Internet Appendix for “Money Doctors”
NICOLA GENNAIOLI, ANDREI SHLEIFER, AND ROBERT VISHNY
We provide proofs of the results in the paper (part I), as well as an alternative formulation of the
model with additive utility from financial advice (part II).
I. Proofs
Proof of Proposition 1: Consider the expressions for 𝜋𝐴 (𝑓𝐴 , 𝑓𝐵 ) and 𝜋𝐵 (𝑓𝐴 , 𝑓𝐵 ) in equations
(8) and (9). For any 𝜃 ≥ 0, in equilibrium we must have
𝑅−𝑓𝐴
𝑅−𝑓𝐵
∈ [√1 − 𝜃,
1
√1−𝜃
],
(IA.1)
otherwise only one manager makes zero profits. This manager could cut his fee and make some
positive profits as well. This condition alone implies that when 𝜃 = 0, the unique equilibrium
features 𝑓𝐴∗ = 𝑓𝐵∗ = 0. When 𝜃 > 0, the equilibrium must be inside the above interval and satisfy
the manager’s first-order conditions. When 𝑓𝐴 ≥ 𝑓𝐵 these first-order conditions are
𝑓𝐴 :
𝑓𝐵 :
𝑅−𝑓𝐴 2
)
𝑅−𝑓𝐵
(𝑅 − 2𝑓𝐴 ) ∙ [(
(𝑅 − 2𝑓𝐵 ) ∙
1
+
2
[
𝑅−𝑓
2
− (1 − 𝜃)] − 2𝑓𝐴 (𝑅−𝑓𝐴 ) = 0,
(IA.2)
𝐵
1−(
𝑅 − 𝑓𝐴 4
)
𝑓𝐵 𝑅 − 𝑓𝐴 4
𝑅 − 𝑓𝐵
− (
) = 0.
4𝜃
𝜃 𝑅 − 𝑓𝐵
]
𝑅−𝑓
These two equations cannot be jointly satisfied for 𝑓𝐴 > 𝑓𝐵 . To see this, set 𝑦 ≡ (𝑅−𝑓𝐴 ) and solve
𝐵
the above first-order conditions for 𝑓𝐴 and 𝑓𝐵 as a function of 𝑦. Next, impose the condition 𝑓𝐴
𝑅−𝑓
> 𝑓𝐵 . This identifies a quadratic equation in 𝑦 that cannot be satisfied for (𝑅−𝑓𝐴 ) > √1 − 𝜃. As a
𝐵
consequence, the only possible equilibrium featuring 𝑓𝐴 ≥ 𝑓𝐵 is symmetric, that is, 𝑓𝐴∗ = 𝑓𝐵∗. It is
*
Citation format: Gennaioli, Nicola, Andrei Shleifer, and Robert Vishny, Internet Appendix for “Money
Doctors,” Journal of Finance, DOI: 10.1111/jofi.12188. Please note: Wiley-Blackwell is not responsible for the
content or functionality of any supporting information supplied by the authors. Any queries (other than
missing material) should be directed to the authors of the article.
1
straightforward to check that in that equilibrium equation (8) of Proposition 1 is met. When 𝑓𝐴 ≤
𝑓𝐵 , the same argument shows that 𝑓𝐴∗ = 𝑓𝐵∗ is the only equilibrium satisfying the above first-order
conditions. In this symmetric equilibrium, the second-order conditions are also met (i.e., managers’
objective functions are locally concave).
Proof of Corollary 1: Without managers, investors do not obtain any excess return. With
𝑅
2+𝜃
money managers, each investor invests 𝑥 = 2𝑎𝜎 (1+𝜃) at expected excess return 𝑅 under his most
trusted manager. With quadratic utility, the aggregate welfare gain is given in Corollary 1.
