x) u A r ( x) = x u ( x) ( x) : J.W. Pratt (*1931) K.J. Arrow (*1921) 6

7 Behavioral Finance Driven Investment Strategies Expected Utility A decision maker is called risk-averse, if his utility function u is concave and (here also) strictly monotone increasing. Power-utility function: u γ x γ ( x) =, γ < 1, γ 0 Utility u(x) Wealth x Then it holds: and A A r ( x) = 1γ x ( x) =1γ 7

8 Behavioral Finance Driven Investment Strategies Expected Utility (Relative) Risk tolerance: ( r )( x) If there exists a (unique) real number CE = CE(X) for X with ( CE) = E[ u( X )] then CE is called the certainty equivalent of X. For a risk-averse decision maker the certainty equivalent of X is always smaller than the expected value of X, i.e. The following approximation holds: λ u CE = E A 1 ( r )( x) [ X ] 1 ( X ) E[ X ] ( ) ar[ X ] CE CE X V = 2 λ H.M. Markowitz (*1927) 8

9 Behavioral Finance Driven Investment Strategies Dynamic investment strategies 9

10 Dynamic Investment Strategies Constant mix The Constant Mix Strategy is a dynamic investment strategy. - The portfolio is readjusted depending on the specific market developments. - Trading filter: Readjustment if the portfolio weights changed significantly. - Timing filter: Regular readjustments, e.g. monthly. At the beginning the relative weight of the assets in the portfolio is defined. The goal of each readjustment is the re-establishment of the original weights. Well-performing assets will be sold, non-performing assets will be bought. 10

11 Dynamic Investment Strategies CPPI At the beginning of a CPPI (Constant Proportion Portfolio Insurance) Strategy the investor defines a certain floor. The so-called Cushion is calculated and multiplied by a factor m > 1 at the beginning and (at certain times) during the runtime. Exposure t = m ( PortfolioValue V(t) Floor F(t) ) Cushion The resulting exposure is invested in the risky assets, the remaining capital is invested riskfree or is raised. Portfolio value Initial capital Cushion Current portfolio value Discounted floor Terminal portfolio value Zeit 11

12 Dynamic Investment Strategies Maximization of expected utility Investment strategy with relative portfolio weights: Reallocation possible at any time. Value of the portfolio at time t: Remaining wealth is allocated to the riskfree asset. The portfolio value is then determined by the following SDE: dv ( t) = V ( t) 1 ( t) 1 r ( t) µ dt ( t) σdw ( t) 0 = V = V ( ) v ( t) r ( t) ( µ r 1) dt ( t) σdw ( t) Goal: Maximization of the expected utility of the wealth at T, i.e. [ ( V ( T ))] max E u ( t) = ( ( t) ( t) ), 0 t T, plus ( ) cash ( t) 1,..., d d 1 ( t) V ( t, ) V = Fisher Black (* ) Myron Scholes (*1941) 12

13 Dynamic Investment Strategies Maximization of Expected Utility Optimal investment strategy for Power utility function EUT ( t) EUT = λ EUT with risk tolerance parameter 1 = 1γ Growth Optimum Portfolio (only risky investments) G -1 ( σσ ) ( r1) = µ The remaining wealth is invested in the riskfree asset, i.e. Cash EUT = 1 λ 1 G G EUT λ Cash, λ EUT This is a Constant Mix Strategy. The higher the risk tolerance, the higher the investment in the risky portfolio. 13

14 Behavioral Finance Driven Investment Strategies Behavioral Finance Daniel Kahneman (*1934) Amos Tversky (* ) Human beings cannot comprehend very large or very small numbers. It would be useful for us to acknowledge that fact. Daniel Kahneman (*1934) 14

