Abstract:
The amount of risk in weather derivatives is calculated properly by taking into account a correlation between weather observation sites and a correlation between meteorological elements. A weather prediction method by creating a weather scenario from historical weather data includes a first step of estimating parameters of a weather time-series model based on historical weather data including past plural meteorological elements, a meteorological element correlation between sites, and a correlation between meteorological elements, and a second step of converting random numbers into meteorological elements based on the estimated parameters. The second step is executed a given number of times to create a plurality of weather scenarios.

Description:
CLAIM OF PRIORITY  
       [0001]     The present application claims priority from Japanese application P20041-0573143 filed on Mar. 2, 2003, the content of which is hereby incorporated by reference into this application.  
       BACKGROUND OF THE INVENTION  
       [0002]     This invention relates to a method of estimating weather risks, typically, risks in weather derivatives.  
         [0003]     Weather conditions such as temperature, precipitation, and the amount of snow cover significantly affect business activities of a company. For instance, a heat wave in summer boosts sales for air conditioning manufacturers and electric power companies whereas it dents profit of department stores and railway companies by raising cooling cost. To give another example, extraordinarily low precipitation brings more customers or visitors to travel agencies and theme parks whereas it is adverse to electric power companies since their hydraulic power units cannot run efficiently and the cost of alternative power generation increases. Fluctuations in sales or profit due to such weather conditions are called weather risks.  
         [0004]     In recent years, financial derivatives called weather derivatives are attracting attention as an instrument to reduce weather risks and ensure a steady profit. A weather derivative is a contract between a business entity subject to weather risks and a property insurance company or the like in which the business entity pays the insurance company contract money first and receives a compensation payment if certain set weather conditions are met in future times. Several methods have been proposed to calculate contract money of a weather derivative (refer to JP 2001-222605 A and JP 2003-122918 A).  
         [0005]     Proposed as the basis for calculating contract money of a weather derivative are a weather forecast based on a multi-site temperature model (refer to “Multivariate long memory modeling of daily surface air temperatures and the valuation of weather derivative portfolios”, written by Rodrigo Caballero et al., intermetURL: http://stephenjewson.com/articles/), a weather forecast based on a multi-site precipitation model (refer to “Multisite generalization of a daily stochastic precipitation generation model”, D. S. Wilks, Journal of Hydrology, 1998, 210, pp. 178-191), and a weather forecast based on a single-site precipitation-temperature correlation model (see, for example, Richardson, C. W., “Stochastic simulation of daily precipitation, temperature, and solar radiation”, Water Resources Research, 17, pp. 182-190).  
         [0006]     A provider of a weather derivative (a property insurance company, a trade firm, or a bank) holds a portfolio of the contract, and needs to calculate the amount of risk and analyze factors about the portfolio. Conventionally, those needs have been taken care of in the following manner. First, a time-series model built for each site or for each meteorological element is used to create a weather scenario. The 99% VaR (Value at Risk) is calculated from the weather scenario created and the sum of the 99% VaR is evaluated as the total amount of risk ( FIG. 12 ).  
       SUMMARY OF THE INVENTION  
       [0007]     The above-mentioned method, which calculates VaR for each individual weather derivative contract, overestimates the total amount of risk since there are significant correlations between weather observation sites and between meteorological elements.  
         [0008]     It is therefore an object of this invention to estimate the amount of risk in weather derivatives properly by taking into account a correlation between weather observation sites and a correlation between meteorological elements.  
         [0009]     According to a embodiment of this invention, there is provided a method of predicting weather on a computer by creating a weather scenario from historical weather data, including: a first step of estimating parameters of a weather time-series model based on historical weather data including past plural meteorological elements, a meteorological element correlation between sites, and a correlation between meteorological elements; and a second step of converting random numbers into meteorological elements based on the estimated parameters, wherein the method creates a plurality of weather scenarios by executing the second step predetermined times.  
         [0010]     According to this invention, the amount of risk in weather derivatives can be estimated properly. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0011]      FIG. 1  is a block diagram showing the structure of a prediction device according to an embodiment of this invention.  
         [0012]      FIG. 2  is a flow chart of a weather scenario creating processing according to the embodiment of this invention.  
         [0013]      FIG. 3  is an explanatory diagram of weather scenario creating conditions according to the embodiment of this invention.  
         [0014]      FIG. 4  is an explanatory diagram of historical weather data according to the embodiment of this invention.  
         [0015]      FIG. 5  is an explanatory diagram of a weather scenario according to the embodiment of this invention.  
         [0016]      FIG. 6  is a flow chart showing details of parameter estimating processing according to the embodiment of this invention.  
         [0017]      FIG. 7  is a conceptual diagram of processing of estimating independent parameters, which do not have a correlation between sites, according to the embodiment of this invention.  
         [0018]      FIG. 8  is a conceptual diagram of processing of estimating parameters that indicate the magnitude of correlation between sites according to the embodiment of this invention.  
         [0019]      FIG. 9  is a flow chart showing details of weather scenario creation simulation processing according to the embodiment of this invention.  
         [0020]      FIG. 10  is a flow chart showing details of d-th day precipitation creating processing according to the embodiment of this invention.  
         [0021]      FIG. 11  is a flow chart showing details of d-th day temperature creating processing which utilizes the precipitation according to the embodiment of this invention.  
         [0022]      FIG. 12  is an explanatory diagram of VaR calculation. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0023]     An embodiment of this invention will be described below with reference to the accompanying drawings.  