Proof of Lemma 1: Given investment 𝑥𝑖 , the optimal mixture (𝑥1,𝑖 , 𝑥2,𝑖 ) of assets 1 and 2
solves
𝑎
2
2
𝑚𝑎𝑥𝑥1,𝑖,𝑥2,𝑖 [𝑅1 𝑥1,𝑖 + 𝑅2 𝑥2,𝑖 ] − 2 [𝜎1 𝑥1,𝑖
+ 𝜎2 𝑥2,𝑖
],
(IA.3)
subject to 𝑥1,𝑖 + 𝑥2,𝑖 = 𝑥𝑖 . The first order conditions of the problem are given by
𝑅𝑧 − 𝑎𝜎𝑧 𝑥𝑧,𝑖 = 𝜆 𝑓𝑜𝑟 𝑧 = 1,2,
(IA.4)
where 𝜆 is the Lagrange multiplier attached to the constraint 𝑥1,𝑖 + 𝑥2,𝑖 = 𝑥𝑖 . It is easy to see that
the above first-order conditions are satisfied at the portfolio of Lemma 1.
Proof of Proposition 2: Because investors separately choose the manager to invest in a
specific asset and the amount to invest, the profit of each manager j is separable in the two assets
and equal to 𝜋𝑗 (𝑓1,𝐴 , 𝑓1,𝐵 ) + 𝜋𝑗 (𝑓2,𝐴 , 𝑓2,𝐵 ). Managers thus compete in the two assets separately and,
for each of the assets, equilibrium fees and investments follow from Proposition 1, yielding
equations (13), (14), and (15).
Proof of Corollary 2: The equilibrium under investors’ misperception is found by replacing
in Proposition 2 the true return of asset z with investors’ expected return. The properties described
in Corollary 2 then follow.
Proof of Proposition 3: Suppose that there is no trust, that is, 𝜃 = 0. In this case, Bertrand
competition among money managers prevails. Suppose that we are in an equilibrium in which both
2
managers pander at 𝑡 = 0; formally, 𝜔𝐴 = 𝜔𝐵 = 1. In this equilibrium, managers make zero profits
at t = 0. At t = 1, the manager with higher assessed ability captures the entire market while the
other makes zero profits. Consider, for instance, the outcome attained at t = 1 by manager A. When
𝑉̃𝐴 > 𝑉̃𝐵 , the manager captures the full market. If 𝑉̃𝐴 ≤ 𝑅1,𝑒 + 2𝑉̃𝐵 , the fee that manager A can
charge is restricted by what investors can obtain with manager B. As a result, manager A charges a
fee 𝑓𝐴 = 𝑉̃𝐴 − 𝑉̃𝐵 so that investors are just indifferent (but they all invest with A). At this fee, the
manager’s profit is proportional to (𝑅1,𝑒 + 𝑉̃𝐵 )(𝑉̃𝐴 − 𝑉̃𝐵 ). If instead 𝑉̃𝐴 > 𝑅1,𝑒 + 2𝑉̃𝐵 , manager A is
so talented that he can act as a monopolist. As a result, he charges a fee 𝑓𝐴 = (𝑅1,𝑒 + 𝑉̃𝐴 )/2 and his
2
profit is proportional to (𝑅1,𝑒 + 𝑉̃𝐴 ) /4. Thus, when 𝜃 = 0 the future expected profit of manager A
are proportional to
𝛿𝑊𝐴 (𝜔𝐴 , 𝜔𝐵 ) =
+∞
𝛿∙∫
−∞
̃𝐵
𝑅1,𝑒 +2𝑉
[∫
̃𝐵
𝑉
(IA.5)
(𝑅1,𝑒 + 𝑉̃𝐴 )2
] ℎ(𝑉̃𝐴 , 𝑉̃𝐵 |𝜔𝐴 , 𝜔𝐵 )𝑑𝑉̃𝐴 𝑑𝑉̃𝐵 ,
4
̃𝐵
𝑅1,𝑒 +2𝑉
+∞
(𝑅1,𝑒 + 𝑉̃𝐵 )(𝑉̃𝐴 − 𝑉̃𝐵 ) + ∫
where ℎ(𝑉̃𝐴 , 𝑉̃𝐵 |𝜔𝐴 , 𝜔𝐵 ) is the 𝑡 = 0 (normal) distribution of average abilities at t = 1 conditional
on managers choosing portfolios 𝜔𝑗 for 𝑗 = A, B at t = 0. Given statistical independence of
managerial skills, this distribution is the product of two densities ℎ(𝑉̃𝑗 |𝜔𝑗 ) for 𝑗 = A, B. We can
express the problem of the manager at t = 0 as follows. Denote by 𝜋𝑗 (𝜑𝐴 , 𝜑𝐵 , 𝜔𝐴 , 𝜔𝐵 ) the profit of
manager j at t = 0 and and by 𝜋𝑗 (𝑉̃𝐴 , 𝑉̃𝐵 ) the manager’s equilibrium profits at t = 1 when the
assessed abilities are (𝑉̃𝐴 , 𝑉̃𝐵 ). At t = 0, then, the manager’s optimal strategy solves
max𝜑𝑗,𝜔𝑗 𝜋𝑗 (𝜑𝐴 , 𝜑𝐵 , 𝜔𝐴 , 𝜔𝐵 ) + 𝛿 ∬ 𝜋𝑗 (𝑉̃𝐴 , 𝑉̃𝐵 )ℎ(𝑉̃𝐴 |𝜔𝐴 ) ℎ(𝑉̃𝐵 |𝜔𝐵 )𝑑𝑉̃𝐴 𝑑𝑉̃𝐵 .
