Restrepo et al.: Monte Carlo simulation applied to Xiphias gladius and Gadus morhua 



739 



(Fgtatus quo) 's a quota of 220,000 mt, but the actual 

 value might be anywhere from ~170,000 to 260,000 

 mt. If the TAG is set at 220,000 mt, then these results 

 suggest there is roughly a 50% chance of the fishing 

 mortality increasing and a 50% chance of it decreas- 

 ing. Suppose one is risk averse and chooses a TAG of 

 210,000mt instead. What would be the perceived risk, 

 or probability of exceeding the target fishing mortal- 

 ity, under this quota? 



The risk of exceeding the target fishing mortality 

 (Fstatusquo) is givcn by the proportion of the area under 

 the histogram to the left of the TAG chosen (Fig. 1). 

 Thus, 



I 



Prob(Fachieved>Ftarget) = Z. 



i-1 



P(i) 



where p(i) is the probability mass (relative frequency 

 of outcomes) associated with the ith bar of the histo- 

 gram, and I is the number of bars to the left of the 

 chosen TAG. This probability can be computed for any 

 value of the TAG. In practice, the risk would be com- 

 puted by sorting in ascending order the 1000 catch 

 values obtained from the simulation, and then plotting 

 the cumulative count of outcomes less than any value 

 of the TAG versus that value of the TAG (Fig. 2). One 

 can also derive a family of risk curves. For example, 

 separate curves could be generated for the risk of ex- 

 ceeding Fgtatus quo by each of several amounts (in ab- 

 solute or relative numbers). For each of the 1000 

 simulation runs, one computes the value of Fgtatus quo 

 and the catch that causes current F to exceed the status 

 quo by the specified amount. The resulting histogram 

 of catches is summed to obtain the risk curve. 



Estimating cost as yield foregone 



If we choose a conservative value for the TAG in order 

 to ensure that risk of exceeding the target fishing mor- 

 tality will be small, then we are probably passing up 

 some of the yield we could have had in the short term 



while still meeting our objective (e.g., see Bergh and Butterworth 1987). It is possible to describe this cost in 

 economic or biological terms. Here, we express the cost as the expected value of the potential yield foregone, 

 which we define as follows. For any TAG, x, let 



160 



140 



120 



100- 



80 



60 



40 



20 







100 120 140 160 180 200 220 240 

 total catch (1000 mt) 



160 180 200 

 total catch (1000 mt) 



Figure 2 



(a) Probability of exceeding the current (1990) cod Gadus 

 morhua fishing mortality and expected value of the potential 

 yield foregone (with 95% confidence hand determined as the 

 2.5th and 97.5th percentiles of the distribution resulting from 

 1000 simulations) as functions of total catch selected for 

 1991. (b) Probability of the 1991 fishing mortality exceeding 

 the 50% rule fishing mortality, and expected value of the yield 

 foregone (with 95% confidence band), as functions of the total 

 catch selected for 1991. 



d(i) 



0, yield associated with ith interval of histogram < x 



1 , yield associated with ith interval of histogram > x. 



Then, 



E (potential yield foregone) = ^ p(i) d(i) (y(i) - x) 



1 = 1 



where E(ll) denotes the expectation operator, the summation is over all intervals of the histogram (Fig. 1), and 



