3. Simulation
3.5 Insurer's Activities
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3.5 Insurer's Activities
In this part we introduce the activities of the insurer step by step, and how does the activity affect the insurer's balance sheet. We divided each year t into two periods:
beginning period and end period. Assume that all activities happen only at the beginning or the end of each policy year.
At the beginning of each year t, the insurer receives premiums from the policyholders, simultaneously pays the commissions and the expenses. The insurer increases the reserve listed on the balance sheet accordingly, then makes an asset allocation based on their investment strategy.
At the end of year t, the insurer renews the asset prices marked to the market (i.e.
marked to the price of t+1). Then the insurer pays claims through selling the asset proportionally and deducts the reserve. In follow subsections we describe detailed descriptions of each activity.
We denotes as the value of asset, liability and shareholder's equity on the insurer's balance sheet at the beginning of year t.
And , and as the value of asset, liability and shareholder's equity on the insurer's balance sheet at the end of year t.
(a) At the Beginning Period
The activities of the beginning period includes: collecting premium, paying expenses and commissions, increasing reserve and allocating capitals. At the beginning period, the insurer receives the net premium:
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Then the insurer increases the reserve based on the premium already collected.
= at the beginning period.
The variation in reserve increases the liability value on the balance sheet:
Then the insurer allocates their capital (i.e. premium received) to the financial assets, which are bonds and stock, based on market price at year t. The initial capital of the insurer equals to the net premium:
Then the insurer decides its asset position according to the investment strategy:
, where is the weighted matrix for year t.
The market price the insurer observes at the beginning of the year t is denoted as follow:
, where denotes the stock price at year t, denotes the price of zero coupon bond with one-year maturity at year t, the price of zero coupon bond with two-year maturity at year t, etc.
The insurer then purchases the market assets by the weighted and the market price:
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：
, where is the volume of stocks purchased at the beginning period of year
t,
denotes the volume of zero coupon bonds with one-year maturity purchased at the beginning period of year, denotes the volume of zero coupon bonds with two-year maturity purchased at the beginning period of year, etc.Thus, after the beginning period of year t the insurer has the portfolio of , the asset value and the shareholder's equity value as follows:
(b) At the Ending period
At the end of each period the insurer needs to sell the assets to pay the claim. The insurer also needs to change the reserve value on the balance sheet, repurchase bonds, reports balance sheet and recognizes profit and loss of the financial assets held.
In the ending period, the insurer first receives free capital of one-year
zero-coupon bonds which purchased in the beginning period. Then the maturity of the zero-coupon bonds is reduced by one year. The temporary position of the ending period is expressed as:
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, where is the temporary position of the ending period.
Then the insurer starts to recognize the profit and loss of year t. The market price observed at the end of the year t is the market price of year t+1. The profit and loss of the assets is the changes of market prices between the beginning and the ending period times the insurer's asset position, which can be expressed as:
Then the insurer recognizes the loss of the liability:
The total profit and loss is therefore:
After recognizing the profit and loss, the insurer uses free capital received by matured bonds to repurchase the bonds of the longest maturity, which is 20 years.
Notice that the matured bonds are treated as available-for-sell items on the balance sheet. We therefore assume the repurchased bonds are treated as available-for-sell items. The new position of the 20 years zero-coupon bonds is:
, where is the face value of zero-coupon bonds.
Then the insurer should sell the assets to pay the claims. We denotes the expected
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This change the liability value in the balance sheet:
= at the ending period.
Then we assume that the insurer sells their asset in proportions so that asset allocation would not be affected by paying claims. The insurer therefore sells assets by the weighted: But notice that the current position of 20 years zero-coupon bond is zero (Since the insurer just purchases it), we transfer the short position of 20 years zero-coupon bond to the shortest maturity bonds, which is 1 years zero-coupon bond.
The short position for each asset is therefore adjusted as:
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This determines the final asset position of the insurer in the ending period:
Then if the insurer uses buy-and-hold strategy, the asset value is the position of available-for-sell items times the market price plus the amortized value of
hold-to-maturity items.
If the insurer uses duration matching strategy, the asset value at the ending period is:
We summarize the activities of each period in figure 5.