(IA.6)
We replace the double integral in the above expression by the expression for 𝑊𝐴 (𝜔𝐴 , 𝜔𝐵 ) that we
just calculated. Since the current profit of manager A is zero, and his future profit increases in 𝑉̃𝐴 ,
the manager wishes to boost his assessed ability at t = 1 as much as possible. As a result, full
pandering is never an equilibrium. By deviating to contrarianism 𝜔𝐴 = 0, the manager does not
lose any current profit (which is zero anyway), but boosts future profits by moving ℎ(𝑉̃𝐴 |𝜔𝐴 ) to the
3
right. The same argument holds for manager B. As a result, a full pandering equilibrium cannot
exist in the absence of trust.
Proof of Proposition 4: From equation (1), the static profit of a monopolist manager
delivering an expected excess return 𝑅𝑗 at fee 𝑓𝑗 is proportional to 𝑓𝑗 ∙ (𝑅𝑗 − 𝑓𝑗 )/𝜎. For given
expected return 𝑅, the optimal fee charged by the manager is given by 𝑓𝑗 = 𝑅𝑗 /2. At this optimal
fee, the manager’s static profit is equal to
𝑅𝑗2
4𝜎
.
If the manager panders, his clients perceive an
expected first period return of 𝑅1,𝑒 and a first period risk of (𝑣 + 𝜂). In the second period,
pandering generates an expected excess return that is normally distributed with mean 𝑅1,𝑒 − (𝑅1,𝑒 −
𝑣
𝑣𝜂
𝑅1 ) (𝑣+𝜂) and variance 𝑣+𝜂 . The variance of returns perceived by investors as of the second period
is instead equal to 𝜂
2𝑣+𝜂
.
𝑣+𝜂
Note that we assume here that asset class 1 continues to be the
overvalued one in the second period, but this assumption does not qualitatively affect our results.
The expected discounted payoff obtained by the manager from pandering at t = 0 is then given by
2
𝑣
𝑣𝜂
2
[𝑅1,𝑒 − (𝑅1,𝑒 − 𝑅1 ) (𝑣 + 𝜂 )] + 𝑣 + 𝜂
𝑅1,𝑒
+𝛿
.
2𝑣 + 𝜂
4 ∙ (𝑣 + 𝜂)
4𝜂 𝑣 + 𝜂
(𝐼𝐴. 7)
If instead the manager acts contrarian, his clients perceive an expected first-period return of 𝑅2,𝑒
and a first period risk of (𝑣 + 𝜂). In the second period, contrarianism generates an expected excess
𝑣
𝑣𝜂
return that is normally distributed with mean 𝑅1,𝑒 + (𝑅2 − 𝑅2,𝑒 ) (𝑣+𝜂) and variance 𝑣+𝜂 (where the
term 𝑅1,𝑒 comes from the fact that both managers pander at 𝑡 = 1). The variance of returns
perceived by investors as of the second period is instead equal to 𝜂
2𝑣+𝜂
.
𝑣+𝜂
The expected discounted
payoff obtained by the manager from being contrarian at t = 0 is then given by
2
𝑅2,𝑒
4 ∙ (𝑣 + 𝜂)
+𝛿
2
𝑣
𝑣𝜂
[𝑅1,𝑒 + (𝑅2 − 𝑅2,𝑒 ) (𝑣 + 𝜂 )] + 𝑣 + 𝜂
2𝑣 + 𝜂
4𝜂 𝑣 + 𝜂
4
.