15 Behavioral Finance Driven Investment Strategies Behavioral Finance 15

16 Behavioral Finance Cumulative Prospect Theory Considered currency within the following questions: Israeli pound Median of the monthly net income of a family in 1979 approx. 3,000 pounds Which options would you choose? Option 1: Guaranteed payment of 500 with probability 1 Option 2: Uncertain payment of 1,000 with probability 0.5 and 0 with probability % of participants choose Option 1. Which option would you choose? Option 1: Guaranteed loss of 500 with probability 1 Option 2: Uncertain loss of 1,000 with probability 0.5 and 0 with probability % of participants choose Option 2. 16

17 Behavioral Finance Cumulative Prospect Theory Investors evaluate their wealth relative to a reference wealth B. Investors behave risk-averse on gains. Investors behave risk-seeking on losses. Investors are more sensitive to losses than to gains. Extension of the concave utility function in Composite utility function at the reference wealth B: u ( x) u = u ( x B) ( B x) with usual utility functions u (x) and u - (x), in our case u (x) = x γ and u - (x) = β x γ, 0<γ<1 and β>1.,, falls falls x B x < B 17

18 Behavioral Finance Cumulative Prospect Theory Allais Paradox (1953): Which option would you choose? Option 1: Guaranteed payment of 5 with probability 1 Option 2: Uncertain payment of 5,000 with probability % of participants choose Option 2 Which option is more valuable for you? Option 1: Minor increase in the payment from a probable to a guaranteed event Option 2: Equal increase in the payment from a probable to a slightly more probable event The majority of participants value Option 1 higher. 18

19 Behavioral Finance Cumulative Prospect Theory Investors overweight small and underweight large probabilities. Introduction of a probability distortion function. Requirements regarding the probability distortion ww: Twice differentiable Strictly monotone increasing with [ 0,1] [ 0,1], w( 0) = 0, w( 1) = 1, w 0 w : > Inverse S-shaped Additional technical conditions Prelec Distortion [1998]: w ( ln p) ( p) e b =, 0 < b < 1 19

20 20 Behavioral Finance Cumulative Prospect Theory Jin & Zhou Distortion [2008]: for ( ) ( ) ( ) ( ) ( ) < < < = 1, 0, 0 1 ~ ~ p z p F c k z p p F k p w b Z b a a Z ( ) ( ) 0, :,, ~ ~ ln 0, 1, 0, ~ 0 2 ~ ~ 0 > = > < < < k c F z N Z c b a Z µ Z σ Z

21 Behavioral Finance Cumulative Prospect Theory Goal: Maximization of the expected utility of the relative wealth compared to the reference wealth B at time T, i.e. with V ( ( T ) B) = V ( V ( T ) B) V ( V ( T ) B) max V - [ ( )] ( V ( T ) B) = u ( V ( T ) B) w 1 F ( ) ( V ( T ) B) V ± E ± ± V T B and Pricing Kernel ~ ~ Z : = Z µ r 1-1 ( T ) = exp{ ( r θ θ ) θ W ( T )}, θ = σ ( 1) 2 with distribution function F Z ~ 21

22 Behavioral Finance Cumulative Prospect Theory Technical Assumptions For 0 < c let ϑ ~ ~ 1 ( ) ( ( ( ) ) 1 ~ 1 1 c γ Z ~ w γ γ = F~ Z, ( 0) = 0 and G( c) γ γ E 1 ϑ = E[ Z 1 ~ ] Z c Z Z c Further let k ( c) = ϑ β w ( 1 F~ ( c) ) Z 1γ ~ ( c) ( E[ Z 1 ~ ] ) Z > c γ and ϑ χ = k ( ) 1 1 γ Assumption 1: inf c> 0 k ( c) > 1 and 0 < χ < Assumption 2: There exists an optimal solution c* to the optimization problem inf β ~ E[ Z ( 1 F~ ( c) ) Z 0 c< 1 ~ Z > c ] 1 1 γ G( c) 22

23 Behavioral Finance Cumulative Prospect Theory Case V (t) > e -r (T-t) B: For the optimal portfolio value it always holds V r ( T t ) ( t) > e and the optimal investment strategy is given by = CPPI DH B Cash with the following parts: 1. CPPI Part (equal to a standard CPPI in case of no distortion): CPPI V 1 b 1γ G [ B] r ( T t ) ( t) = V ( t) e Multiplier m < 1/(1-γ) Cushion 23

