JP2014029583A - Curve estimation method and device - Google Patents

Abstract

[PROBLEMS] To draw an appropriate spot rate curve from interest-bearing bond data.
[MEANS FOR SOLVING PROBLEMS] One interest-bearing bond is extracted for each maturity date from the market price including the maturity date, interest, redemption amount, and paid interest for each interest-bearing bond, and the maturity date for the extracted interest-bearing bond issue Generates a matrix that enumerates the payment amount at each payment period for each extracted interest-bearing bond issue, and converts this matrix into a regular matrix, and extracts the market for each extracted interest-bearing bond issue Executes the process of calculating the discount rate at the maturity date for each of the extracted interest-bearing bond issues from the matrix including the price and the regular matrix, and the maturity date for each of the extracted interest-bearing bond issues from the discount rate at the maturity date The interest rate on the day is calculated, and the curve is calculated using the regression model from the interest rate on the maturity date. The regression model represents the interest rate for the period corresponding to the maturity date as a weighted sum of a plurality of basis functions, and the center value of the basis function is set according to the distribution of the period corresponding to the maturity date. .
[Selection] Figure 6

Description

  The present technology relates to a technology for estimating a curve representing a relationship between a spot rate and a remaining period.

  The spot rate (SR) is an average yield according to a period (remaining period) until the maturity date, which is currently predicted by the market. Strictly speaking, this is the average yield of bonds (discounted bonds) without interest payments until the maturity date. Generally, it is calculated from the price of government bonds without default risk.

  However, in the case of Japan, a government bond is an interest-bearing bond that has multiple interest payments until the maturity date, so it is not possible to obtain a spot rate directly from a government bond. Therefore, the interest rate bond is divided into the interest and the principal part, and the spot rate is calculated on the assumption that the bond is a combination of bonds that mature at the time of each payment and do not pay interest on the way.

  There are approximately 300 Japanese government bonds, with a maximum period of 40 years. A problem is that the number of issues of government bonds (for example, 30 to 40 years) with a long period until maturity is small. Interest and redemption amounts are determined at the time of issuance and are not changed midway.

FIG. 1 schematically shows interest payment and redemption of a bond of issue α and calculation of present value. For example, every half year, a constant interest c _α continues L _α times until the maturity date, and the redemption amount R _α is also paid on the maturity date. More specifically, the interest payment amount c _α is paid from the first interest payment to L _α −1, and c _α + R _α is paid at the last L _α time.

On the other hand, the present value of these payment amounts is a value obtained by multiplying the value shown at the bottom of each payment period. Note that r (t) represents the spot rate after time t, and in the case of time t, the discount rate 1 / (1 + r (t)) ^t is multiplied. In other words, the present value increased by compound interest is considered as the future value, and the present value is calculated backward from the future value. Note that t _k ^α represents a period until the k-th payment time of the brand α.

Assuming that the market value of government bonds = the sum of the present value of interest and principal, the market price p _α is expressed as follows.

  Further, the relationship between the discount rate d (t) and the spot rate is defined as follows. That is, discount rate d (t) = ratio between present value and future value. Therefore, it is specifically shown as follows.

  Conventionally, the spot rate is calculated by the following procedure. That is, transaction data of government bonds is collected, and a discount rate is calculated based on a statistical model of market prices of government bonds. The statistical model is represented by the following formula.

Note that ε _α represents an error of the normal distribution N (0, σ ² ). N represents the number of brands.

  However, as mentioned above, Japanese government bonds have a small number of issues with a long remaining period. FIG. 2 shows the relationship between the remaining period of government bonds and the price of government bonds. As can be seen from the part surrounded by the dotted line, the number of samples decreases for the part with a longer remaining period. For this reason, if a spot rate curve is drawn based on these data, the curve will meander unnaturally in a portion where the remaining period is long as shown in FIG. The spot rate curve is a curve representing the relationship between the remaining period and the spot rate.

Kawasaki Noritori, Yasuchi Tomohiro (2002) Yield Curve Estimation by Regularized Nonlinear Regression Model, Statistical Mathematics, Vol. 50, No. 2, 149-164 Mizuho Report “Understanding the Structure of Government Bond Spot Rate Curves by Principal Component Analysis and Examining its Predictability: Using a Common Factor Model Based on Macroeconomic and Financial Variables”, published on September 21, 2010, Mizuho Research Institute Konishi Sadanori, Kitagawa Genshiro, Information Criterion, Asakura Shoten, 2004, pages 80-106

  Therefore, an object of the present technology is to provide a technology for enabling an appropriate spot rate curve to be drawn from interest-bearing bond data as one aspect.

  The curve estimation method according to one aspect of the present technology includes: (A) one issue per maturity date from a data storage unit that stores the maturity date, interest, redemption amount, and market price including paid interest for each issue of interest-bearing bonds. (B) Generate a matrix that lists the payment amount at each payment period for each extracted interest-bearing bond issue from the maturity date, interest and redemption amount for the extracted interest-bearing bond issue (C) Converting the generated matrix into a regular matrix, and discounting the maturity date for each of the extracted interest-bearing bond issues from the extracted matrix including the market price for each interest-bearing bond issue and the regular matrix (D) Calculate the interest rate at the maturity date for each of the extracted interest-bearing bonds from the discount rate at the calculated maturity date, and (E) calculate the interest rate at the maturity date. From a song using a regression model Including the process of calculating the. The above regression model expresses the interest rate for the period corresponding to the maturity date as a weighted sum of a plurality of basis functions, and the central value of the plurality of basis functions has a distribution of the period corresponding to the maturity date. Set accordingly.

  Appropriate spot rate curves can be drawn from interest-bearing bond data.