(𝐼𝐴. 8)
The monopolist manager thus prefers to pander than to be contrarian when (𝐼𝐴.7) is higher than
(𝐼𝐴.8), which occurs when
2
2
𝑅1,𝑒
− 𝑅2,𝑒
𝑣(𝑣 + 𝜂)
>𝛿∙
.
𝑣
𝜂(2𝑣 + 𝜂)
[(𝑅2 − 𝑅2,𝑒 ) + (𝑅1,𝑒 − 𝑅1 )] [2𝑅1,𝑒 + [(𝑅2 − 𝑅2,𝑒 ) − (𝑅1,𝑒 − 𝑅1 )] (𝑣 + 𝜂 )]
When, as assumed, both assets are misvalued by the same amount, that is, (𝑅2 − 𝑅2,𝑒 ) =
(𝑅1,𝑒 − 𝑅1 ), the above condition becomes as reported in Proposition 4:
2
2
𝑅1,𝑒
− 𝑅2,𝑒
[(𝑅2 − 𝑅2,𝑒 ) + (𝑅1,𝑒 − 𝑅1 )]2𝑅1,𝑒
5
>𝛿∙
𝑣(𝑣 + 𝜂)
.
𝜂(2𝑣 + 𝜂)
II. Trust as an Additive Boost to Utility
In this setup, managers are rewarded not for helping investors take risk, but more generally
for helping them invest. Formally, delegating to the manager acts as an additive boost to the
investor’s utility. To see how this works, suppose that investor 𝑖 delegates to manager 𝑗 investments
𝑠
𝑟
(𝑥𝑖,𝑗
, 𝑥𝑖,𝑗
) in the safe and risky asset, respectively. In addition, the same investor makes safe and
𝑠
𝑟
𝑠
𝑟
𝑠
𝑟
risky investments (𝑥𝑖,𝑖
, 𝑥𝑖,𝑖
) on his own, where we have that 𝑥𝑖,𝑗
+ 𝑥𝑖,𝑗
+ 𝑥𝑖,𝑖
+ 𝑥𝑖,𝑖
= 1. Then the
utility of investor 𝑖 is given by
𝑠
𝑟
𝑠
𝑟
𝑈𝑖 (𝑥𝑖,𝑗
, 𝑥𝑖,𝑗
, 𝑥𝑖,𝑖
, 𝑥𝑖,𝑖
, 𝑓𝑗 ) =
(IA.9)
𝑠
𝑟 ̂
𝑥𝑖,𝑗
(𝑅𝑓 + 𝑎𝑖,𝑗 − 𝑓𝑗 ) + 𝑥𝑖,𝑗
(𝑅 + 𝑎𝑖,𝑗 − 𝑓𝑗 ) +
𝑠
𝑟 ̂
+𝑥𝑖,𝑖
(𝑅𝑓 + 𝑎𝑖,𝑖 ) + 𝑥𝑖,𝑖
(𝑅 + 𝑎𝑖,𝑖 )
2
𝑟
𝑟
+ 𝑥𝑖,𝑖
(𝑥𝑖,𝑗
)
−
𝜎.
2
Here 𝑅̂ is the expected return of the risky asset. Parameter 𝑎𝑖,𝑗 ≥ 0 captures the utility boost
obtained by the investor when hiring manager 𝑗. Parameter 𝑎𝑖,𝑖 is the utility obtained by the
investor from investing on his own. When 𝑎𝑖,𝑗 ≫ 𝑎𝑖,𝑖 for all 𝑖 and 𝑗 = A, B, the investor never
invests on his own. Consistent with our previous analysis, this is the case we consider here. As in
our basic setup, the critical choice for the investor is whether to hire manager A or B.
