Consider the management of a productive asset that generates a return

stream of S, where

dS = ¹dt + ¾dW

¹ is negative, implying a tendency for the asset to generate lower (pos-

sibly negative) returns over time. The asset can be replaced at a ¯xed

cost of K, with the new asset generating a return of Sn. Suppose you

want to maximize the discounted expected °ow of returns over time,

using a discount rate or r.

The value of the discounted expected °ow of returns, V (S) satis¯es the

complementarity conditions

rV (S) ¸ S + LV (S)

V (S) ¸ V (Sn) ¡ K

with one of these two condition satis¯ed with equality at each value

of S. When the ¯rst condition is satis¯ed with equality it is optimal

to keep the current asset; when the second condition is satis¯ed with

equality it is optimal to replace the asset.

Suppose that r = 0:05, ¹ = ¡0:03, ¾ = 0:1, K = 5 and Sn = 1. Ap-

proximate V using Á(S)c for some choice of Á and solve the associated

complementarity problem. Determine the optimal rule for replacing

the asset. You should ¯nd that it is optimal to replace the asset when

S falls below a certain value. Determine this value.

Be sure to experiment with the number of nodes and the approximation

interval to be sure that you solution is not adversely a®ected by these choices.

stream of S, where

dS = ¹dt + ¾dW

¹ is negative, implying a tendency for the asset to generate lower (pos-

sibly negative) returns over time. The asset can be replaced at a ¯xed

cost of K, with the new asset generating a return of Sn. Suppose you

want to maximize the discounted expected °ow of returns over time,

using a discount rate or r.

The value of the discounted expected °ow of returns, V (S) satis¯es the

complementarity conditions

rV (S) ¸ S + LV (S)

V (S) ¸ V (Sn) ¡ K

with one of these two condition satis¯ed with equality at each value

of S. When the ¯rst condition is satis¯ed with equality it is optimal

to keep the current asset; when the second condition is satis¯ed with

equality it is optimal to replace the asset.

Suppose that r = 0:05, ¹ = ¡0:03, ¾ = 0:1, K = 5 and Sn = 1. Ap-

proximate V using Á(S)c for some choice of Á and solve the associated

complementarity problem. Determine the optimal rule for replacing

the asset. You should ¯nd that it is optimal to replace the asset when

S falls below a certain value. Determine this value.

Be sure to experiment with the number of nodes and the approximation

interval to be sure that you solution is not adversely a®ected by these choices.

## This question was asked on May 01, 2010.

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