FIG. 1 is a diagram for explaining interest payment and redemption of government bonds and calculation of present value. FIG. 2 is a diagram schematically showing the relationship between the remaining period of government bonds and the price. FIG. 3 is a diagram showing an example of a spot rate curve drawn in the prior art. FIG. 4 is a functional block diagram of the information processing apparatus according to the embodiment of the present technology. FIG. 5 is a diagram illustrating an example of data stored in the first data storage unit. FIG. 6 is a diagram showing a main processing flow according to the present embodiment. FIG. 7 is a diagram illustrating a processing flow of the filtering process. FIG. 8 is a diagram illustrating an example of a cash flow matrix. FIG. 9 is a diagram showing a processing flow of discount rate calculation processing. FIG. 10 is a diagram for explaining a conversion process into a regular matrix. FIG. 11 is a diagram showing an example of government bond compound yield data. FIG. 12 is a diagram illustrating an example of a discount rate curve. FIG. 13 is a diagram illustrating an example of a regular matrix. FIG. 14A is a diagram illustrating a numerical example of the spot rate. FIG. 14B is a diagram illustrating an example of a spot rate plot. FIG. 15 is a diagram for explaining a first variation in the spot rate curve estimation process. FIG. 16 is a diagram illustrating a processing flow of spot rate curve estimation processing for the first variation. FIG. 17 is a diagram for explaining the second variation. FIG. 18 is a diagram for explaining the third variation. FIG. 19 is a diagram for explaining the fourth variation. FIG. 20 is a diagram illustrating a processing flow of spot rate curve estimation processing for the second to fourth variations. FIG. 21 is a diagram illustrating an example of output data. FIG. 22 is a diagram illustrating another example of output data. FIG. 23 is a functional block diagram of a computer.

  FIG. 4 is a functional block diagram of the information processing apparatus 100 that estimates the spot rate curve according to the present embodiment. The information processing apparatus 100 includes a first data storage unit 101, an extraction unit 102, a second data storage unit 103, a matrix generation unit 104, a third data storage unit 105, a discount rate calculation unit 106, and a fourth The data storage unit 113, the fifth data storage unit 107, the spot rate calculation unit 108, the sixth data storage unit 109, the curve estimation unit 110, the seventh data storage unit 111, and the output unit 112 are included.

  The first data storage unit 101 stores, for example, government bond data as shown in FIG. In the example of FIG. 5, data such as a brand identifier, maturity date, interest (or interest rate), redemption amount, payment interval, market price, transaction volume, and bidding date are stored for each brand of the government bond. The market price is an amount obtained by adding accrued interest (paid interest) to the market price. In some cases, the number of transactions is also stored.

  The extraction unit 102 extracts data used in the following processing from the government bond data stored in the first data storage unit 101 and stores the data in the second data storage unit 103. The matrix generation unit 104 generates a cash flow matrix described below from the government bond data stored in the second data storage unit 103 and stores it in the third data storage unit 105.

  The discount rate calculation unit 106 includes a regular conversion unit 1061, and uses the data stored in the second data storage unit 103 and the fourth data storage unit 113 to store the cache stored in the third data storage unit 105. The flow matrix is converted into a regular matrix and a discount rate matrix is calculated and stored in the fifth data storage unit 107. The fourth data storage unit 113 stores data for calculating a forward discount factor, such as government bond compound yield data.

  The spot rate calculation unit 108 calculates the spot rate at each maturity date from the discount rate matrix stored in the fifth data storage unit 107 using the data stored in the second data storage unit 103, and calculates The spot rate thus obtained is stored in the sixth data storage unit 109. The curve estimation unit 110 uses the data stored in the second data storage unit 103 to execute a process of estimating a spot rate curve from the spot rate stored in the sixth data storage unit 109, and the processing result Is stored in the seventh data storage unit 111. The output unit 112 outputs spot rate curve data to an output device (display device, printing device, or other computer connected via a network).

  Next, processing contents of the information processing apparatus 100 will be described with reference to FIGS. 6 to 20.

  First, the extraction unit 102 performs a filtering process of extracting one brand of government bonds for each maturity date or generating one brand of virtual government bonds from the government bond data stored in the first data storage unit 101, The processing result is stored in the second data storage unit 103 (step S1). For example, a processing flow as shown in FIG. 7 is performed.

  First, the extraction unit 102 sorts the government bond data in the order of early maturity dates (step S21). And the extraction part 102 groups for every government bond of the same maturity date (step S23). Thereafter, the extraction unit 102 selects one brand of government bonds for each group according to the rules described below, or generates one brand of virtual government bonds as described below, and is selected or generated. The government bond data is stored in the second data storage unit 103 (step S25).

  For step S25, various variations are conceivable. As a first variation, one issue of a government bond with a maximum transaction volume is selected. This is because it is estimated that the larger the transaction volume, the smaller the error from the appropriate price. In addition, as a second variation, one issue is selected as a government bond with the latest bid date. This is also because the fluidity is generally high.

Furthermore, as a third variation, for a market price, redemption amount, and interest of a government bond with the same maturity date, one issue of a virtual government bond is generated by a weighted sum. The transaction amount is used for the weight. For example, if the weight of the brand i is a _i , the market price of the brand i is p _i , the redemption amount of the brand i is R _i, and the interest of the brand i is c _i , the following is expressed.

  p represents the market price of the virtual government bond, c represents the interest of the virtual government bond, and R represents the redemption amount of the virtual government bond.

  As for the first to third variations, one kind of government bond is extracted for government bonds that exceed the threshold for at least one of the transaction volume, the number of transactions, and the remaining period (period until the maturity date). Also good. For example, those with a remaining period of less than 2 months are excluded, or those with a transaction volume of less than a threshold are excluded.

In addition, as a fourth variation, sorting may be performed according to the following procedure.
(1) Limit the types of government bonds to be selected for each remaining period zone.
If the remaining period is 2 years or less, only the 2-year bond is selected. If the remaining period is 2 years or more and 5 years or less, only the 5-year bond is selected. If the remaining period is 5 years or more and 10 years or less, the 10-year bond If only the remaining option period exceeds 10 years and is 20 years or less, only the 20-year bond period is longer than 20 years and if it is 30 years or less, only the 30-year bond period is longer than 30 years and less than 40 years. If so, only 40-year bonds are selected. Japanese government bonds are not newly issued, so if there are no 5-year bonds, 10-year bonds may be used instead. Also, 2-year bonds with less than 30 days remaining are excluded.
(2) After limiting as in (1), if a plurality of issues of government bonds remain for the same maturity date, the issue of the latest bid date is selected.

  Returning to the description of the processing in FIG. 6, the matrix generation unit 104 generates a cash flow matrix (CF matrix) for the government bonds included in the filtering processing result stored in the second data storage unit 103, and generates third data Store in the storage unit 105 (step S3).

  Each row of the cash flow matrix corresponds to one issue of government bonds, and each column corresponds to the payment period of either interest or redemption amount (principal). In the cash flow matrix ij, a payment amount (interest or interest and redemption amount) to be paid for the issue i of the government bond at the payment time j is set.