         [0024]      FIG. 1  is a block diagram showing the structure of a prediction device according to this embodiment.  
         [0025]     The prediction device of this embodiment is composed of a display device  111 , a CPU  112 , a ROM  113 , input devices (a keyboard  114  and a mouse  115 ), a RAM  116 , an external storage device (hard disk)  117 , and a communication bus  118 , which connects the above components to one another.  
         [0026]     The external storage device  117  stores an operating system  1171 , a weather scenario creating program  1172 , and a portfolio calculation program. The CPU  112  calls up these programs to execute various processing. The external storage device  117  also stores historical weather data  1102  and a weather scenario  1103 .  
         [0027]     In the prediction device of this embodiment, the weather scenario creating program  1172  creates the weather scenario  1103  from scenario creating conditions  1101 , which are inputted through the input devices  114  and  115 , referring to the historical weather data  1102 .  
         [0028]      FIG. 2  is a flow chart of weather scenario creating processing according to this embodiment. The weather scenario creating processing is executed by the weather scenario creating program  1172 .  
         [0029]     The weather scenario creating program  1172  first controls the display device  111  to display a message that prompts a user to input scenario creating conditions. Then the weather scenario creating program  1172  obtains the scenario creating conditions  1101  inputted by the user through the input devices  114  and  115  (Step S 11 ). The weather scenario creating program  1172  then obtains the historical weather data  1102  necessary to create the weather scenario  1103  that accommodates the conditions inputted by the user (Step S 12 ).  
         [0030]     Thereafter, the weather scenario creating program  1172  estimates parameters of a time-series model which reflects a correlation between sites and a correlation between meteorological elements (Step S 13 ), and creates as many weather scenarios as specified in the scenario creating conditions  1101  (Step S 14 ).  
         [0031]     The weather scenario  1103  created by weather simulation is outputted to the external storage device  117  (Step S 15 ).  
         [0032]      FIG. 3  is an explanatory diagram of the weather scenario creating conditions  1101 .  
         [0033]     The weather scenario creating conditions  1101  inputted by a user through the input devices  114  and  115  include a scenario creation number  21 , a scenario creation period  22 , and a scenario creation site  23 .  
         [0034]      FIG. 4  is an explanatory diagram of the historical weather data  1102 .  
         [0035]     The historical weather data  1102  is a record of weather data (temperature and precipitation) collected every day on plural sites for many years. Alternatively, the historical data  1102  may be weather data collected every hour instead of every day, or may include snowfall amount, wind velocity, wind direction and the like in addition to temperature and precipitation.  
         [0036]     Although  FIG. 4  shows only historical weather data of the period and site that correspond to the inputted weather scenario creating conditions  1101  ( FIG. 3 ), the historical weather data  1102  can be a record of weather data of more periods and sites which is extracted as the weather scenario creating conditions  1101  demand.  
         [0037]      FIG. 5  is an explanatory diagram of the weather scenario  1103 .  
         [0038]     The weather scenario  1103  contains 10,000 weather scenarios (#0001 to #10000) created for the period (June 1 to July 20) and site (Osaka, Nagoya, and Tokyo) that correspond to the weather scenario creating conditions  1101  ( FIG. 3 ).  
         [0039]      FIG. 6  is a flow chart showing details of the parameter estimating processing (Step S 103  of  FIG. 2 ).  
         [0040]     Independent parameters, which do not have a correlation between sites, are estimated first (Step S 51 ). Specifically, parameters pi, μi, mi(0)(d), σi(0)(d), mi(1)(d), and σi( 1 )(d) are estimated. These parameters are used in formulas (1) to (7), which are described later.  
         [0041]     Next estimated are parameters that indicate the magnitude of the correlation between sites (Step S 52 ). Specifically, parameters ρ0(i, j), ρ1(i, j), φ0(i, j), and φ1(i, j) are estimated. These parameters are used in formulas (8) to (13), which are described later.  
         [0042]      FIG. 7  is a conceptual diagram of the processing of estimating independent parameters which have no correlation between sites (Step S 51  of  FIG. 6 ).  
         [0043]     The processing of estimating independent parameters which have no correlation between sites is made up of three parts. One is precipitation parameter estimating processing  61 . The precipitation parameter estimating processing  61  is for estimating, from the precipitation historical data, parameters that characterize a fluctuation in precipitation on a site for which a weather scenario is created. Specifically, the parameters pi and μi which are used in formulas (1) and (3) described below are estimated.  
         [0044]     The other two of the three parts are fine day temperature parameter estimating processing  621  and rainy day temperature parameter estimating processing  622  which are collectively referred to as temperature parameter estimating processing  62 . The temperature parameter estimating processing  62  is for estimating, from the temperature historical data, parameters that characterize a fluctuation in temperature of a site for which a weather scenario is created. The use of different parameters for a fine day and a rainy day gives more accurate depiction of a temperature fluctuation since the temperature movement in clear weather and the temperature movement in wet weather differ from each other. Specifically, the parameters mi(0)(d), σi(0)(d), mi(1)(d), and σi(1)(d) which are used in formulas (2), (4) and (5) described below are estimated.  
         [0045]      FIG. 8  is a conceptual diagram of the processing of estimating parameters that indicate the magnitude of correlation between sites (Step S 52  of  FIG. 6 ).  
         [0046]     The act of estimating parameters that indicate the magnitude of correlation between sites is made up of two parts, inter-site precipitation correlation parameter estimating processing  71  and inter-site temperature correlation parameter estimating processing  72 .  
         [0047]     Estimated in the inter-site precipitation correlation parameter estimating processing  71  are a parameter that characterizes the magnitude of correlation between the precipitation of a site i and the precipitation of a site j on the same day and a parameter that characterizes the magnitude of correlation between the precipitation of the site i and the precipitation of the site j on the previous day. Specifically, the parameters ρ0(i, j) and ρ1(i, j) which are used in formulas (8), (10), and (12) described below are estimated. The inter-site temperature correlation parameter estimating processing  72  calculates estimation of a parameter that characterizes the magnitude of correlation between the temperature of the site i and the temperature of the site j on the same day and a parameter that characterizes the magnitude of correlation between the temperature of the site i and the temperature of the site j on the previous day. Specifically, the parameters φ0(i, j) and φ1(i, j) which are used in formulas (9), (11), and (13) described below are estimated.  
