Pricing Parametric Insurance on Tourist Arrivals
1 Motivation
This paper explores the pricing of a parametric insurance product linked to tourist arrivals. The Maldivian tourism industry is highly vulnerable to exogenous factors, and severe drawdowns in this sector alone can have spillover effects across the broader economy. A parametric insurance product linked to aggregate arrivals could provide quick liquidity to support resort operators, local guesthouses, related businesses (e.g., airlines, travel agents, dive centers) and even the Government during periods of systemic stress.
Note: At its current state, this is just an exploratory study designed as a proof-of-concept for an insurance product. It is not to be considered as an exhaustive or heavily-researched academic or professional paper.
2 Exploratory Data Analysis
To understand the baseline behavior of the market, we first pull monthly tourist arrival data from the Maldives Monetary Authority (MMA) API and apply a natural logarithmic transformation to narrow down the range of the observations. A time series visualization is then generated to illustrate long-term structural patterns inherent to the tourism industry.
Figure 1 illustrates that the industry exhibits a clear, consistent growth trend periodically interrupted by two significant, but short-term structural breaks (the 2004 Tsunami and COVID-19). This indicates that though the market is very strong, it is heavily exposed to sudden tail-risk events.
Next, we analyze the intra-year seasonality profile to understand the cyclical peaks and troughs of tourist arrivals throughout the calendar year.
In Figure 2 it can be seen that the seasonality profile is mostly deterministic, with the highest variance observable during the start of the year. Regardless, peak and off-peak seasons seem to be highly predictable. From a technical standpoint, this does however mean that, any trigger for an insurance payout must be dynamically adjusted each month. Because the volume of arrivals can fluctuate so significantly within a year, a proportional drop in arrivals during peak months would result in a much larger financial loss compared to the same percentage drop in the off-season.
Finally, we calculate a 24-month rolling standard deviation to evaluate historical volatility and check for heteroskedasticity across different cycles.
As shown in Figure 3, barring severe structural breaks, the variance across the time series remains mostly constant. The spikes on the chart correspond directly to isolated black-swan events. This indicates that these extreme shocks must be stripped out before we can accurately model standard market conditions.
3 Time Series Decomposition
To isolate the underlying stochastic noise we must first remove the more predictable, deterministic components of the time series. We do so by applying a robust Seasonal and Trend decomposition using Loess (STL) to separate the data into trend, seasonal, and residual noise components.
Known structural breaks, specifically the massive drop-offs caused by the 2004 Tsunami and the 2020 COVID-19 pandemic, are masked with null values. This prevents these extreme historical outliers from skewing the baseline trend and seasonality.
The decomposition in Figure 4 reveals three very important details.
Trend: Underlying trend is strong and positive, though with recent indications of flattening, suggesting the industry is transitioning from the growth stage into a more mature state.
Seasonality: Amplitude of the seasonality has become notably less variant, indicating that tourism activity is now more evenly distributed throughout the year.
Noise: Residual noise is fat-tailed and negatively skewed, even after adjusting for structural breaks. In practical terms, this means that unexpected shocks happen more frequently than standard models may predict, and these surprises are mostly drop-offs in demand (i.e., surges are rarer).
| Test Method | Test Statistic | P-Value |
|---|---|---|
| Shapiro-Wilk Normality Test | 0.8295 | < 0.0001 |
To verify the non-normality of the residuals, a Shapiro-Wilk test was performed on the isolated noise. Yielding a test statistic of 0.8295 and a p-value of < 0.0001, the test firmly rejects the null hypothesis of a normal distribution (Table 1).
Furthermore, when comparing the empirical distribution of the residual noise against a fitted Normal Distribution, both the histogram (Figure 5) and the Q-Q plot (Figure 6) demonstrate a poor fit. Specifically, the Q-Q plot highlights a pronounced deviation in the left tail, confirming that extreme negative shocks occur far more frequently than a standard Gaussian model predicts.
4 Trend and Seasonality Model
The trend and seasonality components isolated in Section 3 are deterministically projected forward over a 24-month horizon to establish a central baseline forecast such that:
\[\hat{Y}_{t+j}=\hat{T}_{t+j}+\hat{S}_{t+j}\]
Where \(\hat{Y}_{t+j}\) is the baseline forecast at future horizon \(j\), and \(\hat{T}\) and \(\hat{S}\) represent the trend and seasonality expectations, respectively.
Trend: Seasonally adjusted trend is projected using a Random Walk w/ Drift. Mathematically, the trend projection at future horizon \(j\) from the final historical observation \(t=0\) is defined as: \[ \hat{T}_{t+j} = T_t + jc \]
where \(c\) represents the constant drift parameter corresponding to the average historical period-over-period change.
Seasonality: The isolated seasonal component is projected forward using a Seasonal Naïve Method. This assumes the intra-year monthly profile remains constant such that: \[ \hat{S}_{t+j} = S_{t+j-m} \] where \(m = 12\) represents the annual monthly cycle.