For example, a cash flow matrix as shown in FIG. 8 is generated. The example of FIG. 8 shows an example in which N issues of government bonds are extracted, and these N issues of government bonds have M payment periods in total. For example, interest payments on issue N government bonds are made at payment periods t ₁ and t _M-3 , and interest and redemption payments on issue N government bonds are made at payment time t _M. . For other payment periods, “0” is set because there is no payment for the national bond of the issue N. The same applies to other brands.

The payment time t _i is calculated by maturity date−payment interval × n (n is a natural number) for each issue of the government bond. However, the calculation is stopped when the payment time t _i becomes a negative value. In other words, starting from the maturity date, the payment date interval is traced back to the present. The payment amount is the interest of each issue itself, or the interest plus the redemption amount at the maturity date. In addition, since it is used in the processing described below, t _{1 to} t _M are also stored in the third data storage unit 105.

  Next, the discount rate calculation unit 106 executes a discount rate calculation process for calculating the discount rate at each maturity date from the cash flow matrix stored in the third data storage unit 105, and stores the processing result in the fifth data storage. The data is stored in the unit 107 (step S5). The discount rate calculation process will be described with reference to FIGS. 9 to 13.

  First, the regular conversion unit 1061 of the discount rate calculation unit 106 converts the cash flow matrix into a regular matrix using the initial forward discount factor (FD) (step S31).

In this step, first of all, in column j (j = 1, 2,..., M) of the cash flow matrix, the corresponding payment time t _j is the maturity of the issue i (i = 1,..., N). Day T = {T _i | i = 1,. . . , N} is specified. In the example schematically shown in FIG. 10, t ₁ , t ₂ , and t ₅ are identified because they are not maturity dates.

Then, the column identified in this way is added to the column of the maturity date immediately after taking the forward discount factor D into consideration. For the columns t ₁ and t ₂ , add to the column of t ₃ = T ₁ , and for the column of t ₅ add to the column of t ₆ = T ₃ .

  Further, the inverse value 1 / D (t, T) of the forward discount factor is multiplied by the column of t and added to the column of T so as to approximately store the present value of the cash flow at the payment time t. . D (t, T) represents a forward discount factor from the payment time t to the payment time T.

  The initial forward discount factor is calculated, for example, from government bond compound yield data stored in the fourth data storage unit 113. The government bond compound yield data is, for example, data as shown in FIG. 11, and the discount rate is calculated from the government bond compound yield using the above-described equation (1). Then, the forward discount factor D is calculated from the discount rate by the following formula.

d (τ _t ) is a discount rate for the period τ _t from the present time to the time t. Similarly, d (τ _T ) is a discount rate for the period τ _T from the current time point to the time point T.

However, as shown in FIG. 11, since only a discrete discount rate can be obtained, the forward discount factor D (t ₁ , t ₂ ) used to convert the cash flow matrix described above into a regular matrix is used. In some cases, d (τ _t1 ) and d (τ _t2 ) may not be directly calculated. In such a case, the current discount rate is set to “1” in addition to the calculated discount rate at each time point, and a discount rate curve is drawn by an interpolation method such as cubic spline interpolation, for example, and τ _t1 and discount rate obtain d (tau _t1) and d (tau _t2) for tau _t2.

The discount rate curve is a curve representing a monotone decreasing function as shown in FIG. 12, for example. FIG. 12 shows only the points plotted from the multi-yield data for government bonds, as shown in FIG. 12, which shows the change in the discount rate from the remaining period “0” (ie, the current time) to the longer remaining period. The value between is calculated by interpolation calculation. In FIG. 12, the discount rate for the period τ _t from the current time corresponding to the time point t is d (τ _t ), and the discount rate for the time period τ _T from the current time point T is d (τ _T ).

In the example of FIG. 10, for the row of _{t 1, D (t 1,} t 3 = T 1) = d (τ t1) / d (τ t3) is added to the column of T ₁ after multiplying. Furthermore, for the columns of t _2, and adds the multiplied by _{D (t 2, t 3)} = d (τ t2) / d (τ t3) to the columns of T _3. For the column of t ₅ , D (t ₅ , t ₆ = T ₃ ) = d (τ _t5 ) / d (τ _t6 ) is multiplied and added to the column of T ₃ .

  If a column other than the maturity date is deleted, a regular matrix of N rows and N columns as shown in FIG. 13 is obtained from the cash flow matrix as shown in FIG. In other words, the payment timing of the extracted issues of government bonds is converted into a cash flow matrix that is aggregated on the maturity date of any of these issues of government bonds.

  Since the initial forward discount factor is a temporary value, it is not necessary to use compound yield data on government bonds. For example, an arbitrary curve that decreases monotonically from 1 as schematically shown in FIG. 12 may be prepared and used.

  Returning to the description of the processing in FIG. 9, the discount rate calculation unit 106 calculates a discount rate matrix from the regular matrix and stores it in the fifth data storage unit 107 (step S33).

The market price of government bonds is a value obtained by discounting each cash flow by a discount rate and adding up the discount rate matrix d from the following relationship. It is assumed that the regular matrix is C.
Cd = p
d = (d ₁ , d ₂ ,..., d _N ) ^T
d _k is a discount rate at time T _k .
p = (p ₁ , p ₂ ,..., p _N ) ^T
p _k is the market price at time T _k . Such data is stored in the third data storage unit 105.

Therefore, the discount rate matrix d can be obtained by modifying as follows.
d = C ^-1 p

  Then, the discount rate calculation unit 106 determines whether the difference between the previous discount rate matrix d ′ stored in the fifth data storage unit 107 and the currently calculated discount rate matrix d is within an allowable range. (Step S35). The requirement is that the absolute value of the difference in discount rates at any maturity date is less than the threshold. The first time does not satisfy this condition. If this condition is satisfied, the process returns to the caller process.

  On the other hand, when this condition is not satisfied, the regular conversion unit 1061 converts the cash flow matrix into a regular matrix using the forward discount factor calculated from the discount rate matrix calculated in step S33 (step S37). ). The conversion method is the same as in step S31, but only the value of the forward discount factor used is different. Then, the process returns to step S33.

  In this way, steps S33 to S37 are repeated until the discount rate converges. That is, the regular matrix and the discount rate matrix are reset until the market price is correctly calculated.

