In a complete market, we find optimal portfolios for an investor whose satisfaction stems from both a payoff's intrinsic utility and its comparison with a reference, as specified by Koszegi and Rabin. In the regular regime, arising when reference-dependence is low, the marginal utility of the optimal payoff is proportional to a twist of the pricing kernel. High reference-dependence leads to the anchors regime, whereby investors reduce disappointment by concentrating significant probability in one or few fixed outcomes, and multiple personal equilibria arise. If stocks follow geometric Brownian motion, the model implies that younger investors have larger stocks positions than older investors, highlighting the suggestion that reference-dependence helps explain this typical recommendation of financial planners.

1. Reference Dependence:
Endogenous Anchors & Life-Cycle Investing
Paolo Guasoni1 Andrea Meireles-Rodrigues2
Dublin City University1
University of York2
Toulouse School of Economics – MADSTAT Seminar
December 3rd , 2020

2. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Outline
• Aim:
optimal portfolios for reference-dependent preferences.
• Model:
complete market.
• Results:
• reference-adjusted utility;
• first-order condition with singularities;
• quantile characterisation;
• twisted pricing kernel with endogenous curls;
• regular regime vs. anchors regime;
• life-cycle investment.

3. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Outline
2 4 6 8 10
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
Horizon
Initial
optimal
stock
weight
Merton ratio

4. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Reference-Dependent Preferences
B. Kőszegi and M. Rabin (2006).
A model of reference-dependent preferences. Quart. J. Econom., 121(4).
Main tenets:
• Reference B (possibly stochastic)
• Utility u : D ⊆ R → R (globally concave).
• Gain-loss utility ν : R → R (S-shaped).
Max. U(Z| B) = EP[u(Z)] +
Z
R
Z
R
ν u(z) − u(b)
dPB(b) dPZ (z) .

5. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Reference-Dependent Preferences
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
ν

6. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Piecewise-Linear Gain-Loss
-1.0 -0.5 0.0 0.5 1.0
-2
-1
0
1
2
ν

7. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Piecewise-Linear Gain-Loss
Assumption 1
ν(x) =







η
1 − η
x, if x 0,
λη
1 − η
x, if x ≥ 0,
for some η, λ ∈ (0, 1) .
U(Z| B) = EP[u(Z)] +
η
1 − η
GL(Z, B) ,
where
GL(Z, B) = λ
Z
R
Z
R
u(z) − u(b)
1
1{zb} dPB(b) dPZ (z)
+
Z
R
Z
R
u(z) − u(b)
1
1{zb} dPB(b) dPZ (z) .

8. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Personal Equilibria
Definition
A payoff X is a personal equilibrium with initial wealth w0 if
v(w0, X) B sup
Z∈C(w0)
U(Z| X) = U(X| X) ,
and PE(w0) denotes the set of personal equilibria.
‘If I expect to do it, then I will indeed do it!’

9. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Preferred Personal Equilibria
Definition
A payoff X ∈ PE(w0) is a preferred personal equilibrium if
sup
Z∈PE(w0)
U(Z| Z) = U(X| X) ,
and PPE(w0) denotes the set of preferred personal equilibria.
‘Of all the plans that I will actually follow through on,
I choose the best one.’

10. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Complete Market
• Horizon T ∈ (0, +∞).
• Safe asset, with interest rate r = 0.
• Unique pricing kernel ζ.
• Set of non-negative payoffs at the horizon T that are affordable
from initial capital w0 0 is
C (w0) B
n
X ∈ L0
+ : EP[ζX] ≤ w0
o
.

11. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Standing Assumptions
Assumption 2
• u(·) is str. increasing, str. concave, cont. differentiable, and
u0
(0+) B lim
x→0+
u0
(x) = +∞ and u0
(+∞) B lim
x→+∞
u0
(x) = 0.
• ζ has a cont. and str. positive density fζ(·) on (0, +∞).
• For all y ∈ (0, +∞),
EP[ζI(yζ)] +∞ and EP[u(I(yζ))] +∞,
where I(·) B (u0)−1
(·).
Remark: Unique EUT optimiser e
X B I(ỹζ), with ỹ 0 s.t. EP
h
ζ e
X
i
= w0.

12. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Reference-Adjusted Utility
Lemma
For all w0 0,
v(w0, B) = sup
Z∈C(w0)
EP[ūB(Z)] ,
with reference-adjusted utility
ūB(x) B
u(x)
1 − η
(1 − η (1 − λ) FB(x))
−
ηλ
1 − η
EP[u(B)] −
η (1 − λ)
1 − η
EP
h
u(B) 1
1{Bx}
i
.
Remark:
• right deriv. ∂+ūB(·) has cont., non-increasing generalised inverse ĪB(·);
• solve using standard methods of convex analysis;
• main difficulty: endogenise references.

13. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Reference-Adjusted Utility
ūB
0 50 100 150 200 250
-1
0
1
2
3
4
5
6
B = w0
ūB
0 50 100 150 200 250
-1
0
1
2
3
4
5
6
B = e
X
ĪB
0.00 0.01 0.02 0.03 0.04 0.05
0
50
100
150
200
250
300
350
ĪB
0.0 0.1 0.2 0.3 0.4 0.5
0
20
40
60

14. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Reference-Adjusted Utility
Lemma
For all w0 0,
v(w0, B) = sup
Z∈C(w0)
EP[ūB(Z)] ,
with reference-adjusted utility
ūB(x) B
u(x)
1 − η
(1 − η (1 − λ) FB(x))
−
ηλ
1 − η
EP[u(B)] −
η (1 − λ)
1 − η
EP
h
u(B) 1
1{Bx}
i
.
Remark:
• right deriv. ∂+ūB(·) has cont., non-increasing generalised inverse ĪB(·);
• solve using standard methods of convex analysis;
• main difficulty: endogenise references.

15. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Twist and Envelopes
Definition
Define the twist w : [0, +∞) → [0, +∞) as
w(x) ≡ w(η, λ; x) B
(1 − η) x
1 − η (1 − λ)
1 − Fζ(x)
.
Denote the set of increasing envelopes of w(·) by
W B





w?(·) ∈ C([0, +∞)) : w?(·) is non-decreasing,
w(·) ≤ w?(·) ≤ w(·) , and
if w?(x0) , w(x0) then w?(·) const. around x0





, ∅,
with w(·) and w(·) the lower and upper envelopes of w(·), resp.
Remark: H. Jin and X. Y. Zhou (2008). Behavioural Portfolio Selection in
Continuous Time. Math. Finance, 18(3).

16. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Twist and Envelopes
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w

17. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Twist and Envelopes
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w
w
w

18. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Twist and Envelopes
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w
w
w
w?
1
w?
2

19. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
A Sea of Anchors
Theorem
1. (Regular regime)
If w is increasing, then
PE(w0) = {I(y∗
w(ζ))} ,
where y∗ 0 uniquely solves EP[ζX∗] = w0.
2. (Anchors regime)
If w is not monotonic, then
PE(w0) = {I(y?
w?
(ζ)) : w?
∈ W} ,
where each y? 0 uniquely solves EP[ζI(y?w?(ζ))] = w0.
Remark:
• twisted pricing kernel ζ?
B w?
(ζ);
• no safe PE!

20. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Properties of Personal Equilibria
‘Proposition’
• Regularity and monotonicity with respect to pricing kernel.
• Sensitivity to initial capital and reference-dependence.
• Asymptotic behaviour as reference-dependence vanishes.
• Upper bound on number of atoms of anchors.
• Relation between regimes and elasticity of pricing kernel.

21. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
1-2-1 Rule
Corollary
Assume that fζ(·) is strictly unimodal with mode θ 0.
1. (Regular regime)
If
1 − η (1 − λ) ≥ 1 −
1
1 − Fζ(θ) + fζ(θ) θ
, (1)
holds for all θ, then PE(w0) = {X∗}, where
X∗
B I(y∗
w(ζ)) .

22. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
1-2-1 Rule
Corollary
Assume that fζ(·) is strictly unimodal with mode θ 0.
2. (Anchors regime)
If (1) fails, then PE(w0) = {X?
α : α ∈ [x1, x̄2]}, where
X?
α B I(y?
α w(α)) 1
1{ζ∈[α,ᾱ]} + I(y?
α w(ζ)) 1
1{ζ[α,ᾱ]}.
Here, x̄2 θ x̄1 denote the two solutions of
1 − η (1 − λ)
1 − Fζ(x) + fζ(x) x
= 0,
xi is the unique solution of w(xi ) = w(x̄i ) for each i ∈ {1, 2},
and ᾱ is the unique number in [x̄1, x2] s.t. w(α) = w(ᾱ).

23. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
1-2-1 Rule
λ
η
0
1
1
PE(w0) = {X∗
}
PE(w0) = {X?
α : α ∈ [x1, x̄2]}

24. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes
Assumption 3
• St = s0 exp
n
µ − σ2
2
t + σWt
o
for some s0 0, µ ∈ R {0},
and σ 0;
• u(x) B log(x), for all x ∈ (0, +∞).
ζ = exp
(
−
µ
σ
WT −
µ2T
2σ2
)
=
ST
s0
− µ
σ2
e
µT
2
µ
σ2 −1
P
∼ LogN 1, exp
(
µ2T
σ2
− 1
)!
.

25. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes: Regular Regime
0.0 0.5 1.0 1.5 2.0
0
100
200
300
400
500
ζ
X∗
T
e
X
0.0 0.5 1.0 1.5 2.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ζ
π∗
T
µ
σ2
PE gives up wealth in good states
to make up for losses in bad ones.
PE depresses demand for stocks.
Parameters: w0 = 100, µ = 3%, σ = 29%, T = 1, λ = 0.33, η = 0.33.

26. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes: Anchors Regime
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X?
α,T
w
w
w?
1
w
0.0 0.5 1.0 1.5 2.0
0
50
100
150
200
250
ζ
X?
α,T
e
X
Each atom is endogenous.
Parameters: w0 = 100, µ = 3%, σ = 29%, T = 1, λ = 0.33, η = 0.66.

27. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes: Anchors Regime
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X?
α,T
w
w
w?
1
w
0.0 0.5 1.0 1.5 2.0
0
50
100
150
200
250
ζ
X?
α,T
e
X
Each atom is endogenous.
Parameters: w0 = 100, µ = 3%, σ = 29%, T = 1, λ = 0.33, η = 0.66.

28. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes: Anchors Regime
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X?
α,T
w
w
w?
1
w
0.6 0.8 1.0 1.2 1.4
99.6
99.8
100.0
100.2
100.4
ζ
X?
α,T
Each atom is endogenous.
Parameters: w0 = 100, µ = 3%, σ = 29%, T = 1, λ = 0.33, η = 0.66.

29. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes: Anchors Regime
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X?
α,T
w
w
w?
1
w
0.0 0.5 1.0 1.5 2.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ζ
π?
α,T
µ
σ2
Terminal optimal stock weight is
discontinuous.
Parameters: w0 = 100, µ = 3%, σ = 29%, T = 1, λ = 0.33, η = 0.66.

30. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes: Anchors Regime
0.70 0.75 0.80
4.61
4.62
4.63
4.64
4.65
α
U(X?
α| X?
α)
Parameters: w0 = 100, µ = 3%, σ = 29%, T = 1, λ = 0.33, η = 0.66.

31. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Black-Scholes: Critical Horizon
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
5
10
15
20
25
30
35
40
45
λ
η
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
η
T∗
η = 0.66
λ = 0.33
Parameters: w0 = 100, µ = 3%, σ = 29%, T = 1.

32. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Conclusion
• Personal equilibria
• Regular regime (low reference-dependence): unique PE.
• Anchors regime (high reference-dependence): multiple PE with
endogenous atoms.
• Black-Scholes
• Long horizons: regular regime with diffuse PE.
• Shorter horizons: anchors regime with lower stock holdings.
2 4 6 8 10
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
T
Initial
optimal
stock
weight
µ
σ2
T∗
≈ 5.06

33. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Available at: https://ssrn.com/abstract=3658342 or
http://dx.doi.org/10.2139/ssrn.3658342

34. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Thank You!
Questions?

35. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
More general ν(·)
Assumption 1’
For all y 0,
x 7→ u0
(x)
1 + ν0
−(u(y) − u(x))
is decreasing on (0, y).
Back

36. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Key Lemmata: Reference-Adjusted Utility
Lemma 1
For all w0 0, the optimisation problem
v(w0, B) = sup
Z∈C(w0)
EP[ūB(Z)]
has unique solution ẐB BĪB(ŷζ), with ŷ 0 s.t. EP
h
ζẐB
i
=w0.
Remark:
• ūB(·) is cont., str. increasing, str. concave, and differentiable outside
∆B B {x ∈ (0, +∞) : P{B = x} 0};
• ūB(·) is more risk averse than u(·).
Back

37. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Key Lemmata: Quantile Characterisation of PE
Lemma 2
Let B ∈L0
+ be non-degenerate, ess inf ζ =0 and ess sup ζ =+∞.
Then, B ∈ PE(w0) if and only if all conditions below hold.
1. FB(·) is strictly increasing.
2. For all x ∈ (0, +∞),
qB(FB(x)) = I
(1 − η) ŷqζ(1 − FB(x))
1 − η (1 − λ) FB(x)
!
,
where ŷ ∈ (0, +∞) solves EP
h
ζqB
1 − Fζ(ζ)
i
= w0.
3. B = qB
1 − Fζ(ζ)
a.s..
Remark: H. Jin and X. Y. Zhou (2008). Behavioural Portfolio Selection in
Continuous Time. Math. Finance, 18(3).
Back

38. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Safe PE
Lemma
Let Xf B w0. Then, Xf ∈ PE(w0) if and only if
ess sup ζ
ess inf ζ
≤
1
1 − η (1 − λ)
.
Remark: A.E. Bernardo and O. Ledoit (2000). Gain, loss, and asset pricing.
J. Polit. Econ., 108(1).
Back

39. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Properties (I)
Proposition 1
1. For all X ∈ PE(w0), ζ 7→ X is cont. and decreasing, with
lim
ζ→0
X = +∞ and lim
ζ→+∞
X = 0.
2. If w is str. increasing, then the unique PE
X∗
≡ X∗
(λ, η) B I(y∗
w(ζ)) , (1)
where y∗
0 uniquely solves EP[ζX∗
] = w0, is abs. continuous. Moreover,
X∗
e
X a.s. on
ζ ζ̃
(and X∗
e
X a.s. on
ζ ζ̃
), with
ζ̃ B qζ
(1 − η) y∗
− (1 − η (1 − λ)) ỹ
η (1 − λ) ỹ
.
In addition, y∗
% in λ, η; in w0.
3. If η is suff. small or λ is suff. close to 1, then PE(w0) = {X∗
} with X∗
as in
(1). Moreover,
lim
η→0
X∗
(λ, η) = e
X and lim
λ→1
X∗
(λ, η) = e
X.

40. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Properties (II)
Proposition 2
Define the concentration function H : (0, +∞) → R as
H(x) B 1 − η (1 − λ) 1 − Fζ(x) + fζ(x) x
,
and the underwater set
N B {x ∈ (0, +∞) : H(x) ≤ 0} .
1. w(·) is str. increasing if and only if int(N) = ∅.
2. If int(N) , ∅, then
int(N) = ∪i∈I (ai , bi ) , for some ∅ , I ⊆ N, 0 ai bi ,
and (ai , bi ) ∩ (aj , bj ) = ∅ for i , j,
whence any personal equilibrium is constant on each {ai ≤ ζ ≤ bi }.
Moreover, if I is finite, then any PE has at most |I| atoms.
Remark: sup
x∈(0,+∞)
fζ(x) x
Fζ(x)
≤ 1 ⇔ int(N) = ∅ for all λ, η ∈ (0, 1).
Back

41. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Decreasing Density
Corollary
Assume that fζ(·) is decreasing. Then,
PE(w0) = {I(y∗
w(ζ))} for all λ, η ∈ (0, 1) ,
where y∗ ∈ (0, +∞) uniquely solves EP[ζI(y∗ w(ζ))] = w0.
Moreover, lim(η,λ)→(1,0) X∗ = X̄, where
X̄ B I
ȳζ
Fζ(ζ)
!
and ȳ ∈ (0, +∞) uniquely solves EP
h
ζX̄
i
= w0.
Back

42. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Black-Scholes
Proposition 1
Let κ B µ/σ, Φ(·) and φ(·) be the std. normal cdf and pdf, resp.
1. (Regular regime)
If
1 − η (1 − λ) ≥ 1 −
√
κ2T
√
κ2T Φ
√
κ2T
+ φ
√
κ2T
, (2)
then PE(w0) = {X∗
T }, where
X∗
t = e
Xt −
η (1 − λ) w0
(2 − η (1 − λ)) ζt
1 − 2 Φ(d1(ζt , t, T))
.
Moreover,
π∗
t =
µ
σ2
1−η (1 − λ)
φ(−d1(ζt , t, T))
p
κ2 (2T − t) (1 − η (1 − λ) Φ(−d1(ζt , t, T)))
!
.
Here, d1(x, t, T) B
log
(x)+κ2t/2
p
κ2(2T−t)
.

43. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Black-Scholes
Proposition 1
2. (Anchors regime)
If (2) fails, then PE(w0) = {X?
α : α ∈ [x1, x̄2]}, where
X?
α =
w0
kα
1 − η (1 − λ) Φ(−d1(α, T, T))
α
1
1{ζ∈[α,ᾱ]}
+
1 − η (1 − λ) Φ(−d1(ζ, T, T))
ζ
1
1{ζ[α,ᾱ]}
.
Moreover,
π?
α,T =
µ
σ2
1−η (1−λ)
φ(−d1(ζ, T, T))
√
κ2T(1−η (1−λ)Φ(−d1(ζ, T, T)))
1
1{ζ[α,ᾱ]}.
Here, d2(x, t, T) B
log
(x)−κ2(T−t)/2
p
κ2(T−t)
, and
kα B1−
η (1 − λ)
2
+
1−η (1 − λ) Φ(−d1(α, T, T))
α
Φ(d2(ᾱ, 0, T))−Φ(d2(α, 0, T))
+ Φ(d1(α, T, T))−Φ(d1(ᾱ, T, T))
1−η (1 − λ)
Φ(−d1(α, T, T))+Φ(−d1(ᾱ, T, T))
2
.
Back

44. Outline Introduction PE and PPE Model Main Results Example Conclusion Link to Paper Appendix
Appendix
Black-Scholes: Critical Horizon
Condition (2) holds if and only if T ≥ T∗ B %2/κ2, where
% ≡ %(λ, η) ∈ (0, +∞) is the unique solution of
x (1 − η (1 − λ) Φ(x)) − η (1 − λ) φ(x) = 0.
Remark:
• T∗
w.r.t. κ;
• limλ→1 T∗
= 0 and limη→0 T∗
= 0;
• lim(λ,η)→(0,1) T∗
= +∞;
• limT→+∞ X∗
t = e
Xt for all t ∈ [0, T).
Back