         [0048]      FIG. 9  is a flow chart showing details of the processing of creating a weather scenario by simulation (Step S 14  of  FIG. 2 ).  
         [0049]     First, counters n and d are set to the initial value “1” (Step S 81 ). The counter n is used to create a necessary number (a scenario creation number N) of weather scenarios. The counter d is used to create a weather scenario of a necessary period (D days).  
         [0050]     The next processing is for creating an n-th weather scenario (Step S 82 ). In the n-th weather scenario creating processing (Step S 82 ), the precipitation on a d-th day is created first (Step S 821 ) (if d=1, it is the first day and the weather scenario creating processing is conducted for the first time). Then the temperature on the d-th day is created with the use of precipitation correlation random numbers (Xi(d)) which have been used to create the precipitation (Step S 822 ).  
         [0051]     Then whether the counter d was equal to the necessary period D is judged (Step S 823 ).  
         [0052]     When d=D, it is judged that a weather scenario for the necessary period (for D days) has been completed and the processing goes to Step S 83 . On the other hand, when d≠D, it means that the weather scenario being created has not yet covered all of the necessary period (D days) specified. In this case, “1” is added to the counter d through an update and the process returns to Step S 821  to create the precipitation for the (d+1)-th day.  
         [0053]     In Step S 83 , the created weather scenario for D days is stored. Then whether the counter n was equal to the scenario creation number N is judged (Step S 84 ).  
         [0054]     When n=N, it is judged that as many weather scenarios as necessary, namely, N (scenario creation number) weather scenarios, have been created and the processing goes to Step S 15 . On the other hand, when n≠N, the count of weather scenarios created up to this point falls short of the necessary scenario number N. This calls for an update in which the counter d is set to “1” and “1” is added to the counter n. The process then returns to Step S 82  to create the (n+1)-th weather scenario.  
         [0055]      FIG. 10  is a flow chart showing details of the n-th day precipitation creating processing (Step S 821  of  FIG. 9 ).  
         [0056]     The first step of the n-th day precipitation creating processing is to create as many independent random numbers as the number of sites (Step S 91 ). The independent random numbers are converted into correlation random numbers which reflect the inter-site precipitation correlation (Step S 92 ). The correlation random numbers are converted into precipitation for the respective sites (Step S 93 ).  
         [0057]     Specifically, stochastic variables ui are created which are independent of one another and which conform to the standard normal distribution (Step S 91 ). Then the precipitation correlation random numbers Xi(d) are obtained from the constant ρ0(i, j), which indicates the magnitude of correlation between the precipitation of the site i and the precipitation of the site j on the same day, and the constant ρ1(i, j), which indicates the magnitude of correlation between the precipitation of the site i and the precipitation of the site j on the previous day, with the use of formulas (12), (10), (8), and (6) (Step S 92 ). Then formula (1) is used to convert the precipitation correlation random numbers Xi(d) into precipitation ri(d) (Step S 93 ).  
         [0058]      FIG. 11  is a flow chart showing details of the processing of creating the temperature of the d-th day from the precipitation (Step S 822  of  FIG. 9 ).  
         [0059]     The first step of the d-th day temperature creating processing is to create as many independent random numbers as the number of sites (Step S 101 ). The independent random numbers are converted into correlation random numbers which reflect the inter-site temperature correlation (Step S 102 ).  
         [0060]     The correlation random numbers are converted into temperature for the respective sites based on the precipitation (Step S 103 ). Here the precipitation is used by judging whether the precipitation is “0” or not (Step S 1031 ). When the precipitation is “0”, it is judged that the day in question is clear and the correlation random numbers are converted into a fine day temperature (Step S 1032 ). When the precipitation is not “0”, on the other hand, it is judged that the day in question is rainy and the correlation random numbers are converted into a rainy day temperature (Step S 1033 ).  
         [0061]     Specifically, stochastic variables vi are created which are independent of one another and which conform to the standard normal distribution (Step S 101 ). Then inter-site temperature correlation random numbers Yi(d) are obtained from the constant φ0(i, j), which indicates the magnitude of correlation between the temperature of the site i and the temperature of the site j on the same day, and the constant φ1(i, j), which indicates the magnitude of correlation between the temperature of the site i and the temperature of the site j on the previous day, with the use of formulas (13), (11), (9), and (7) (Step S 102 ). Then the precipitation correlation random numbers Xi(d) are used to judge whether the day in question is a fine day or a rainy day and to determine which conversion function Φ is to be employed in conversion to temperature (Step S 1031 ). With the conversion function Φ specified and formula (2), the inter-site temperature correlation random numbers Yi(d) are converted into a temperature ti(d) (Steps S 1032  and S 1033 ).  
         [0062]     A specific calculation method of the weather scenario creating processing described above will be explained next by presenting numerical formulas.  
         [0063]     The precipitation and temperature on the d-th day on the site i are given as ri(d) mm and ti(d)° C., respectively. In this model, ri(d) is expressed by formula (1) and ti(d) is expressed by formula (2).  
                 r   i     ⁡     (   d   )       =     {         0           for   ⁢           ⁢       X   i     ⁡     (   d   )         ≤     p   i                   Ψ     -   1       ⁡     (         X   i     ⁡     (   d   )       ,     p   i       )               for   ⁢           ⁢       X   i     ⁡     (   d   )         ≥     p   i                       (   1   )                   t   i     ⁡     (   d   )       =     {             Φ     (   0   )       -   1       ⁡     (       Y   i     ⁡     (   d   )       )               for   ⁢           ⁢       X   i     ⁡     (   d   )         ≤     p   i                   Φ     (   1   )       -   1       ⁡     (       Y   i     ⁡     (   d   )       )               for   ⁢           ⁢       X   i     ⁡     (   d   )         ≥     p   i                       (   2   )             
 
