Mean Variance Optimization of Non-Linear Systems and Worst-case Analysis



because km >kc = ksq. Consider now the aggregate liquidity risk. When Dm = 2Dc, this
is given by

N

Φsq = prob

i=1


N

δiDsq >    Rsq = prob(X0 < ksq)

i=1

in the status quo, and by

N

Φm = prob

i=1


N

δiDc >Rm +   Rc = prob(X0

i=3

after the merger, where X0 = PN= NN. Since Km > ksq, it follows Φm Φsq.

We can then express the expected aggregate liquidity needs in the status quo as

Ωsq = ∕,NDsq (Xsq ksqNDsq)f (Xsq)d(Xsq) = NDsq    (X0 - ksq)f (X0)d(X0).

ksq NDsq                                                ksq

Applying the same logic, the post-merger expected aggregate liquidity needs are

Ωm = NDc [1 (X0 Km)f(X0)d(X0)

Km

= NDsq (1 + (Kmksq))    (X0 Km)f(X0)d(X0),

Km

where we have used Dm =2Dc and Dm +(N2)Dc = NDc = NDsq +(Kmksq)NDsq.
Given
Km >ksq , we can write the expected aggregate liquidity needs as

Ωsq = NDsq


NDsq


1 (X0 ksq)f(X0)d(X0)+   Km (X0 ksq)f(X0)d(X0)

Km                           ksq

RK1m(X0 Km)f(X0)d(X0)+(Kmksq) RK1m f(X0)d(X0)+
RkKsqm (X0 Km)f(X0)d(X0),

and, after rearranging and simplifying, we have
because (X
Km 1) < 0. Analogous steps can be followed for the case rrD > ρ. Q.E.D.

Ωm ^-sq — NDsq


(Km


ksq) RK1m (X


Km   0

ksq (Xsq


Km1)f(X0)d(X0)
Km)f(X0)d(X0)


<0


Proof of Proposition 5

Proposition 4 implies that if km = ksq, then Φm Φsq and Ωm Ωsq for any rrD > ρ. A
fortiori this must be true in equilibrium where km < ksqm and Ωm are decreasing in Km,
which falls with
km).                                                              Q.E.D.

37



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