At the beginning of year t At the end of a year t
(market price = ) (market price = )
net premium collected bonds matured
reserve changed profit/loss recognition
=
asset allocation reserve changed
=
assets sold for claim
asset/liability recognition asset/liability recognition
Figure 5. Insurer's activities at each period
4. Simulation Results
In this section we present the results of our simulation. First we introduce the parameters we use in the models which are mentioned in previous sections, and
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several assumptions for the simulation. We then exhibit the simulation result of the balance sheet, which includes shareholder's equity and the profit and loss with the volatility and average. We also run several test for the sensitivity of interest rate, observe the consequences under different scenarios, including some extreme situations.
We adopt the parameter values and simulation set up used in Chan (2010) with some adjustment to simulate paths of annual interest rate and discount factors. They set the mean reverting speed . Consider that the financial market have suffered the low-level interest rate environment since 2008, we set and the initial interest rate as 1.5%.
And for the parameter of stock market, we refer to Yu, Tsai and Hunag (2010) to simulate the path of stock price. Initial stock price is set at 1,500, return rate of the stock price , stock volatility . We summarize the global parameters in table 5.
According to the , we can estimate a set of weighted that generate the efficient frontier portfolio. The result is shown in table 6.
The insurer seldom invests their capital on the stock over 10%, we therefore take for the buy-and-hold strategy.
As for duration matching, we fix the stock position at 0.1, and the lower bound of each financial asset is set as 1% which means the insurer should at least hold 1%
position of each financial asset. We also want to discuss about that what the
consequence is if an insurer uses wrong duration as the risk indicator. We therefore
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assume the insurer may use book-duration or fair-duration under both regulations. The results are presented in follow subsections.
Table 5. Parameters of the simulation
Items Value
Variable Expense rate Surrender rate
Table 6. The weighted of efficient frontier portfolio
0.888888888888889 0.111111111111111
4.1 Balance Sheet Results
First we show the results of our simulation on the balance sheet. We focus on two
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variables, shareholder's equity (E) and profit and loss (P/L), on the balance sheet since that the shareholders may care about these two items most. We observe several
statistic variables of these two items and compute the value at risk (VaR) under different circumstances.
(a) Equity
We calculate the average and the standard deviation of the equity at the end of each year. The result is shown in figure 6 and table 7.
Figure 6. Average equity under different reserve regulation (n=10,000)
Table 7. The numerical result of average equity under different reserve regulation (n=10,000)
Average E under Book Reserve ($) Average E under Fair Reserve ($)
Year Buy-and-Hold Book-Duration Fair-Duration Year Buy-and-Hold Book-Duration Fair-Duration
0 555,970,378 550,992,191 550,992,191 0 528,985,136 524,006,950 524,006,950
1 571,265,600 582,963,398 583,021,088 1 535,841,733 547,539,531 547,597,221
2 575,470,159 594,109,481 594,168,355 2 537,257,431 555,896,753 555,955,627
3 575,985,106 600,417,253 600,477,115 3 535,137,396 559,569,543 559,629,405
4 571,815,230 600,998,816 601,059,215 4 528,574,542 557,758,127 557,818,527
5 564,234,455 598,726,438 598,787,408 5 518,866,371 553,358,354 553,419,324
6 552,480,899 592,761,883 592,823,097 6 505,289,079 545,570,062 545,631,276
0 5 10 15 20 25
Average Equity under Book value Duration (at the end)
Fair-Duration
Average Equity (at the end)
Average Equity under Fair value Duration (at the end)
Fair-Duration Book-Duration Buy-and-Hold
‧ Average E under Book Reserve ($) Average E under Fair Reserve ($)
7 536,833,176 582,238,459 581,606,001 7 488,166,476 533,571,759 532,939,301
8 516,709,530 567,670,813 567,310,694 8 466,970,862 517,932,145 517,572,026
9 493,188,033 550,787,440 550,142,917 9 442,801,772 500,401,179 499,756,656
10 466,687,841 529,212,202 529,158,954 10 416,151,908 478,676,269 478,623,021
11 435,650,382 503,312,986 504,001,730 11 385,568,136 453,230,740 453,919,484
12 400,489,833 473,659,968 474,496,319 12 351,556,187 424,726,322 425,562,673
13 361,871,434 441,114,805 443,262,318 13 314,803,810 394,047,181 396,194,694
14 319,179,498 405,003,957 407,611,225 14 274,893,083 360,717,541 363,324,810
15 272,273,182 365,021,261 368,089,362 15 231,760,969 324,509,048 327,577,149
16 221,523,557 323,488,647 326,443,912 16 185,995,655 287,960,746 290,916,011
17 166,360,071 277,331,405 279,529,813 17 137,122,433 248,093,766 250,292,175
18 106,488,881 226,280,193 228,500,894 18 85,165,482 204,956,793 207,177,495
19 89824512.34 171,811,167 174045898.6 19 29,066,412 160,156,848 162,391,580
20 79512475.54 112,068,641 114326364.3 20 19,066,412 112,068,641 114,326,364
The results show that the equity pattern is unaffected by the allocation strategy. It is only affected by the reserve calculation method, if the reserve is calculated in fair value, which is relatively higher, the equity would be lesser and vice versa.