         [0064]     In formulas (1) and (2), the symbol pi represents the probability of the weather being fine on the site i. Ψ−1 is a function for converting stochastic variables Xi(d) into precipitation and Φ−1 is a function for converting stochastic variables Yi(d) into temperature. Xi(d) and Yi(d) are stochastic variables ranging from 0 to 1.  
         [0065]     Although there are various other functions that represent a temperature distribution and a precipitation distribution, the explanation below employs the inverse function of an exponential function for the precipitation and the inverse function of a normal distribution function for the temperature for the sake of simplification. A modification in function form does not affect in any way application of this invention.  
                 X   i     ⁡     (   d   )       =       (     1   -     p   i       )     +       p   i     ⁢           ⁢       ∫   0   Ψ     ⁢       1     μ   i       ⁢           ⁢     exp   ⁡     (     -     ξ     μ   i         )       ⁢     ⅆ   ξ                     (   3   )                   Y   i     ⁡     (   d   )       =       ∫     -   ∞       Φ     (   0   )         ⁢       1         2   ⁢   π       ⁢       σ   i     (   0   )       ⁡     (   d   )           ⁢           ⁢   exp   ⁢     {     -         (     ξ   -       m   i     (   0   )       ⁡     (   d   )         )     2       2   ⁢       (       σ   i     (   0   )       ⁡     (   d   )       )     2           }     ⁢     ⅆ   ξ                 (   4   )                   Y   i     ⁡     (   d   )       =       ∫     -   ∞       Φ     (   1   )         ⁢       1         2   ⁢   π       ⁢       σ   i     (   1   )       ⁡     (   d   )           ⁢           ⁢   exp   ⁢     {     -         (     ξ   -       m   i     (   1   )       ⁡     (   d   )         )     2       2   ⁢       (       σ   i     (   1   )       ⁡     (   d   )       )     2           }     ⁢     ⅆ   ξ                 (   5   )             
 
