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



More intriguing information

1. The name is absent
2. The Shepherd Sinfonia
3. The name is absent
4. A Duality Approach to Testing the Economic Behaviour of Dairy-Marketing Co-operatives: The Case of Ireland
5. From Communication to Presence: Cognition, Emotions and Culture towards the Ultimate Communicative Experience. Festschrift in honor of Luigi Anolli
6. CROSS-COMMODITY PERSPECTIVE ON CONTRACTING: EVIDENCE FROM MISSISSIPPI
7. The name is absent
8. The storage and use of newborn babies’ blood spot cards: a public consultation
9. Corporate Taxation and Multinational Activity
10. The Effects of Attendance on Academic Performance: Panel Data Evidence for Introductory Microeconomics