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

because k_m >k_c = k_sq. Consider now the aggregate liquidity risk. When D_m = 2D_c, this
is given by

^N

δiDsq >    Rsq = prob(X⁰ < ksq)

i=1

in the status quo, and by

^N

δiDc >Rm +   Rc = prob(X⁰

i=3

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

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

Ωsq = ∕^,ND^sq (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 + (Km — ksq))    (X⁰ — Km)f(X⁰)d(X⁰),

Km

where we have used D_m =2D_c and D_m +(N — 2)D_c = ND_c = ND_sq +(K_m — k_sq)ND_sq.
Given K_m >k_sq , we can write the expected aggregate liquidity needs as

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

Km                           ksq

^R_K¹_m(X⁰ — Km)f(X⁰)d(X⁰)+(Km — ksq) ^R_K¹_m f(X⁰)d(X⁰)+
^R_k^K_sq^m (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.

Proof of Proposition 5

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

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