  Returning to the description of the processing in FIG. 6, the spot rate calculation unit 108 uses the data stored in the second data storage unit 103 to calculate the government bond from the discount rate matrix stored in the fifth data storage unit 107. The spot rate for the maturity date of each brand is calculated and stored in the sixth data storage unit 109 (step S7).

  Similar to the equation (2) described above, it is calculated by the following equation.

Here, r _i is the spot rate on the maturity date T _i . D _i represents a discount rate at T _i .

The maturity date T _i can be converted into a remaining period from the present. Then, for example, data as shown in FIG. 14A is obtained. In the example of FIG. 14A, data of the remaining period and the spot rate (SR) are included. When the data of FIG. 14A is graphed, it is as shown in FIG. 14B. In the example of FIG. 14B, the horizontal axis represents the remaining period, and the vertical axis represents the spot rate. Thus, it can be seen that spot rates are not calculated for some remaining periods, particularly between 30 and 40 years. In the present embodiment, by executing the processing as described above, data suitable for performing the regression calculation described below is prepared.

  Then, the curve estimation unit 110 executes a spot rate curve estimation process on the spot rate stored in the sixth data storage unit 109, and stores the processing result in the seventh data storage unit 111 (step S9). The spot rate curve estimation process will be described with reference to FIGS.

(A) First Variation In the first variation, for example, a spot rate curve is estimated using a regression model including a weighted sum of Gaussian basis functions. More specifically, the spot rate r _α after the period T _α is represented by the following regression model.

ε _α represents an error of the normal distribution N (0, σ ² ). σ ² represents the error variance. W _k represents the weight of the k th basis function. Furthermore, Equation (4) represents a Gaussian basis function. In Equation (4), μ _k represents the center value of the kth basis function, and s ² represents the spread of the basis function. It is a parameter.

  In the first variation, the center value of the basis function is set as shown in FIG. That is, if the number of basis functions is m, the center value of the first basis function and the center value of the mth basis function are set to the remaining period “0” and the maximum remaining period, respectively, and the remaining period “0” is set. ”To the maximum remaining period, the set number of remaining periods between the center values of the basis functions (remaining period corresponding to the maturity date; hereinafter referred to as a sample. In FIG. 15, a black circle represents a sample) is equal. Set as follows. As shown in FIG. 15, when the number of samples is N, the center values of a plurality of basis functions are such that N / (m−1) samples exist between adjacent center values of the basis functions. Set. m is set in the process described below.

  In this way, a large number of basis functions are set in the range A where the sample density is high. In the range B where the sample density is low, the basis function is set to be small. As described above, in the range A where the sample density is high, many basis functions are set so that the basis function can be set according to the sample, and in the range B where the sample density is low, a small number of bases are drawn so as to draw a smooth curve. Set the function.

The parameter μ _k representing such a center value is expressed as follows.

However, T ₁ , T ₂ ,. . . , T _N represent the remaining period corresponding to the maturity date. Further, the subscript T in equation (5) represents the maximum integer that does not exceed N (k−1) / (m−1). However, T ₀ = 0.

  And the probability density function for such a regression model is expressed as follows.

  A logarithmic likelihood with a penalty including a penalty term from such a probability density function is defined as follows.

  Note that λ represents a regularization parameter. In addition, the first and second terms in equation (7) represent the log likelihood of the regression model error, and the third term represents a penalty term set so that the weight matrix w changes smoothly. .

The weight matrix w and σ ² that maximize the log-likelihood with such penalties are assumed to be an estimated value w ^ (“^” represents a superscript hat) and σ ² ^ (“^” is a superscript (Representing a hat), the following is solved.

  Then, the solution is expressed as follows.

Note that the spot rate matrix r = (r ₁ , r ₂ ,..., R _N ).

Parameters m, s ² and λ such as a regression model are determined so as to minimize the value of the generalized information criterion defined by the weight matrices w ^ and σ ² ^ obtained in this way.

  Note that tr in the third term on the right side of equation (9) represents the sum of the diagonal components.

In summary, the curve estimation unit 110 performs a process as shown in FIG. First, the curve estimation unit 110 sets initial values of m, s ^2, and λ (step S41). Then, the curve estimation unit 110 calculates the estimated values of the weight matrix w and σ ² using m, s ^2, and λ (step S43). The estimated value is calculated using equation (8).

Then, the curve estimation unit 110 calculates a generalized information criterion value from the estimated values of the weight matrix w and σ ² (step S45). The value of the generalized information criterion is calculated using equation (9).

  Thereafter, the curve estimation unit 110 determines whether or not the end condition of the spot rate curve calculation process is satisfied (step S47). The termination condition for minimizing the value of the generalized information criterion is determined according to the nonlinear optimization method to be used. If the end condition is satisfied, the process returns to the caller process.

On the other hand, when the process termination condition is not satisfied, the curve estimation unit 110 sets new values of m, s ^2, and λ using a predetermined algorithm (step S49). The values of m, s ² and λ are determined according to a well-known nonlinear optimization method such as a genetic algorithm. Then, the process returns to step S43. However, the nonlinear optimization method is not limited to the genetic algorithm.

When m, s ^2, and λ are set in advance by some method, the weight matrix w and σ ² are estimated by using the values as they are without performing the process of FIG. Calculate and use the value.

By performing the processing as described above, in addition to m, s ² and λ that minimize the value of the generalized information criterion, the weight matrices w and σ ² at that time are also obtained. Therefore, the spot rate for an arbitrary remaining period can be calculated by the following formula and formula (4).

If there is m, weight matrix w, σ ² , center position μ _k (1 ≦ k ≦ m) of each basis function, and parameter s ² representing the spread, equation (10) can be calculated, so curve estimation unit 110 Stores these data in the seventh data storage unit 111.

(B) Second Variation In the first variation, the parameter s ² representing the spread of the basis function is the same for each basis function, but this can be set for each basis function.

  For example, as shown in FIG. 17, the spread is reduced in the range A where the sample density is high, and the spread is increased in the range B where the sample density is low. As a result, the smoothness can be improved in the area B. The center value of the basis function is the same as that of the first variation.

Therefore, in the second variation, the parameter s ² _k representing the spread of the kth basis function is set as follows.

Here, Cs is a constant determined by the following processing, and Δt is a constant set in advance. Further, the denominator on the right side of the equation (11) represents the number of samples included between μ _k −Δt and μ _k + Δt, as schematically shown in FIG. Again, the sample is represented by a black circle. That is, the parameter s _k ² representing the spread is set so as to be proportional to the square of the reciprocal of the number of samples that fall within the range of Δt before and after the center value μ _k of the basis function.