The volatility of equity is shown in figure 7 and table 8.
Figure 7. Volatility of equity under different reserve regulation (n=10,000)
0 5 10 15 20 25
Volatility of E under Book value Reserve
Fair-Duration
Volatility of E under Fair value Reserve
Fair-Duration Book-Duration Buy-and-Hold
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Fair-duration represents insurers with fair-duration matching ALM strategy, Book-duration represents insurers with fair-duration matching ALM strategy and Buy-and-hold is the insurers without ALM.
Table 8. The numerical result of volatility equity under different reserve regulation (n=10,000)
Volatility of E under Book Reserve ($) Volatility of E under Fair Reserve ($)
Year Buy-and-Hold Book-Duration Fair-Duration Year Buy-and-Hold Book-Duration Fair-Duration
0 555,970,378 550,992,191 550,992,191 0 528,985,136 524,006,950 524,006,950
1 571,265,600 582,963,398 583,021,088 1 535,841,733 547,539,531 547,597,221
2 575,470,159 594,109,481 594,168,355 2 537,257,431 555,896,753 555,955,627
3 575,985,106 600,417,253 600,477,115 3 535,137,396 559,569,543 559,629,405
4 571,815,230 600,998,816 601,059,215 4 528,574,542 557,758,127 557,818,527
5 564,234,455 598,726,438 598,787,408 5 518,866,371 553,358,354 553,419,324
6 552,480,899 592,761,883 592,823,097 6 505,289,079 545,570,062 545,631,276
7 536,833,176 582,238,459 581,606,001 7 488,166,476 533,571,759 532,939,301
8 516,709,530 567,670,813 567,310,694 8 466,970,862 517,932,145 517,572,026
9 493,188,033 550,787,440 550,142,917 9 442,801,772 500,401,179 499,756,656
10 466,687,841 529,212,202 529,158,954 10 416,151,908 478,676,269 478,623,021
11 435,650,382 503,312,986 504,001,730 11 385,568,136 453,230,740 453,919,484
12 400,489,833 473,659,968 474,496,319 12 351,556,187 424,726,322 425,562,673
13 361,871,434 441,114,805 443,262,318 13 314,803,810 394,047,181 396,194,694
14 319,179,498 405,003,957 407,611,225 14 274,893,083 360,717,541 363,324,810
15 272,273,182 365,021,261 368,089,362 15 231,760,969 324,509,048 327,577,149
16 221,523,557 323,488,647 326,443,912 16 185,995,655 287,960,746 290,916,011
17 166,360,071 277,331,405 279,529,813 17 137,122,433 248,093,766 250,292,175
18 106,488,881 226,280,193 228,500,894 18 85,165,482 204,956,793 207,177,495
19 89824512.34 171,811,167 174045898.6 19 29,066,412 160,156,848 162,391,580
20 79512475.54 112,068,641 114326364.3 20 19,066,412 112,068,641 114,326,364
The result shows that the equity volatility is significant high for fair-duration matching insurer under book value reserve while the equity volatility is relatively low for fair-duration matching insurer under fair value reserve. There is no significant difference between book-duration matching and buy-and-hold strategy.
