4  Performance Metrics

4.1 Management Objectives

The IOTC has adopted interim target and limit reference points for albacore: targets of SBMSY and FMSY, and limits of 0.4 SBMSY and 1.4 FMSY(IOTC, 2015). As SBMSY is less than 20% of SB0 in the OMs, the limit reference point corresponds to less than 10% of SB0 (Section 2.3), and therefore performance relative to SB0 is also reported.

Consistent with the tuning objectives used for the earlier albacore MSE work (Mosqueira & Hillary, 2025, 2026), the CMPs are evaluated against the following candidate management objectives:

  1. Status: Maintain the stock in the green quadrant of the Kobe plot (SB > SBMSY and F < FMSY) with at least 60% probability;
  2. Safety: Spawning biomass should not fall below the limit reference point, with at least 90% probability, and fishing mortality should be below the limit reference point with a high probability;
  3. Yield: Maximise catches;
  4. Stability: Changes in the TAC between management periods should be relatively gradual.

The CMPs are tuned for the Base Case OM to maximise the mean annual catch over the years the CMPs are active (2029–2058), at two tuning levels (Section 5.9):

  1. T1: a probability of at least 0.6 of being in the Kobe green quadrant over the last 15 years of the projection period (2044–2058);

  2. T2: the T1 constraint plus a requirement of a probability of at least 0.9 that spawning biomass stays above the LRP (0.4 SBMSY) in every year over 2034–2058.

Table 4.1: The candidate management objectives and the corresponding performance metrics.
Management Objective Performance Metric
Status
Maintain the stock in the Kobe green quadrant with at least 60% probability Probability that SB > SBMSY and F < FMSY over 2044–2058 (tuning constraint, T1 and T2), over 2029–2058, and in each time window
Probability that SB < SBMSY and F > FMSY (Kobe red quadrant)
Mean SB/SBMSY and mean F/FMSY
Minimum SB/SBMSY
Safety
SB should not fall below the LRP, with at least 90% probability, and F should be below the limit with high probability Probability that SB > 0.4 SBMSY in every year over 2034–2058 (tuning constraint, T2)
Probability that SB > 0.4 SBMSY (proportion of simulations and years)
Probability that F < 1.4 FMSY
Yield
Maximise catches Mean annual catch over 2029–2058 (maximised in tuning)
Mean annual catch in each time window
Stability
Change in TAC between management periods should be relatively gradual Mean absolute proportional change in the TAC between management periods
Probability that the TAC change limit is binding

4.2 Performance Metrics

All performance metrics are calculated over the 30 years in which the CMPs set the TAC (2029–2058), unless a different set of years is specified, for each OM and CMP. The following notation is used:

  • \(s = 1, \ldots, S\) indexes the simulations and \(y \in \mathcal{Y}\) the \(N_Y\) years;
  • \(SB_{s,y}\) is the female spawning biomass in the reference season of year \(y\);
  • \(F_{s,y}\) is the annual apical fishing mortality, summed over the quarters of year \(y\), on the same basis as FMSY;
  • \(C_{s,y}\) is the annual catch (total removals of all fleets, in t);
  • \(\text{TAC}_{s,c}\) is the TAC set in management cycle \(c = 1, \ldots, N_C\), where \(\text{TAC}_{s,0}\) is the TAC (catch) in the year before the first management cycle, \(\Delta_{s,c} = \lvert \text{TAC}_{s,c} - \text{TAC}_{s,c-1} \rvert / \text{TAC}_{s,c-1}\) is the relative change in the TAC, and \(\delta_{s,c}\) is the maximum TAC change allowed by the CMP in the direction of the change (increases: 15%; decreases: 15% for T1 and 30% for T2);
  • \(\mathrm{I}(\cdot)\) is 1 if the condition is true and 0 otherwise.

4.2.1 Status

The probability of being in the green quadrant of the Kobe plot:

\[ P_{\text{Kobe}} = \frac{1}{S N_Y} \sum_{s} \sum_{y} \mathrm{I}\left(\frac{SB_{s,y}}{SB_{\text{MSY}}} > 1 \ \text{and} \ \frac{F_{s,y}}{F_{\text{MSY}}} < 1\right) \tag{4.1}\]

calculated for all years the CMPs are active (2029–2058), the tuning period (2044–2058), and each time window (Section 4.3). The other Status metrics are calculated over all years the CMPs are active (2029–2058).

The probability of being in the red quadrant:

\[ P_{\text{Red}} = \frac{1}{S N_Y} \sum_{s} \sum_{y} \mathrm{I}\left(\frac{SB_{s,y}}{SB_{\text{MSY}}} < 1 \ \text{and} \ \frac{F_{s,y}}{F_{\text{MSY}}} > 1\right) \tag{4.2}\]

The mean SB/SBMSY:

\[ \overline{SB/SB_{\text{MSY}}} = \frac{1}{S N_Y} \sum_{s} \sum_{y} \frac{SB_{s,y}}{SB_{\text{MSY}}} \tag{4.3}\]

the mean F/FMSY:

\[ \overline{F/F_{\text{MSY}}} = \frac{1}{S N_Y} \sum_{s} \sum_{y} \frac{F_{s,y}}{F_{\text{MSY}}} \tag{4.4}\]

and the minimum SB/SBMSY over the years the CMPs are active in each simulation, \(\min_{y} SB_{s,y}/SB_{\text{MSY}}\).