5 Noise Model
As discussed in Section 3, the residual noise of the tourist arrival time series is fundamentally asymmetric and fat-tailed. To mathematically accommodate this, we partition the noise into a central body and extreme tails at the 7.5th and 92.5th percentiles.
The empirical body is modeled using a Skew-t distribution, which explicitly captures the negative skew identified in the previous section. Extreme Value Theory (EVT) is then applied to the boundaries. Specifically, two separate Generalized Pareto Distributions (GPD) are fitted to the tail exceedances on either side of the body. This aims to address the fat-tailed nature of the data, allowing the model to accurately price the higher-than-normal probability of severe, business-cycle related drawdowns in tourist arrivals.
As such, the cumulative distribution function (CDF) of the simulated stochastic noise \(F(x)\) is defined as a piecewise function:
\[ F(x)=\begin{cases} p_{L}\left(1+\xi_{L}\dfrac{\vert{}x\vert{}-u_{L}}{\sigma_{L}}\right)^{-\frac{1}{\xi_{L}}} & \text{if } x \le -u_{L} \\ p_{L} + (1 - p_{L} - p_{R})\left(\dfrac{F_{\text{Skew-}t}(x \mid \xi, \omega, \alpha, \nu) - F_{L}}{F_{R} - F_{L}}\right) & \text{if } -u_{L} < x \le u_{R} \\ 1 - p_{R}\left(1+\xi_{R}\dfrac{x-u_{R}}{\sigma_{R}}\right)^{-\frac{1}{\xi_{R}}} & \text{if } x > u_{R} \end{cases} \]
Where \(u_L\) and \(u_R\) are the absolute threshold boundaries for the left and right tails, \(p\) represents the empirical tail probability, while \(\sigma\) and \(\xi\) dictate the scale and shape of the GPD fitted to the respective tail exceedances. The Maximum Likelihood Estimates (MLE) for these parameters are detailed in Table 2.
| Distribution | Parameter | Estimate |
|---|---|---|
| Skew-t (Body) | \(\xi\) | 0.0160 |
| Skew-t (Body) | \(\omega\) | 0.0407 |
| Skew-t (Body) | \(\alpha\) | -0.4478 |
| Skew-t (Body) | \(\nu\) | 1.9681 |
| GPD (Left Tail) | \(\sigma_L\) | 0.1072 |
| GPD (Left Tail) | \(\xi_L\) | 0.0507 |
| GPD (Right Tail) | \(\sigma_R\) | 0.0840 |
| GPD (Right Tail) | \(\xi_R\) | -0.0550 |
6 Monte Carlo Simulation
To price the insurance product, we generate 10,000 sample paths of possible future states over a 24-month period using a Monte Carlo simulation. The simulated stochastic noise, drawn from the fitted Skew-t + GPD model in Section 5, is then added to the deterministic baseline forecast in Section 4 to create a distribution of future scenarios.
The “log-scale” simulation paths are then exponentiated to revert them back to “absolute” tourist arrival numbers, before calculating the simulated mean and 95% confidence intervals.
7 Model Diagnostics
To validate the reliability of the stochastic model as a whole, we first compare the simulated noise against the empirical historical noise.
The histogram (Figure 7) and Q-Q plot (Figure 8) demonstrate that the combined Skew-t + GPD model is a good fit for the empirical data. Crucially, the tight fit along the lower quantiles of the Q-Q plot confirms that the model accurately captures left-tail events. This ensures that significant business-cycle drawdowns are properly captured in the simulations.
Finally, the fan chart (Figure 9) visualizes a 24-month forecast. It preserves the previously observed trend and seasonality while accounting for wide and asymmetric uncertainty bands, correctly reflecting the bias toward significant, and unexpected drop-offs in tourist arrivals.
8 Product Case Study
Simulations in Section 6 are then used to price a hypothetical parametric insurance product. The contract comes with a fixed notional payout (\(N\)) per unit deficit whenever actual arrivals in a specific month (\(Y_t\)) fall below a predetermined strike threshold (\({K}_{t} = k \cdot \hat{Y}_{t}\)). This is defined as a percentage (\(k\)) of the expected baseline forecast (\(\hat{Y}_{t}\)). For a given simulated path \(i\) over a contract duration of \(T\) months, the present value of the payout (\(PV_i\)) is calculated using a continuously compounded zero-coupon yield curve (\(r_t\)) bootstrapped from US Treasuries:
\[ PV_i = \sum_{t=1}^{T} N \cdot \max({K}_{t} - Y_{i,t}, 0) \cdot e^{-r_t \frac{t}{12}} \]
The Expected Loss (\(EL\)) of the contract is simply the average of these discounted payouts across all 10,000 simulated paths:
\[ EL = \frac{1}{10,000} \sum_{i=1}^{10,000} PV_i \]
However, charging only the \(EL\) as premium could leave an insurer insolvent during a more severe market shock. So, to ensure solvency, they must hold some capital reserves to cover tail-risk scenarios as well. One such way to determine the capital requirement is using the Conditional Tail Expectation (CTE), which calculates the average financial loss within the worst \((1 - \alpha)\) percentile of simulated outcomes (e.g., the worst 5% of paths).