24 Behavioral Finance Cumulative Prospect Theory 2. Distortion Hedging Part: DH ( t) V ( t) = λ ( t) G with λ as well as d b a r ( T t ) ( t) = [ Call( V, K, t, T ) e ( K B) N( d ( V, K, t, T )] 2 1γ ( V, K, t, T ) and v B e K = B χ lnv = 1 2 ( t) ln K ( r σ ) ( T t) σ T t 3. Cash Account (includes the protection B e -r (T-t) of the reference point): ( ) CPPI DH ( t) = 1 ( t) ( t) Cash 1 r ( T t ) c 0 in case of no distortion a 1 γ

25 Behavioral Finance Cumulative Prospect Theory Reminder: : [ B] λ ( t) 1 b 1 γ r ( T t ) ( t) = V ( t) e V w ( t ) G CPPI: CPPI V CPPI [ B] 1 1 γ CPPI r ( T t ) ( t) = V ( t) e w CPPI ( t ) G Example: Compare w (t) and w CPPI (t) graphically by means of Δw G (t) := w (t)-w CPPI (t) for the following parameters: r=1%, μ=5%, σ=20%, γ=0.49, a =-0.3, b =0.3, z 0 =0.5, b - =0.7 v=10, B=7, t=0.5, T=1 25

26 Behavioral Finance Cumulative Prospect Theory Underweighting Overweighting 26

27 Behavioral Finance Cumulative Prospect Theory CPPI 27

28 Behavioral Finance Cumulative Prospect Theory CPPI 28

29 Behavioral Finance Cumulative Prospect Theory 29

30 Behavioral Finance Cumulative Prospect Theory Case V (t) < e -r (T-t) B: The optimal investment strategy is given by with the following parts: 1. CPPI Part (corresponds to proportion of CPPI on V t ): with CPPI V CPPI DH 1 b 1γ Put PS Cash ( ) = V G V [ ] ( t) B = r ( T t ) ( t) = V ( t) e ( t) V ( t) Put( V, B, t T ) > B =, ( ) V ( t) CPPI Proportion of CPPI < 1 V ( t) and CPPI V 1 b 1γ [ B] r ( T t ) ( t) = V ( t) e Multiplier m Cushion 30

31 Behavioral Finance Cumulative Prospect Theory 2. Distortion Hedging Part: with λ as well as d and ~ K with DH 2 ~ χ ( t) V ( t) = λ ( t) G b a ~ r ( T t ) ~ ( t) = [ Call( V, K, t, T ) e ( K B) N( d ( V, K, t, T )] 1γ ( ~ 2 lnv ) ( t) ln K ( r 2 ) ( ),,, 1 σ T t V K t T = = ( c ) ( min{ c, c }) ~ σ T t 2 Change Point (discounted) to the Good times strategy for increasing stock prices r ( T t ) v B ~ χ 0 in case of no distortion 0 a 1 γ 1 > B, v = B e v 1 ( ( ) 1 γ k c 1 ( ) { } a = γ a b b γ ~ c Z 1 ~ γ 1 γ 1 γ E 1 ~ c Z ~ Z min c, c 0 E 1 0 c0 < Z min{ c0, c } 31

32 Behavioral Finance Cumulative Prospect Theory 3. Put Option Hedging Part: Put ( t) is the Hedge Portfolio of the (short) Put Option Put ( V,B,t,T ) 4. Performance-seeking Portfolio: PS with λ PS PS ( t) V ( t) = λ ( t) ( t) = σ ~ Z c 3. Cash Account (includes the protection B e -r (T-t) of the reference point): ( ) CPPI DH Put PS ( t) = 1 ( t) ( t) ( t) ( t) Cash 1 ( t) Z ( t) ln c N G ~ ln Z ( t) µ ~ ( t) Z ( t) ~ χ ( c ) ~ 0 σ ~ Z v a ( { }) b ( ) γ b min c, c c γ 32