  The basis function is expressed as follows.

(C) Third Variation In the second variation, the spread of the basis function is symmetric, but as already shown in FIG. 18, the sample density may not be symmetric.

  Therefore, the basis function is defined as follows.

Parameter s ² _k for the following Thus mu _{k, -} and the parameters s ² _k for more than mu _k, introducing and _+, is defined as follows.

However, for the basis functions at both ends, there is no sample on one side, so s ² _{1, −} = s ² _{1, +} and s ² _{m, +} = s ² _{m, −} .

The right-hand side denominator of equation (14) represents the number of samples included between μ _k −Δt and μ _k, and the right-hand side denominator of equation (15) is included between μ _k and μ _k + Δt. Represents the number of samples to be played.

(D) the base in the fourth variation the third variation, the parameter indicating the spread of the basis functions have been determined according to the number of samples coming from mu _k in Delta] t, because it sets a Delta] t, the adjacent Different parameter values are set even between the center values of the functions.

In this embodiment, as shown in FIG. 19, the basis function phi _k centered value mu _k, the basis function phi _{k + 1} centered value mu _{k + 1,} the μ _k μ _{k + The} same sample density (= number of samples / (μ _{k + 1} −μ _k )) in the range up to ₁ . Therefore, a parameter representing the spread of the basis function is defined as follows.

  The absolute value portion on the right side of the equations (16) and (17) represents the length between the center values of adjacent basis functions.

  In the second to fourth variations, unlike the first variation, the parameter Cs related to the spread of the basis function is determined. However, the log likelihood including the probability density function and the penalty term is the same as the first variation.

  Therefore, the generalized information criterion is expressed by equation (18) instead of equation (9).

In summary, in the case of the second to fourth variations, the curve estimation unit 110 performs processing as shown in FIG. First, the curve estimation unit 110 sets initial values of m, Cs, and λ (step S51). Then, the curve estimation unit 110 calculates the estimated values of the weight matrix w and σ ² using m, Cs, and λ (step S53). The estimated value is calculated using equation (8).

Then, the curve estimation unit 110 calculates the value of the generalized information criterion based on the estimated values of the weight matrix w and σ ² (step S55). The value of the generalized information criterion is calculated using equation (18).

  Thereafter, the curve estimation unit 110 determines whether or not the end condition of the spot rate curve calculation process is satisfied (step S57). The termination condition for minimizing the value of the generalized information criterion is determined according to the nonlinear optimization method to be used. If the end condition is satisfied, the process returns to the caller process.

  On the other hand, when the process termination condition is not satisfied, the curve estimation unit 110 sets new values of m, Cs, and λ using a predetermined algorithm (step S59). For example, values of m, Cs, and λ are determined according to a well-known nonlinear optimization method such as a genetic algorithm. Then, the process returns to step S53. However, the nonlinear optimization method is not limited to the genetic algorithm.

When m, Cs, and λ are set in advance by some method, the estimated values of the weight matrices w and σ ² are obtained by using the values as they are without performing the processing of FIG. Is calculated and used.

By performing the processing as described above, in addition to m, Cs, and λ that minimize the value of the generalized information criterion, the weight matrices w and σ ² at that time are also obtained. Therefore, the spot rate for an arbitrary remaining period can be calculated by the equations (5), (10), and (13).

M and Cs, weight matrix w, σ ² , center position μ _k (1 ≦ k ≦ m) of each basis function, parameters s ² _{k, −} and s ² _{k, +} (1 ≦ k ≦ m) representing the spread ), The equation (13) can be calculated, and the curve estimation unit 110 stores these data in the seventh data storage unit 111.

  Returning to the description of the processing in FIG. 6, the output unit 112 outputs the data itself stored in the seventh data storage unit 111 or the expression (10) and the data stored in the seventh data storage unit 111. Then, the remaining period is changed in a predetermined increment, the spot rate for the remaining period is calculated, and numerical data is output according to the calculation result or the curve itself is output (step S11).

  For example, data as shown in FIG. 21 may be calculated and output as it is. In the example of FIG. 21, the remaining periods are listed in increments of 0.01, and the spot rates of these remaining periods are listed.

  When the spot rate curve is used, a curve as shown in FIG. 22 is obtained. In the example of FIG. 22, the horizontal axis represents the remaining period, and the vertical axis represents the spot rate (SR). In the circled part, the number of JGBs is small (see Fig. 2), so in the conventional technology, only a curve that changed greatly as shown in Fig. 3 could be drawn. A changing curve is drawn.

  Although the embodiment of the present technology has been described above, the present technology is not limited to this. For example, the functional block diagram shown in FIG. 4 is an example, and implementation in which the program module configurations do not match is possible.

  Furthermore, the processing flow is also an example, and as long as the processing result does not change, the order of the processing steps may be changed or a plurality of processing steps may be executed in parallel.

  The information processing apparatus 100 that is the curve estimation apparatus described above is a computer apparatus, and as shown in FIG. 23, a memory 2501, a CPU (Central Processing Unit) 2503, a hard disk drive (HDD). 2505, a display control unit 2507 connected to the display device 2509, a drive device 2513 for the removable disk 2511, an input device 2515, and a communication control unit 2517 for connecting to a network are connected by a bus 2519. An operating system (OS) and an application program for executing the processing in this embodiment are stored in the HDD 2505, and are read from the HDD 2505 to the memory 2501 when executed by the CPU 2503. The CPU 2503 controls the display control unit 2507, the communication control unit 2517, and the drive device 2513 according to the processing content of the application program, and performs a predetermined operation. Further, data in the middle of processing is mainly stored in the memory 2501, but may be stored in the HDD 2505. In an embodiment of the present technology, an application program for performing the above-described processing is stored in a computer-readable removable disk 2511 and distributed, and installed from the drive device 2513 to the HDD 2505. In some cases, the HDD 2505 may be installed via a network such as the Internet and the communication control unit 2517. Such a computer apparatus realizes various functions as described above by organically cooperating hardware such as the CPU 2503 and the memory 2501 described above and programs such as the OS and application programs. .

  The above-described embodiment can be summarized as follows.