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We assume that the distribution of equity is normal distributed, the value at risk at 95% confidence level is shown in table 9.
Table 9. The numerical result of VaR equity under different reserve regulation (α=95%)
VaR of E under Book Reserve ($) VaR of E under Fair Reserve ($)
Year Buy-and-Hold Book-Duration Fair-Duration Year Buy-and-Hold Book-Duration Fair-Duration
0 528,430,869.69 531,318,156.89 520,724,446.12 0 490,449,831.40 502,214,173.07 504,938,270.12 1 539,468,358.88 557,367,687.73 543,760,934.82 1 493,067,077.92 519,187,359.46 522,863,324.47 2 540,921,479.42 563,599,580.82 547,317,827.53 2 491,990,728.32 522,101,171.20 526,439,794.73 3 538,797,656.24 565,719,072.90 547,160,313.54 3 486,974,512.91 521,134,635.86 526,039,820.12 4 532,171,483.50 562,625,495.81 542,090,612.57 4 478,531,245.20 515,252,295.78 520,668,306.94 5 523,017,995.05 557,448,248.16 535,361,443.88 5 466,151,566.21 507,634,820.81 513,460,966.81 6 508,604,517.41 547,957,444.88 523,998,761.68 6 450,312,421.03 495,940,531.02 502,271,945.05 7 490,476,905.17 534,777,808.27 508,552,729.52 7 431,287,005.24 480,999,960.98 486,915,739.73 8 466,708,903.27 516,685,380.12 488,900,858.43 8 407,458,459.02 461,455,973.18 468,173,829.46 9 440,137,841.31 497,633,817.87 468,703,186.38 9 381,965,248.44 441,523,320.43 448,449,625.57 10 410,322,824.85 472,910,900.91 443,608,786.73 10 353,729,766.87 416,311,750.88 424,726,415.74 11 375,840,987.44 448,860,295.47 416,873,162.73 11 322,193,735.20 392,913,913.47 399,028,486.53 12 338,096,897.03 416,738,606.64 384,649,514.86 12 287,752,400.38 361,674,967.63 368,959,186.24 13 296,196,166.27 386,033,952.72 353,356,714.85 13 250,603,966.45 333,034,544.74 339,554,164.03 14 251,064,075.25 348,012,730.12 317,483,289.11 14 212,028,675.35 297,588,798.03 306,544,210.20 15 202,606,413.70 310,529,872.36 279,721,355.07 15 170,374,633.81 264,149,356.15 271,905,304.68 16 149,362,782.94 268,553,535.21 239,827,564.62 16 126,707,689.67 227,109,544.88 236,347,711.87 17 91,519,317.85 223,448,179.32 194,403,455.60 17 78,045,779.69 188,407,732.08 196,662,569.56 18 30,309,379.19 173,286,049.69 146,806,329.49 18 25,213,594.54 146,255,588.89 155,709,919.26 19 13,472,809.75 116,459,686.72 90,188,187.75 19 -35,734,550.03 98,844,439.33 109,561,222.04 20 2,461,259.47 54,107,272.02 27,410,773.76 20 -57,984,803.62 47,865,278.46 59,569,542.27(b) Profit and Loss
The result of average profit and loss at each year is shown in figure 8, and the numerical result is shown in table 10. We consider that the last year of the simulation may have some simulation errors since we do not have any allocation strategy and activity at year 20 in the simulation. We therefore exclude the profit and loss result of
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Figure 8. Average P/L under different reserve regulation (n=10,000)
Table 10. The numerical result of average P/L under different reserve regulation (n=10,000)
Average P/L under Book Reserve ($) Average P/L under Fair Reserve ($)
Year Buy-and-Hold Book-Duration Fair-Duration Year Buy-and-Hold Book-Duration Fair-Duration
0 18,868,805 17,805,477 17,805,477 0 18,868,805 18,413,868 18,678,223
1 9,651,775 10,345,043 10,425,825 1 3,052,092 8,296,659 10,295,685
2 -13,177,025 -12,146,881 -12,146,462 2 -13,464,638 -3,352,836 539,555
3 -16,557,732 -15,262,311 -15,262,189 3 -16,470,047 -5,325,730 -1,104,702
4 -20,776,364 -19,241,641 -19,241,999 4 -20,227,461 -7,858,772 -3,215,858
5 -21,712,514 -19,781,265 -19,781,602 5 -20,695,738 -7,910,432 -3,258,896
6 -23,570,264 -21,329,476 -21,330,166 6 -22,066,299 -8,674,368 -3,895,721