         [0066]     In the formulas, μi represents the average precipitation in wet weather on the site i, the symbol mi(0)(d) represents the mean value of the temperature when the d-th day is a fine day, σi(0)(d) represents the standard deviation of the temperature when the d-th day is a fine day, mi(1)(d) represents the mean value of the temperature when the d-th day is a rainy day, and σi(1)(d) represents the standard deviation of the temperature when the d-th day is a rainy day.  
         [0067]     In the formulas, Xi(d) and Yi(d) can be expressed as functions of stochastic variables xi(d) and yi(d), which conform to the standard normal distribution, by formulas (6) and (7), respectively.  
                 X   i     ⁡     (   d   )       =       ∫     -   ∞         x   i     ⁡     (   d   )         ⁢       1       2   ⁢           ⁢   π         ⁢           ⁢   exp   ⁢     {     -       ξ   2     2       }     ⁢           ⁢     ⅆ   ξ                 (   6   )                   Y   i     ⁡     (   d   )       =       ∫     -   ∞         y   i     ⁡     (   d   )         ⁢       1       2   ⁢           ⁢   π         ⁢           ⁢   exp   ⁢     {     -       ξ   2     2       }     ⁢     ⅆ   ξ                 (   7   )             
 
         [0068]     The stochastic variables xi(d) and yi(d) are expressed by the following multivariable autoregressive formulas:  
               (             x   1     ⁡     (     d   +   1     )               ⋮               x   m     ⁡     (     d   +   1     )             )     =       A   ⁡     (             x   1     ⁡     (   d   )               ⋮               x   m     ⁡     (   d   )             )       +     B   ⁡     (             u   1     ⁡     (   d   )               ⋮               u   m     ⁡     (   d   )             )                 (   8   )                 (             y   1     ⁡     (     d   +   1     )               ⋮               y   m     ⁡     (     d   +   1     )             )     =       C   ⁡     (             y   1     ⁡     (   d   )               ⋮               y   m     ⁡     (   d   )             )       +     D   ⁡     (             v   1     ⁡     (   d   )               ⋮               v   m     ⁡     (   d   )             )                 (   9   )             
 
         [0069]     In the expressions, ui(d) and vi(d) are each stochastic variables which are independent of one another and which conform to the standard normal distribution. Constant matrices A, B, C, and D, which take into account the inter-site correlation, are defined as follows: 
 
 A=G   1   G   0   −1  
 
 BB   t   =G   0   −G   1   G   0   −1   G   1   t   (10) 
 
         [0070]     Expression 11 
 
 C=H   1   H   0   −1  
 
 DD   t   =H   0   −H   1   H   0   −1   H   1   t   (11) 
 