  The curve estimation method according to the present embodiment stores (A) a maturity date, interest (may be an interest rate), redemption amount, and market price including paid interest for each issue of interest-bearing bonds (for example, government bonds). From the data storage unit, for each maturity date extracted from the maturity date, interest and redemption amount for each maturity date, and (B) the step of extracting one maturity bond for each maturity date A step of generating a matrix enumerating payment amounts at each payment period; and (C) converting the generated matrix into a regular matrix, and from a matrix including a market price for each symbol of the interest-bearing bond and a regular matrix, A step of calculating a discount rate at the maturity date for each of the extracted interest-bearing bond issues, and (D) a maturity date for each of the extracted interest-bearing bond issues from the calculated discount rate at the maturity date. Interest rate in Comprising the steps of: leaving, and calculating a curve using the interest rate, the regression model in maturity calculated (E). The above regression model expresses the interest rate for the period corresponding to the maturity date as a weighted sum of a plurality of basis functions, and the central value of the plurality of basis functions has a distribution of the period corresponding to the maturity date. Set accordingly.

  By carrying out such processing, an appropriate curve can be obtained even when the interest rate data for a certain period is insufficient.

  In addition, in the range from the period corresponding to the maturity date of 0 to the period corresponding to the latest maturity date, the center values of the plurality of basis functions described above are included in the range between the adjacent center values. The number of periods corresponding to the day may be set to be the same number. In this way, a center value corresponding to the density at which a period corresponding to the maturity date exists is set.

  In addition, the parameter representing the spread of each of the plurality of basis functions described above is set according to the number of periods corresponding to the maturity date included in a predetermined range from the center value of the basis function. Also good. Thus, the parameter representing the spread of the basis function is also set according to the density at which a period corresponding to the maturity date exists. For example, the lower the density in which there is a period corresponding to the maturity date, the wider the basis function may be.

  Further, the predetermined range described above includes a first range for a value smaller than the center value of the basis function and a second range for a value larger than the center value of the basis function, Each parameter representing the spread includes a first parameter representing a spread on the value side smaller than the center value of the basis function and a second parameter representing a spread on the value side larger than the center value of the basis function. Also good. In such a case, the first parameter is set according to the number of periods corresponding to the maturity included in the first range, and the second parameter corresponds to the maturity included in the second range. It may be set according to the number of periods. In this way, a parameter representing the spread of the basis function asymmetrically may be set.

  Further, a parameter representing the spread of each of the plurality of basis functions may be determined according to the length between adjacent center values of the basis function. In this manner, a parameter representing the spread of the basis function asymmetrically may be set.

  Further, in the calculation of the curve described above, the logarithmic likelihood including a penalty term for smoothly changing the probability density function based on the regression model used in the calculation of the curve and including a penalty term is maximized, and a plurality of basis functions The value of the parameter group including the weights may be determined.

  Further, in the calculation of the curve described above, parameters related to the interval between the center values of the basis functions in the regression model are set so that the value of the generalized information criterion defined in relation to the parameter group is minimized. The parameter for controlling the spread of the basis function and the regular parameter included in the penalty term may be determined. By performing the processing as described above, an appropriate regression model can be determined.

  In addition, the data storage unit described above may further store a transaction amount or a bid date for each brand of interest-bearing bonds. In this case, in the step of extracting one brand of interest-bearing bond for each maturity date, one brand of interest-bearing bond may be selected for each maturity date based on the transaction amount or the bidding date. According to the non-obvious knowledge of the inventor, a more appropriate interest rate curve can be obtained by implementing such a selection.

  Further, in the step of extracting one issue of interest-bearing bonds for each maturity date described above, if there are multiple issues of interest-bearing bonds with the same maturity date, a hypothetical 1 You may make it calculate the interest, the redemption amount, and the market price of the interest-bearing bond of one brand. Statistically appropriate data can be generated.

  In the process of calculating the discount rate described above, the payment amount at the maturity date immediately after the payment date is calculated from the payment amount at the payment date other than the maturity date of the extracted interest-bearing bonds in the generated matrix. Then, it may be converted into a regular matrix in which payment amounts for each maturity date are listed for each of the extracted interest-bearing bonds. In this way, the interest payment timing of the extracted interest-bearing bonds is appropriately aggregated on the maturity date of any issue of the interest-bearing bonds.

  Further, the processing for calculating the discount rate described above is (c1) In the generated matrix, from the payment amount at the payment date other than the maturity date of the extracted interest-bearing bond issue, A process of calculating a payment amount at the maturity date immediately after the payment date using the first interest rate at the maturity date, and converting the payment amount at the maturity date for each of the extracted interest-bearing bonds into a regular matrix; (C2) Processing for calculating the second interest rate at the maturity date for each of the extracted interest-bearing bond issues from the product of the inverse matrix of the regular matrix and the matrix including the market price for each extracted interest-bearing bond issue And (c3) replacing the first interest rate with the second interest rate until the difference between the first interest rate and the second interest rate is all equal to or less than a predetermined threshold, converting the regular matrix and the second interest rate To include the process of repeatedly performing the calculation It may be.

  The second interest rate obtained by performing such processing is an appropriate discount rate. Note that when calculating the payment amount at the maturity date, the interest rate for a period not included in the first interest rate may be calculated by interpolation calculation (for example, cubic spline interpolation).

  Note that a program for causing a computer to execute the processing described above can be created, and the program includes, for example, a flexible disk, an optical disk such as a CD-ROM, a magneto-optical disk, and a semiconductor memory (for example, ROM). Or a computer-readable storage medium such as a hard disk or a storage device.

  The following supplementary notes are further disclosed with respect to the embodiments including the above examples.

(Appendix 1)
From the data storage unit that stores the maturity date, interest, redemption amount, and market price including paid interest for each issue of interest-bearing bonds, extract one issue of interest-bearing bonds for each maturity date,
From the maturity date, the interest, and the redemption amount for the extracted interest-bearing bond issue, generate a matrix listing the payment amount for each payment date for each extracted interest-bearing issue issue,
The generated matrix is converted into a regular matrix, and the discount on the maturity date for each of the extracted interest-bearing bond issues is extracted from the extracted matrix including the market price for each interest-bearing bond issue and the regular matrix. Execute the process to calculate the rate,
From the calculated discount rate on the maturity date, calculate the interest rate on the maturity date for each of the extracted interest-bearing bond issues,
Including a process of calculating a curve from the calculated interest rate on the maturity date using a regression model, and is executed by a computer,
The regression model represents an interest rate for a period corresponding to the maturity date as a weighted sum of a plurality of basis functions;
A curve estimation method in which center values of the plurality of basis functions are set according to a distribution of a period corresponding to the maturity date.