7 -25,579,827 -23,131,070 -23,126,693 7 -23,565,249 -9,603,874 -4,667,269
8 -27,772,975 -25,247,028 -25,236,916 8 -25,219,173 -10,739,123 -5,609,868
9 -29,102,974 -26,238,726 -26,214,578 9 -26,018,356 -11,070,958 -5,877,976
10 -30,220,832 -27,004,297 -26,997,265 10 -26,563,304 -11,209,674 -6,003,842
11 -33,762,655 -30,445,232 -30,510,320 11 -29,463,901 -13,225,464 -7,726,940
12 -35,884,569 -32,573,531 -32,586,193 12 -30,898,456 -14,262,941 -8,560,048
13 -37,364,092 -33,720,148 -33,779,530 13 -31,721,116 -14,615,563 -8,881,933
14 -39,471,687 -35,712,044 -35,662,333 14 -33,029,539 -15,474,326 -9,532,112
0 2 4 6 8 10 12 14 16 18
Average P/L under Book value Reserve
Fair-Duration
Average P/L under Fair value Reserve
Fair-Duration Book-Duration Buy-and-Hold
‧ Average P/L under Book Reserve ($) Average P/L under Fair Reserve ($)
15 -41,702,102 -37,988,472 -38,024,038 15 -34,455,954 -16,534,474 -10,466,734
16 -43,545,017 -39,454,659 -39,459,576 16 -35,359,158 -16,913,536 -10,764,231
17 -45,908,212 -42,003,121 -41,992,222 17 -36,788,102 -18,075,768 -11,723,267
18 -48,528,566 -44,218,761 -44,230,433 18 -38,267,650 -18,849,649 -12,381,711
19 -51,252,864 -46,880,404 -46,882,693 19 -39,859,909 -19,950,963 -13,293,842
20 19,837,562 24,125,550 24,128,606 20 19,837,562 22,970,396 22,477,163
The Average P/L shows that the P/L under buy-and-hold strategy is the averagely lowest under both reserve regulation while the P/L under fair duration matching strategy is the averagely highest. For more specification information we observe the volatility of P/L in figure 9 and table 11.
Figure 9. P/L volatility under different reserve regulation (n=10,000)
Table 11. The numerical result of P/L volatility under different reserve regulation (n=10,000)
The Volatility of P/L under Book value Reserve ($) The Volatility of P/L under Fair value Reserve ($) Year Buy-and-Hold Book-Duration Fair-Duration Year Buy-and-Hold Book-Duration Fair-Duration
0 14,000,507 11,436,776 15,249,035 0 14,000,507 11,436,736 9,149,389
1 19,848,709 16,643,978 22,140,103 1 19,849,138 16,629,516 13,302,424
2 20,561,303 17,990,950 23,992,321 2 20,523,768 17,946,211 14,359,933
3 21,514,966 19,418,621 25,894,162 3 21,412,827 19,322,360 15,459,499
0 5 10 15 20 25
The Volatility of P/L under Book value Reserve
Fair-Duration
The Volatility of PL under Fair-value reserve
Buy-andHold Book-Duration Fair-Duration
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4 22,665,191 20,818,536 27,763,157 4 22,495,196 20,662,226 16,532,873
5 23,596,385 22,116,225 29,492,775 5 23,336,222 21,880,015 17,506,693
6 24,470,462 23,330,482 31,110,420 6 24,149,293 23,038,049 18,432,315
7 25,134,982 24,311,173 32,342,317 7 24,719,701 23,940,672 19,110,209
8 25,989,079 25,403,468 33,773,998 8 25,519,584 24,991,111 19,935,657
9 27,050,289 26,502,926 35,071,049 9 26,449,888 25,995,526 20,642,359
10 27,562,264 27,034,139 35,710,975 10 26,944,532 26,534,742 21,035,545
11 28,660,645 25,652,710 36,903,647 11 27,863,540 25,150,998 21,656,232
12 28,289,672 27,953,876 37,213,876 12 27,462,404 27,322,775 21,827,351
13 28,667,717 25,660,062 36,487,008 13 27,849,639 25,270,742 21,470,170
14 29,187,734 28,637,753 37,566,557 14 28,279,176 28,006,326 22,068,511
15 29,822,995 26,820,726 38,044,100 15 28,666,616 26,335,391 22,290,388
16 29,632,644 28,065,341 37,269,599 16 28,600,724 27,492,826 21,925,800
17 29,346,545 26,836,483 36,339,977 17 28,467,263 26,685,111 21,634,550
18 29,625,865 26,753,125 35,725,575 18 28,748,006 26,810,058 21,482,092
19 29,486,769 27,160,127 36,277,854 19 28,601,042 27,263,854 21,851,683
20 28,512,511 27,010,926 36,081,119 20 28,512,511 27,010,926 21,648,671