𝑠
𝑟
𝑠
𝑟
Formally, in this case we have 𝑥𝑖,𝑖
= 𝑥𝑖,𝑖
= 0 and thus 𝑥𝑖,𝑗
= 1 − 𝑥𝑖,𝑗
. The utility of the
investor is then given by
𝑟
𝑈𝑖 (𝑥𝑖,𝑗
, 𝑓𝑗 )
= (𝑅𝑓 + 𝑎𝑖,𝑗 − 𝑓𝑗 ) +
𝑟
𝑥𝑖,𝑗
𝑅
−
𝑟
(𝑥𝑖,𝑗
)
2
2
𝜎,
(IA.10)
where 𝑅 is (as in our basic setup) the expected excess return of the risky asset. The optimal
investment in the risky asset is given by 𝑥̂𝑖,𝑗 = 𝑅/𝜎, and the indirect utility experienced by investor
𝑖 when delegating to manager 𝑗 is given by
𝑅2
𝑈𝑖,𝑗 (𝑥̂𝑖,𝑗 , 𝑓𝑗 ) ≡ 𝑅𝑓 + 2𝜎 + 𝑎𝑖,𝑗 − 𝑓𝑗 .
6
(IA.11)
The investor chooses A over B provided 𝑈(𝑥̂𝑖,𝐴 , 𝑓𝐴 ) ≥ 𝑈(𝑥̂𝑖,𝐵 , 𝑓𝐵 ), which is equivalent to
𝜏̃ 𝑖 ≡ 𝑎𝑖,𝐴 − 𝑎𝑖,𝐵 ≥ 𝑓𝐴 − 𝑓𝐵 .
(IA.12)
Investor 𝑖 chooses manager A when the extra utility boost he obtains from doing so is larger than
the extra fee charged by A.
Suppose that 𝜏̃ 𝑖 is distributed in the population of investors according to a cumulative
distribution function 𝐻(𝜏). Investors with 𝜏̃ 𝑖 > 0 are A-trusting, while investors with 𝜏̃ 𝑖 < 0 are Btrusting. We assume that this distribution has zero mean (i.e., ∫ 𝜏𝑑𝐻(𝜏) = 0), so that there is no
systematic preference for either manager and 𝐻(𝜏) is symmetric around zero.
The profits earned by money managers A and B are then given by
𝜋𝐴 (𝑓𝐴 , 𝑓𝐵 ) = 𝑓𝐴 [1 − 𝐻(𝑓𝐴 − 𝑓𝐵 )]
(IA.13)
𝜋𝐵 (𝑓𝐴 , 𝑓𝐵 ) = 𝑓𝐵 𝐻(𝑓𝐴 − 𝑓𝐵 ),
(IA.14)
respectively. Given the assumed symmetry of 𝐻(𝜏), the Nash Equilibrium is symmetric (𝑓𝐴∗ = 𝑓𝐵∗ =
𝑓 ∗) and the equilibrium fee is identified by the first-order condition
1 − 𝐻(0) − 𝑓 ∗ 𝐻 ′ (0) = 0
⇒ 𝑓∗ =
(IA.15)
1
,
2𝐻 ′ (0)
provided the second-order condition 2𝐻 ′ (0) + [𝐻 ′′ (0)/2𝐻 ′ (0)] > 0 is met.
As an example, suppose that 𝜏 is normally distributed, so that 𝐻(𝜏) is the cumulative
distribution function of the normal distribution. We denote the variance of such distribution as 𝜃̂.
The mapping with parameter 𝜃 is clear. The utility drop experienced by the average investor when
−
switching to his less preferred manager is equal to E(𝜏|𝜏 ≥ 0) =
𝜏2
̂
+∞
𝑒 2𝜃
∫0 𝜏 ∙
̂
√2𝜋𝜃
∙ 𝑑𝜏, which increases
in variance 𝜃̂, as in our main model. The heterogeneity of investors also increases in 𝜃̂. With this
normal distribution we have that the second-order condition is satisfied and equilibrium fees are
equal to
7
̂
𝜋𝜃
𝑓∗ = √ 2 .
(IA.16)
In the case in which trust generates an additive utility boost, managers can charge positive fees, and
these fees increase in the dispersion 𝜃̂ of investors’ trust across the two managers.
There are two important differences between this model and ours. First, since the manager
is not helping the investor take risk, fees do not reflect a sharing of expected market return, in the
sense that 𝑓 ∗ is independent of the expected return (and thus implicitly of the risk 𝑓 ∗) of different
asset classes. Second, and related, the incentives for money managers to pander are weaker in this
model, precisely because money managers cannot charge higher unit fees for placing investors into
hot asset classes with higher perceived returns.
8