(Appendix 2)
The median value of the plurality of basis functions corresponds to the maturity date included in a range between adjacent median values in a range from 0 to a period corresponding to the latest maturity date. The curve estimation method according to appendix 1, wherein the number of periods to be set is the same.

(Appendix 3)
The parameter representing the spread of each of the plurality of basis functions is set according to the number of periods corresponding to the maturity date included in a predetermined range from the center value of the basis function. Estimation method.

(Appendix 4)
The predetermined range includes a first range for values less than the center value of the basis function and a second range for values greater than the center value of the basis function;
A first parameter representing a spread on a value side smaller than the center value of the basis function and a second representing a spread on a value side larger than the center value of the basis function. Including the parameters of
The first parameter is set according to the number of periods corresponding to the maturity included in the first range, and the second parameter corresponds to the maturity included in the second range. The curve estimation method according to appendix 3, which is set according to the number of periods to be performed.

(Appendix 5)
The curve estimation method according to claim 3, wherein a parameter representing a spread of each of the plurality of basis functions is determined according to a length between adjacent center values of the basis function.

(Appendix 6)
In calculating the curve,
The value of the parameter group including the weight of the plurality of basis functions is maximized, and the logarithmic likelihood is defined for the probability density function based on the regression model used in the calculation of the curve and includes a penalty term for smoothly changing. The curve estimation method according to any one of appendices 1 to 5.

(Appendix 7)
In calculating the curve,
Parameters relating to the interval between the center values of the basis functions in the regression model and parameters for controlling the spread of the basis functions so that the value of the generalized information criterion defined in relation to the parameter group is minimized. The curve estimation method according to appendix 6, wherein a regular parameter included in the penalty term is determined.

(Appendix 8)
The data storage unit further stores a transaction amount or a bid date for each issue of the interest-bearing bond,
In the process of extracting one issue of interest-bearing bonds for each maturity date,
The curve estimation method according to any one of appendices 1 to 7, wherein one interest rate bond is selected for each maturity date based on the transaction amount or the bidding date.

(Appendix 9)
In the process of extracting one issue of interest-bearing bonds for each maturity date,
When there are multiple interest-bearing bonds with the same maturity date, the interest, redemption amount, and market price of a hypothetical interest-bearing bond are calculated from the data of interest-bearing bonds of the multiple issues The curve estimation method according to any one of 8.

(Appendix 10)
The process of calculating the discount rate is:
In the generated matrix, the payment amount at the maturity date immediately after the payment date is calculated from the payment amount at the payment date other than the maturity date of the extracted interest-bearing bond issue, and for each of the extracted interest-bearing bond issues The curve estimation method according to any one of appendices 1 to 9, wherein the payment amount for each maturity date is converted into a regular matrix that lists the payment amount.

(Appendix 11)
The process of calculating the discount rate is:
In the generated matrix, immediately after the payment date using the first interest rate on the maturity date of the extracted interest-bearing bond issue from the payment amount at the payment date other than the maturity date of the extracted interest-bearing bond issue. To calculate a payment amount at the maturity date, and convert it into a regular matrix enumerating the payment amounts at the maturity date for each of the extracted interest-bearing bonds.
A second interest rate at the maturity date is calculated for each of the extracted interest-bearing bond issues from the product of the inverse matrix of the regular matrix and the extracted matrix including the market price for each interest-bearing bond issue. ,
The first interest rate is replaced with the second interest rate until the difference between the first interest rate and the second interest rate is equal to or less than a predetermined threshold value, and the transformation of the regular matrix and the second interest rate are performed. The curve estimation method according to any one of appendices 1 to 9, including a process of repeatedly executing the calculation.

(Appendix 12)
From the data storage unit that stores the maturity date, interest, redemption amount, and market price including paid interest for each issue of interest-bearing bonds, extract one issue of interest-bearing bonds for each maturity date,
From the maturity date, the interest, and the redemption amount for the extracted interest-bearing bond issue, generate a matrix listing the payment amount for each payment date for each extracted interest-bearing issue issue,
The generated matrix is converted into a regular matrix, and the discount on the maturity date for each of the extracted interest-bearing bond issues is extracted from the extracted matrix including the market price for each interest-bearing bond issue and the regular matrix. Execute the process to calculate the rate,
From the calculated discount rate on the maturity date, calculate the interest rate on the maturity date for each of the extracted interest-bearing bond issues,
From the calculated interest rate on the maturity date, let the computer execute a process of calculating a curve using a regression model,
The regression model represents an interest rate for a period corresponding to the maturity date as a weighted sum of a plurality of basis functions;
A curve estimation program in which center values of the plurality of basis functions are set according to a distribution of a period corresponding to the maturity date.

(Appendix 13)
An extractor for extracting one issue of interest-bearing bonds for each maturity date from a data storage unit for storing maturity date, interest, redemption amount and market price including paid interest for each issue of interest-bearing bonds;
A generation unit that generates a matrix that lists payment amounts at each payment period for each of the extracted interest-bearing bonds from the maturity date, the interest, and the redemption amount for the extracted interest-bearing bonds;
The generated matrix is converted into a regular matrix, and the discount on the maturity date for each of the extracted interest-bearing bond issues is extracted from the extracted matrix including the market price for each interest-bearing bond issue and the regular matrix. A discount rate calculation unit that executes processing for calculating a rate;
An interest rate calculation unit that calculates an interest rate on the maturity date for each of the extracted interest-bearing bond issues from the calculated discount rate on the maturity date;
A curve calculation unit that calculates a curve using a regression model from the calculated interest rate on the maturity date,
Have
The regression model represents an interest rate for a period corresponding to the maturity date as a weighted sum of a plurality of basis functions;
A curve estimation device in which center values of the plurality of basis functions are set according to a distribution of a period corresponding to the maturity date.