In this case, the initial capital strain is the difference between this tail expectation and the expected loss: \(C_0 = \max(\text{CTE}_\alpha - EL, 0)\). Given that the shareholders charge a specified corporate hurdle rate \(h\) on their capital contributions, the Risk Margin (\(RM\)) represents the opportunity cost of holding these reserves instead of funding other endeavors:
\[ RM = \sum_{t=1}^{T} \left( C_0 \cdot \frac{T - t + 1}{T} \right) \cdot \frac{h-r_t}{12} \cdot e^{-r_t \frac{t}{12}} \]
Our assumption is that the capital can be released linearly over the duration of the contract. As such, the up-front Technical Premium (\(TP\)) charged to the policyholder is:
\[ \text{TP} = EL + RM \]
For instance, consider a standard policy: a 12-month contract duration with an 85% strike threshold, priced by an insurer wanting to stay solvent at a 95% confidence level and with a hurdle rate of 12%. Under these parameters, the model calculates a technical premium of USD 4.32 million.
We then price the parametric insurance product at various configurations to evaluate the fair premium under different pricing assumptions.
Our first matrix (Figure 10) evaluates how the up-front cost of coverage would change based on varying contract terms:
A longer contract duration naturally increases the probability of a severe drawdown materializing while the policy remains active. More time exposed to risk equates to a higher EL, which then directly inflates the premium charged to cover for those losses.
Similarly, the strike threshold dictates how easily a payout is triggered. A higher strike level (e.g., 95%) means the policy pays out for even the relatively minor (and much more frequent) dips in tourist arrivals, leading to the exact same outcome as discussed above.
Next, we evaluate how insurers in different capital regimes may price such a product (Figure 11):
The confidence level reflects how conservative the reserving strategy of an insurer truly is. Targeting a higher confidence level (e.g., 99% CTE) means the insurer is preparing to survive an extreme worst-case scenario (e.g., a 1-in-100 year event), which forces them to hold a much larger capital base to guarantee solvency. The hurdle rate then represents the return demanded by shareholders for tying up that capital. Both of these drive up the RM as they increase.
Ultimately, a highly risk-averse insurer with ability to earn higher returns elsewhere will charge a substantially higher premium, even if the underlying EL of the policy remains exactly the same.
9 Findings
From a commercial standpoint, the pricing sensitivities outlined in Section 8 highlight some practical hurdles for parametric products in the tourism sector.
The inherent tail risk of the industry dictates high technical premiums, which may push the cost of coverage beyond what small, independent operators (e.g., guesthouses) can afford. Consequently, actual market demand would likely be restricted to heavily capitalized multinational hospitality conglomerates, or the Government itself, utilizing the product as a macro-level hedge rather than a micro-level business interruption policy. That being said, it could be argued that notional coverage amounts could be adjusted downward to provide partial relief at a more accessible price point.
Furthermore, local or regional insurers are unlikely to possess the balance sheet required to underwrite this level of highly correlated, localized risk — especially if coverage were expanded beyond cyclical drawdowns to include black-swan events. While securing international reinsurance arrangements could mitigate these capital shortfalls, structuring those specific risk-transfer mechanisms falls outside the scope of this proof-of-concept and remains an area for future exploration.
10 Limitations
Even though the model discussed in this paper provides a quantitative methodology for pricing parametric risk, its real-world validity is subject to several practical limitations:
Moral Hazard: Parametric products require an independent, tamper-proof index. For this reason, such products are usually structured around physical metrics (e.g., temperature). Payouts in this model rely entirely on aggregated government reporting, assuming the data is immune to manipulation or retroactive adjustments.
Basis Risk: Payouts are triggered by national arrivals. A single business (e.g., a specific resort) could suffer severe financial losses from a more localized disruption without triggering an actual national-level payout. On the other hand, a business that experiences no losses could also receive a windfall payout simply because the broader national market performed poorly.
Exclusion of Exogenous Factors: Simulation relies entirely on the historical variance of the arrivals data. It omits external macroeconomic covariates (e.g., inflation, exchange rates, fuel prices) that could be useful leading indicators of the index.
Exclusion of Black Swan Events: Historical structural breaks were masked during decomposition, and consequently the GPD parameters are fitted exclusively to noise from the usual business-cycle. As such, the model accounts for severe economic volatility, but assumes the probability of future, unmodeled black-swan events (e.g., global pandemics or tsunamis) are close to nil.
Lack of Out-of-Sample Validation: Model diagnostics and parameter estimates are evaluated exclusively on in-sample historical data. As a result, the predictive accuracy of the projected baseline and the stability of the simulated tail-risk probabilities have not been verified against a held-out testing set or through historical backtesting.
Future research could expand on this proof-of-concept by conducting rigorous out-of-sample backtesting to quantify real-world forecast error, alongside integrating Generalized Autoregressive Conditional Heteroskedasticity (GARCH) and jump-diffusion processes to better capture black-swan events, and Seasonal Autoregressive Integrated Moving Average with eXogenous variables (SARIMAX) to account for leading macroeconomic indicators.