Abstract.- Monte Carlo simula- 

 tion is used to quantify the uncer- 

 tainty in the results of sequential 

 population analysis and related sta- 

 tistics. Probability density functions 

 describe the measured or perceived 

 uncertainty in the inputs to the 

 assessment model. Monte Carlo sim- 

 ulation is then used to examine the 

 variability in the resulting parameter 

 estimates (stock sizes and fishing 

 mortalities), derived statistics (e.g., 

 Fq i), and in the management regu- 

 lations necessary to achieve various 

 management objectives. We show 

 how relative frequency histograms of 

 the simulation results can be used to 

 describe the risk of not meeting a 

 given management goal as a function 

 of the catch quota selected. We also 

 show how to compute the expected 

 cost, in terms of potential yield fore- 

 gone, associated with picking a con- 

 servative quota. This enables one to 

 balance risks and costs or to allow 

 risk to vary within proscribed limits 

 while keeping the catch quota stable. 

 We illustrate the use of the Monte 

 Carlo approach with examples from 

 two fisheries: North Atlantic sword- 

 fish and northern cod. 



A simple simulation approach to risk 

 and cost analysis, with applications 

 to swordfish and cod fisheries 



Victor R. Restrepo 



University of Miami, Rosenstiel School of Marine and Atmospheric Science 

 4600 Rickenbacker Causeway. Miami, Florida 33149 



John M. Hoenig 



Department of Fisheries and Oceans. Science Branch 



P,0 Box 5667, St John's, Newfoundland AlC 5X1. Canada 



Joseph E. Powers 



Southeast Fisheries Science Center, National Marine Fisheries Service. NOAA 

 75 Virginia Beach Drive. Miami, Florida 33149 



James W. Baird 



Department of Fisheries and Oceans, Science Branch 



PO Box 5667, St John's, Newfoundland AlC 5X1, Canada 



Stephen C. Turner 



Southeast Fisheries Science Center, National Marine Fisheries Service, NOAA 

 75 Virginia Beach Drive, Miami, Florida 33149 



Manuscript accepted 29 July 1992. 

 Fishery Bulletin, U.S. 90:736-748 (1992). 



Fishery managers recognize the 

 dangers of accepting parameter 

 estimates without consideration of 

 the variability inherent in the esti- 

 mates of fish stock status and related 

 parameters. Early strategies for 

 dealing with this were quite simple, 

 such as replacing the estimate of the 

 fishing mortality giving the max- 

 imum yield (Fn,ax) by a more conser- 

 vative value. Sensitivity analyses, in 

 which the effects of various pertur- 

 bations of the inputs are observed, 

 have commonly been employed to ob- 

 tain impressions of the probable 

 bounds on the errors (e.g.. Pope 

 1972, Pope and Garrod 1975). Re- 

 cently, various authors have used the 

 delta method (see Kendall and Stuart 

 1977: 246-248) to obtain analytical 

 expressions or numerical solutions 

 for the variances and covariances of 

 outputs from simple sequential 

 population analyses (SPAs) (Saila et 

 al. 1985, Sampson 1987, Prager and 

 MacCall 1988, Kimura 1989). These 



solutions tend to be complex, only 

 asymptotically valid, and highly 

 model-specific. They have only been 

 worked out for the simplest SPA 

 models and some simple quota- 

 setting procedures (e.g.. Pope 1983). 

 It is possible to measure how well 

 the population estimates correspond 

 to trends in indices of abundance 

 when calibration procedures are ap- 

 plied to sequential population anal- 

 yses. The variance-covariance matrix 

 of the stock sizes estimated directly 

 in the optimization procedure is cal- 

 culated from the inverse Hessian or 

 its approximation (Seber and Wild 

 1989); the variance of any function of 

 these parameters is then approx- 

 imated by the delta method. But 

 estimates of standard errors of 

 population sizes obtained in this way 

 do not reflect all the variability in the 

 inputs and the uncertainty in the 

 model, because they are conditioned 

 on a number of assumptions such 

 as natural mortality being exactly 



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