101 first data storage unit 102 extraction unit 103 second data storage unit 104 matrix generation unit 105 third data storage unit 106 discount rate calculation unit 113 fourth data storage unit 107 fifth data storage unit 108 spot rate calculation unit 109 sixth Data storage unit 110 Curve estimation unit 111 Seventh data storage unit 112 Output unit

Claims (13)

From the data storage unit that stores the maturity date, interest, redemption amount, and market price including paid interest for each issue of interest-bearing bonds, extract one issue of interest-bearing bonds for each maturity date,
From the maturity date, the interest, and the redemption amount for the extracted interest-bearing bond issue, generate a matrix listing the payment amount for each payment date for each extracted interest-bearing issue issue,
The generated matrix is converted into a regular matrix, and the discount on the maturity date for each of the extracted interest-bearing bond issues is extracted from the extracted matrix including the market price for each interest-bearing bond issue and the regular matrix. Execute the process to calculate the rate,
From the calculated discount rate on the maturity date, calculate the interest rate on the maturity date for each of the extracted interest-bearing bond issues,
Including a process of calculating a curve from the calculated interest rate on the maturity date using a regression model, and is executed by a computer,
The regression model represents an interest rate for a period corresponding to the maturity date as a weighted sum of a plurality of basis functions;
A curve estimation method in which center values of the plurality of basis functions are set according to a distribution of a period corresponding to the maturity date. The median value of the plurality of basis functions corresponds to the maturity date included in a range between adjacent median values in a range from 0 to a period corresponding to the latest maturity date. The curve estimation method according to claim 1, wherein the number of periods to be set is set to be equal. The parameter representing the spread of each of the plurality of basis functions is set according to the number of periods corresponding to the maturity date included in a predetermined range from the center value of the basis functions. Curve estimation method. The predetermined range includes a first range for values less than the center value of the basis function and a second range for values greater than the center value of the basis function;
A first parameter representing a spread on a value side smaller than the center value of the basis function and a second representing a spread on a value side larger than the center value of the basis function. Including the parameters of
The first parameter is set according to the number of periods corresponding to the maturity included in the first range, and the second parameter corresponds to the maturity included in the second range. The curve estimation method according to claim 3, which is set according to the number of periods to be performed. The curve estimation method according to claim 3, wherein a parameter representing a spread of each of the plurality of basis functions is determined according to a length between adjacent center values of the basis function. In calculating the curve,
The value of the parameter group including the weight of the plurality of basis functions is maximized, and the logarithmic likelihood is defined for the probability density function based on the regression model used in the calculation of the curve and includes a penalty term for smoothly changing. The curve estimation method according to any one of claims 1 to 5. In calculating the curve,
Parameters relating to the interval between the center values of the basis functions in the regression model and parameters for controlling the spread of the basis functions so that the value of the generalized information criterion defined in relation to the parameter group is minimized. The curve estimation method according to claim 6, wherein a regular parameter included in the penalty term is determined. The data storage unit further stores a transaction amount or a bid date for each issue of the interest-bearing bond,
In the process of extracting one issue of interest-bearing bonds for each maturity date,
The curve estimation method according to any one of claims 1 to 7, wherein one interest-bearing bond is selected for each maturity date based on the transaction amount or a bid date. In the process of extracting one issue of interest-bearing bonds for each maturity date,
When there are multiple issues of interest-bearing bonds with the same maturity date, the interest, redemption amount, and market price of a hypothetical interest-bearing bond are calculated from the data of interest-bearing bonds of the multiple issues. The curve estimation method as described in any one of thru | or 8. The process of calculating the discount rate is:
In the generated matrix, the payment amount at the maturity date immediately after the payment date is calculated from the payment amount at the payment date other than the maturity date of the extracted interest-bearing bond issue, and for each of the extracted interest-bearing bond issues The curve estimation method according to claim 1, wherein the payment amount for each maturity date is converted into a regular matrix that lists the payment amount. The process of calculating the discount rate is:
In the generated matrix, immediately after the payment date using the first interest rate on the maturity date of the extracted interest-bearing bond issue from the payment amount at the payment date other than the maturity date of the extracted interest-bearing bond issue. To calculate a payment amount at the maturity date, and convert it into a regular matrix enumerating the payment amounts at the maturity date for each of the extracted interest-bearing bonds.
A second interest rate at the maturity date is calculated for each of the extracted interest-bearing bond issues from the product of the inverse matrix of the regular matrix and the extracted matrix including the market price for each interest-bearing bond issue. ,
The first interest rate is replaced with the second interest rate until the difference between the first interest rate and the second interest rate is equal to or less than a predetermined threshold value, and the transformation of the regular matrix and the second interest rate are performed. The curve estimation method according to any one of claims 1 to 9, further comprising a process of repeatedly executing the calculation. From the data storage unit that stores the maturity date, interest, redemption amount, and market price including paid interest for each issue of interest-bearing bonds, extract one issue of interest-bearing bonds for each maturity date,
From the maturity date, the interest, and the redemption amount for the extracted interest-bearing bond issue, generate a matrix listing the payment amount for each payment date for each extracted interest-bearing issue issue,
The generated matrix is converted into a regular matrix, and the discount on the maturity date for each of the extracted interest-bearing bond issues is extracted from the extracted matrix including the market price for each interest-bearing bond issue and the regular matrix. Execute the process to calculate the rate,
From the calculated discount rate on the maturity date, calculate the interest rate on the maturity date for each of the extracted interest-bearing bond issues,
From the calculated interest rate on the maturity date, let the computer execute a process of calculating a curve using a regression model,
The regression model represents an interest rate for a period corresponding to the maturity date as a weighted sum of a plurality of basis functions;
A curve estimation program in which center values of the plurality of basis functions are set according to a distribution of a period corresponding to the maturity date. An extractor for extracting one issue of interest-bearing bonds for each maturity date from a data storage unit for storing maturity date, interest, redemption amount and market price including paid interest for each issue of interest-bearing bonds;
A generation unit that generates a matrix that lists payment amounts at each payment period for each of the extracted interest-bearing bonds from the maturity date, the interest, and the redemption amount for the extracted interest-bearing bonds;
The generated matrix is converted into a regular matrix, and the discount on the maturity date for each of the extracted interest-bearing bond issues is extracted from the extracted matrix including the market price for each interest-bearing bond issue and the regular matrix. A discount rate calculation unit that executes processing for calculating a rate;
An interest rate calculation unit that calculates an interest rate on the maturity date for each of the extracted interest-bearing bond issues from the calculated discount rate on the maturity date;
A curve calculation unit that calculates a curve using a regression model from the calculated interest rate on the maturity date,
Have
The regression model represents an interest rate for a period corresponding to the maturity date as a weighted sum of a plurality of basis functions;
A curve estimation device in which center values of the plurality of basis functions are set according to a distribution of a period corresponding to the maturity date.

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