We can observe that the volatility of P/L is affected by the allocation strategy significantly. Under book value reserve, fair-duration matching insurer has higher volatility of P/L while the book-duration matching insurer has more stable P/L. This is reasonable since the fair-duration matching insurer mismatch the asset duration with the liability duration listed on the balance sheet. The book-duration matching insurer manage their interest rate risk appropriately by descent risk indicator, therefore the P/L volatility of such insurers is relatively lower. It seems insignificant between book-duration matching insurers and buy-and-hold strategy insurers.
Similarly we assume that the distribution of P/L is normal distributed, the value at risk at 95% confidence level is shown in table 12.
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Table 12. The numerical result of VaR P/L under different reserve regulation (α=95%)
VaR of P/L under Book Reserve ($) VaR of P/L under Fair Reserve ($)
Year Buy-and-Hold Book-Duration Fair-Duration Year Buy-and-Hold Book-Duration Fair-Duration
0 -9,132,209 -5,068,075 -12,692,593 0 -9,132,209 -4,459,604 379,445
1 -30,045,643 -22,942,914 -33,854,380 1 -36,646,183 -24,962,372 -16,309,162
2 -54,299,630 -48,128,780 -60,131,104 2 -54,512,174 -39,245,257 -28,180,311
3 -59,587,664 -54,099,554 -67,050,512 3 -59,295,700 -43,970,449 -32,023,699
4 -66,106,746 -60,878,713 -74,768,313 4 -65,217,853 -49,183,224 -36,281,603
5 -68,905,284 -64,013,715 -78,767,153 5 -67,368,183 -51,670,462 -38,272,282
6 -72,511,188 -67,990,440 -83,551,005 6 -70,364,885 -54,750,466 -40,760,351
7 -75,849,791 -71,753,416 -87,811,326 7 -73,004,651 -57,485,218 -42,887,687
8 -79,751,132 -76,053,963 -92,784,911 8 -76,258,341 -60,721,345 -45,481,183
9 -83,203,552 -79,244,578 -96,356,676 9 -78,918,133 -63,062,010 -47,162,693
10 -85,345,359 -81,072,574 -98,419,215 10 -80,452,368 -64,279,157 -48,074,932
11 -91,083,945 -81,750,651 -104,317,614 11 -85,190,982 -63,527,460 -51,039,405 12 -92,463,913 -88,481,284 -107,013,946 12 -85,823,264 -68,908,491 -52,214,750 13 -94,699,526 -85,040,271 -106,753,547 13 -87,420,393 -65,157,047 -51,822,272 14 -97,847,154 -92,987,549 -110,795,447 14 -89,587,891 -71,486,977 -53,669,133 15 -101,348,093 -91,629,923 -114,112,238 15 -91,789,185 -69,205,256 -55,047,510 16 -102,810,305 -95,585,340 -113,998,774 16 -92,560,605 -71,899,189 -54,615,831 17 -104,601,302 -95,676,087 -114,672,176 17 -93,722,629 -71,445,990 -54,992,367 18 -107,780,295 -97,725,010 -115,681,584 18 -95,763,663 -72,469,765 -55,345,895 19 -110,226,402 -101,200,659 -119,438,402 19 -97,061,993 -74,478,672 -56,997,209
20 -37,187,460 -29,896,301 -48,033,632 20 -37,187,460 -31,051,455 -20,820,179
4.2 Sensitivity Test Results
We test several interest rate scenarios on each strategy under both reserve regulations. We assume that the long-term interest rate level varies as 1.5%, 2.75%, 4% and 6% while the volatility of interest rate varies from 1% to 4%. The other assumptions remain unchanged. We intend to compare each allocation strategy's performance in different scenarios under each reserve method. We only focus on the
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volatility of P/L since that the volatility of P/L is the most significant indicator in previous sector. We find out two results as follows:
(a) Result 1
We compare the impact of different interest rate volatility under a given long term interest rate . We find out that the Buy-and-Hold strategy is more sensitivity to the interest rate risk than duration matching strategy under both reserve regulations.