         [0071]     G0, G1, H0, and H1 in formulas (10) and (11) are defined as follows:  
                 G   0     =     (             ρ   0     ⁡     (     1   ,   1     )           ⋯           ρ   0     ⁡     (     1   ,   M     )               ⋮       ⋰       ⋮               ρ   0     ⁡     (     M   ,   1     )           ⋯           ρ   0     ⁡     (     M   ,   M     )             )       ,           (   12   )                 G   1     =     (             ρ   1     ⁡     (     1   ,   1     )           ⋯           ρ   1     ⁡     (     1   ,   M     )               ⋮       ⋰       ⋮               ρ   1     ⁡     (     M   ,   1     )           ⋯           ρ   1     ⁡     (     M   ,   M     )             )                                 H   0     =     (             φ   0     ⁡     (     1   ,   1     )           ⋯           φ   0     ⁡     (     1   ,   M     )               ⋮       ⋰       ⋮               φ   0     ⁡     (     M   ,   1     )           ⋯           φ   0     ⁡     (     M   ,   M     )             )       ,           (   13   )                 H   1     =     (             φ   1     ⁡     (     1   ,   1     )           ⋯           φ   1     ⁡     (     1   ,   M     )               ⋮       ⋰       ⋮               φ   1     ⁡     (     M   ,   1     )           ⋯           φ   1     ⁡     (     M   ,   M     )             )                           
        wherein G0 and G1 are matrices that represent the inter-site precipitation correlation. In the formulas, ρ0(i, j) is a constant that indicates the magnitude of correlation between the precipitation of the site i and the precipitation of the site j on the same day. And ρ1(i, j) is a constant that indicates the magnitude of correlation between the precipitation of the site i on one day and the precipitation of the site j on the previous day. Also, H0 and H1 are matrices that represent the inter-site temperature correlation. In the formulas, φ0(i, j) is a constant that indicates the magnitude of correlation between the temperature of the site i and the temperature of the site j on the same day. And φ1(i, j) is a constant that indicates the magnitude of correlation between the temperature of the site i on one day and the temperature of the site j on the previous day.        
 
         [0073]     As described above, weather model of this embodiment has the independent parameters pi, μi, mi(0)(d), σi(0)(d), mi(1)(d), and σi(1)(d) which do not have a correlation between sites, and the parameters ρ0(i, j), ρ1(i, j), φ0(i, j), and φ1(i, j) which indicate the magnitude of correlation between sites. These parameters are statistically estimated from the historical weather data.  
         [0074]     Described next is the procedure of creating a weather scenario using the calculation method explained above.  
         [0075]     First, random numbers ui(d) (i=1, . . . , D) and vi(d) (i=1, . . . , D) are created which are independent of one another and which conform to the standard normal distribution. The random numbers ui(d) and vi(d) are converted into xi(d) and yi(d) by formulas (8) and (9), respectively. Then xi(d) and yi(d) are converted into Xi(d) and Yi(d) by formulas (6) and (7), respectively. Lastly, formula (1) is used to convert Xi(d) into precipitation and formula (2) is used to convert Yi(d) to temperature. The conversion to temperature also uses Xi(d).  
         [0076]     As described above, in this embodiment, many (M) random numbers ui as the number of sites for which weather scenarios are to be created. The created random numbers ui are converted to the random numbers Xi(d) which reflect the inter-site precipitation correlation. Then the converted random numbers Xi(d) are converted to the precipitation ri(d). Many (M) random numbers vi as the number of sites for which weather scenarios are to be created. The created random numbers vi are converted to the random numbers Yi(d) which reflect the inter-site temperature correlation. Then the converted random numbers Yi(d) are converted to the temperature ti(d) with the function Φ. the function Φ is chosen based on the random numbers Xi(d) which reflect the inter-site precipitation correlation. Thus the embodiment can create a weather scenario in which the correlation between sites and the correlation between meteorological elements. Therefore the embodiment is capable of predicting the risk with accuracy.  
         [0077]     It is mathematically very difficult to combine consistently the multi-site temperature model, multi-site precipitation model ((Rodrigo Caballero et al. and D. S. Wilks), and single-site precipitation-temperature correlation model (Richardson, C. W.) of the related art. Instead of using an unprocessed precipitation or temperature value, this invention factors in a correlation between meteorological elements at the stage of stochastic variables, which is the preliminary stage of precipitation or temperature. This invention is therefore capable of providing a weather forecast in which the correlation between precipitation and temperature, the inter-site precipitation correlation, and the inter-site temperature correlation are all consistent with one another.  
         [0078]     While the present invention has been described in detail and pictorially in the accompanying drawings, the present invention is not limited to such detail but covers various obvious modifications and equivalent arrangements, which fall within the purview of the appended claims.