33 Behavioral Finance Cumulative Prospect Theory Example: Compare again w (t) and w CPPI (t) graphically by means of Δw G (t) := w (t) - w CPPI (t) for the following parameters (Underlying V t ): ( ) r=1%, μ=5%, σ=20%, γ=0.75, a =-0.3, b =0.3, z 0 =0.5, b - =0.7 v=10, B=11, t=0.5, T=1, β=2.75 b - b - b - b - b - 33

34 Behavioral Finance Cumulative Prospect Theory CPPI 34

35 Behavioral Finance Cumulative Prospect Theory CPPI 35

36 Behavioral Finance Cumulative Prospect Theory 36

37 Behavioral Finance Cumulative Prospect Theory Goal: Investment strategy which remains above B e -r (T-t) or returns and ends there Assumption: No distortion Leveraged CPPI (LCPPI) 1. If V (t) > F(t) := e -r (T-t) B: CPPI strategy with floor F(t) 2. If V (t) < F(t): Short Put on V with exercise price B Observe: V (t) Put(t) > F(t) CPPI strategy on V (t) Put(t) with floor F(t) Direct risky investment for a faster recovery to a wealth above B When a portfolio value of V (t) = e -r (T-t) KK is achieved, keep F(t), liquidate the remaining portfolio and invest into a CPPI (Case 1.) 37

38 Behavioral Finance Driven Investment Strategies Asset Liability Management Few problems are as important and complex to institutions and individuals as the management of their assets in a way that their liabilities can be covered and their goals achieved. William T. Ziemba (*1941) 38

39 Behavioral Finance Driven Investment Strategies Asset Liability Management ( t) V ( t, ) A = L ( t) : dl( t) = L( t) ( µ dt σ dw ( t) ) L L Goal: The assets shall cover a certain fraction B of the liabilities, i.e. FR ( ) B L( T) V ( T ) A T : = A( T ) L ( T) B V FR (T) corresponds to the value of the funding ratio or short Funding Ratio at time T Solution: Maximization of the expected utility of the relative funding ratio compared to B at time T 39

40 Behavioral Finance Driven Investment Strategies Asset Liability Management Expected Utility Goal: Maximization of the expected utility of V FR (T), i.e. B:=0 The optimal investment strategy (Martellini [2006]) is given by with w EUT ( t) As well as 1 = 1 γ LH Cash ( ) G EUT ( t) 1 w ( t) EUT EUT = w 1 ( σ ) σ L LH = Liability Hedging Portfolio and Cash LH ( ( ) ) EUT G EUT ( t) = 1 1 w ( t) 1 w ( t) 40

41 Behavioral Finance Driven Investment Strategies Asset Liability Management - Distortion: Brummer, Wahl, Zagst [2017] ( ) α 1 1 ( p) = N N ( p) δ σ σ ( r ) w L µ α, δ 1 for 0 < α < 1, 0 < δ 1 41

42 Behavioral Finance Driven Investment Strategies Asset Liability Management - Goal: Maximization of the expected utility of the relative funding ratio compared to B > 0 at time T Assumption: Funded Case V FR (t) > B The optimal investment strategy is given by with = w LH Cash ( ) G ( t) 1 w ( t) w ( t) = 1δ V 1γ V FR ( t) FR B ( t) and CPPI Cash LH ( ( ) ) G ( t) = 1 1 w ( t) 1 w ( t) 42

43 Behavioral Finance Driven Investment Strategies Asset Liability Management - Example (Default Parameter α=1): 43

44 Behavioral Finance Driven Investment Strategies Asset Liability Management Comparison Expected Utility vs. Example (Default Parameter γ=-10, α=1): 44

45 Behavioral Finance Driven Investment Strategies Conclusion If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. John v. Neumann (* ) 45