4.2.2 Safety

The probability that spawning biomass does not fall below the limit reference point (0.4 SBMSY; IOTC (2015)) in any year; i.e., the proportion of simulations in which spawning biomass is above the LRP in every year over 2034–2058:

\[ P_{\text{Safety}} = \frac{1}{S} \sum_{s} \mathrm{I}\left(\min_{y \in \mathcal{Y}_{\text{S}}} \frac{SB_{s,y}}{SB_{\text{MSY}}} > 0.4\right) \tag{4.5}\]

where \(\mathcal{Y}_{\text{S}}\) are the years 2034–2058. A single year below the LRP is a breach. This is the safety constraint of tuning level T2.

The first 5 years the CMPs are active are excluded, so that the stock can recover from the catches of the initial projection years, which are not controlled by the CMPs (Section 5.9).

The probabilities that spawning biomass is above, and fishing mortality is below, the limit reference points (0.4 SBMSY and 1.4 FMSY), as the proportion of simulations and years, are also reported:

\[ P_{SB_{\text{lim}}} = \frac{1}{S N_Y} \sum_{s} \sum_{y} \mathrm{I}\left(\frac{SB_{s,y}}{SB_{\text{MSY}}} > 0.4\right) \tag{4.6}\]

and

\[ P_{F_{\text{lim}}} = \frac{1}{S N_Y} \sum_{s} \sum_{y} \mathrm{I}\left(\frac{F_{s,y}}{F_{\text{MSY}}} < 1.4\right) \tag{4.7}\]

4.2.3 Yield

The mean annual catch:

\[ \bar{C} = \frac{1}{S N_Y} \sum_{s} \sum_{y} C_{s,y} \tag{4.8}\]

calculated over all years the CMPs are active and each time window.

4.2.4 Stability

The mean absolute proportional change in the TAC between management cycles:

\[ \text{AAV} = \frac{1}{S N_C} \sum_{s} \sum_{c} \Delta_{s,c} \tag{4.9}\]

and the probability that the TAC change limit is binding:

\[ P_{\delta} = \frac{1}{S N_C} \sum_{s} \sum_{c} \mathrm{I}\left(\Delta_{s,c} \geq \delta_{s,c}\right) \tag{4.10}\]

As all CMPs limit TAC increases to 15%, and the T1 CMPs also limit decreases to 15%, the probability that the TAC changes by no more than 15% is not reported. The larger decreases allowed in the T2 CMPs are reflected in the mean TAC change.

4.3 Time Windows

The probability of being in the Kobe green quadrant and the mean catch are also calculated for three time windows:

  1. Short: the first 10 years (2029–2038);
  2. Medium: the second 10 years (2039–2048);
  3. Long: the last 10 years (2049–2058).

4.4 Summary of Performance Metrics

The performance metrics reported for every CMP and OM combination are listed in Table 4.2.

Table 4.2: Performance metrics computed for every CMP and OM combination.
Metric Description
P(Kobe green) 2044–2058 Probability of being in the green quadrant of the Kobe plot (SB > SBMSY and F < FMSY) over 2044–2058, the last 15 projection years (the tuning objective).
P(Kobe green) Probability of being in the green quadrant of the Kobe plot (SB > SBMSY and F < FMSY) over all years the MP is active (2029–2058).
P(Kobe green) (years 1–10) Probability of being in the green quadrant of the Kobe plot over the first 10 years the MP is active (2029–2038).
P(Kobe green) (years 11–20) Probability of being in the green quadrant of the Kobe plot over the middle 10 years the MP is active (2039–2048).
P(Kobe green) (years 21–30) Probability of being in the green quadrant of the Kobe plot over the last 10 years the MP is active (2049–2058).
P(Kobe red) Probability of being in the red quadrant of the Kobe plot (SB < SBMSY and F > FMSY) over all years the MP is active.
Mean SB/SBMSY Mean annual female spawning biomass relative to SBMSY over all years the MP is active.
Mean F/FMSY Mean annual fishing mortality relative to FMSY over all years the MP is active.
Minimum SB/SBMSY Lowest annual SB/SBMSY reached in the years the MP is active.
P(SB > 0.4 SBMSY every year) 2034–2058 Probability that spawning biomass is above the limit reference point (0.4 SBMSY) in every year 2034–2058, i.e. the proportion of simulations that never fall below the limit (the safety tuning constraint of T2). The first 5 years the MP is active are excluded so the stock can recover from the interim catches.
P(SB > 0.4 SBMSY) Probability that spawning biomass is above the limit reference point (0.4 SBMSY) over all years the MP is active (proportion of simulations and years).
P(F < 1.4 FMSY) Probability that fishing mortality is below the limit reference point (1.4 FMSY) over all years the MP is active.
Mean catch Mean annual catch (t) over all years the MP is active (2029–2058).
Mean catch (years 1–10) Mean annual catch (t) over the first 10 years the MP is active (2029–2038).
Mean catch (years 11–20) Mean annual catch (t) over the middle 10 years the MP is active (2039–2048).
Mean catch (years 21–30) Mean annual catch (t) over the last 10 years the MP is active (2049–2058).
Mean TAC change Mean absolute proportional change in the TAC between management cycles.
P(TAC change limited) Probability that the change in the TAC between management cycles is at the maximum allowed by the MP (the TAC change limit is binding).