= Solution
Let $F(y)$ be the manager's <utility function> as a function of nonnegative terminal fund wealth. Write
$$
a=\gamma\varepsilon,\qquad b=\gamma(\varepsilon+\alpha)>a.
$$
Then
$$
F(y)=
\begin{cases}
-e^{-ay},&0\leq y\leq w_0,\\
-e^{\gamma\alpha w_0-by},&y\geq w_0.
\end{cases}
$$
Both pieces are increasing and strictly <concave functions>. However, the derivative jumps upward from $ae^{-aw_0}$ to $be^{-aw_0}$ at $w_0$. \b[The payoff utility is increasing but not concave.] A <concave function> must have nonincreasing one-sided slopes.
For <concavification of incentive utility>, the <common tangent for exponential incentive utility> replaces the upward kink by its common tangent. There are contact points $\ell<w_0<h$ and a common slope $s>0$ satisfying
$$
F'(\ell)=F'(h)=s,\qquad
F(h)-F(\ell)=s(h-\ell).
$$
Since $F(\ell)=-s/a$ and $F(h)=-s/b$, the chord condition gives $h-\ell=1/a-1/b$. Solving the derivative matching yields \b[the contact points]
$$
\boxed{\ell=w_0-\frac{b/a-1-\log(b/a)}{b-a},\quad
h=w_0+\frac{\log(b/a)-1+a/b}{b-a},\quad
s=ae^{-a\ell}.}
$$
The assumption that $w_0$ is large enough means, explicitly, that
$$
w_0>\frac{b/a-1-\log(b/a)}{b-a},
$$
so $\ell>0$. The inequalities $\ell<w_0<h$ follow from $\log q<q-1$ and $\log q>1-1/q$ for $q>1$.
The least <concave majorant> on $[0,\infty)$ is
$$
\boxed{\bar U(y)=
\begin{cases}
F(y),&0\leq y\leq\ell,\\
F(\ell)+s(y-\ell),&\ell\leq y\leq h,\\
F(y),&y\geq h.
\end{cases}}
$$
It is increasing and <concave>: the derivative is decreasing outside the interval and equals $s$ inside. The tangent line lies above each original branch. Any <concave majorant> must lie above the chord joining the two contacts, so this one is the least.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-41-concavification.png]
{title=Common tangent replacing the incentive kink in terminal-wealth utility}
{height=420}
The <state-price budget constraint> now reduces the problem to pointwise maximization of $\bar U(y)-qy$, where $q=\eta\zeta_T$ and $\eta>0$ is chosen so that $\mathbb E[\zeta_TY^*]=w_0$. Because $\bar U'(0)=a$ is finite, the nonnegative wealth constraint must be included. \b[The optimal terminal wealth is]
$$
\boxed{Y^*(q)=
\begin{cases}
0,&q\geq a,\\
a^{-1}\log(a/q),&s<q<a,\\
b^{-1}[\gamma\alpha w_0+\log(b/q)],&0<q<s.
\end{cases}}
$$
At $q=s$, any point of $[\ell,h]$ maximizes the concavified objective. If $\kappa\ne0$, the <state-price density> has a continuous <log-normal distribution>, so this event has probability zero. The optimizer then avoids $(\ell,h)$ almost surely and satisfies $\bar U(Y^*)=F(Y^*)$. The supporting-line proof from part (i), with supergradients at the contacts, proves optimality for the concavified problem; the pointwise equality proves \b[the same optimizer and value solve the original manager's problem]. Nonnegative replication is available in the <complete market>.
If $\kappa=0$, the <state-price density> is deterministic. If the required deterministic mean terminal wealth lies in the linear segment, randomize between $\ell$ and $h$ at the common slope instead of choosing an interior wealth. This maintains the budget and attains the same concavified value. The Brownian market can replicate that bounded lottery even when its <market price of risk> is zero.
In the special case $r=\mu=0$, $\zeta_T=1$ and the fund has a fair-game wealth process. Since $\ell<w_0<h$, \b[the manager uses <fair-game gambling induced by an incentive fee>]:
$$
\boxed{\mathbb P(Y^*=h)=\frac{w_0-\ell}{h-\ell},\qquad
\mathbb P(Y^*=\ell)=\frac{h-w_0}{h-\ell}.}
$$
For example choose a threshold event in $W_T$ with the displayed probability and replicate its payoff. Then $w_t=\mathbb E[Y^*\mid\mathcal F_t]$ remains between $\ell$ and $h$, so the strategy respects nonnegative wealth. It earns no risk premium but raises expected incentive <utility function> above the value at the unrandomized $w_0$, because the common tangent lies strictly above the kink.
Back to article page