The result is shown in figure 10 and figure 11.
Figure 10. Volatility of P/L under book value reserve with different σr
The duration-matching in the figures denotes the fair-value duration matching.
The P/L volatility pattern is not significant difference between the fair-value duration matching and book-value duration matching. We therefore only take fair-value duration as an example to compare with the buy-and-hold strategy.
0 5 10 15 20 25
Volatility of P/L under Book value Reserve (mu r=2.75%)
Volatility of P/L under Book value Reserve (mur=2.75%)
sigmar=1%
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Figure 11. Volatility of P/L under fair value reserve with different σr
(b) Result 2
We also find out that the average equity is only affected by the interest rate in our simulation. The result is shown in figure 12.
Figure 12. Average equity under fair reserve in each strategy
We compare the result under fair reserve because of the interest rate scenario do
0 5 10 15 20 25
Volatility of P/L under Fair value Reserve (mur=2.75%)
sigmar=1%
Volatility of P/L under Fair value Reserve (mur=2.75%)
sigmar=1%
The Average Equity under Fair value Reserve
Fair-Duration
Average Equity under Fair value Reserve
Fair-Duration Buy-and-Hold mur=6%
sigmar=4%
Buy-and-Hold Duration-Matching
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not affect book value reserve estimation. The left figure shows the result of low-level interest rate scenario, which is given . The left figure shows the result of high-level interest rate scenario, which is given . The allocation does not affect the pattern of average equity. The only factor affects the equity is the reserve calculation method and the discount rate used. We can therefore confirm that the equity is unaffected by the allocation strategy. This is not consistent with practical situation. We consider that there may be some limitations in our simulation.
5. Conclusions
In this article we compare the impact of fair value reserve between two kinds of insurer via simulation methods. We assume that the insurer with ALM adopts duration matching while the insurer without ALM uses buy-and-hold strategy. We simulate 10,000 scenarios of interest rate and stock price to compare the average and volatility of the shareholder’s equity and profit/loss. We also observe the results under different interest rate parameters as a sensitivity test to strengthen the consequences.
From the result we can observe that if the insurer matches book-duration under book reserve, they face the most stable profit and loss. This is reasonable since they manage their interest risk using appropriate risk indicator. Contrary the ALM insurer who matches fair-duration faces even more volatility than the insurer without ALM under book value reserve. They may overestimate the volatility of interest rate presented on the balance sheet and hedge them unnecessary. Interestingly even the insurer without ALM has more stable profit and loss then the ALM insurer with fair duration matching under book value reserve. This implies the insurer still suffers great volatility under management if the regulations do not support such behavior.
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Under fair value reserve, the insurer with ALM has relatively more stable profit and loss than the insurer without ALM. And the ALM insurers who match
fair-duration faces less volatility than the ALM insurers who match book-reserve since they use decent risk indicator. This is consistent with our expectation. According to the sensitivity test we can also observe that the insurer with ALM faces less effect when the volatility of the interest rate rises, which proves that the insurer should manage their assets well while facing the interest risk. The result also imply that after the adoption of IFRS4 in Taiwan, the insurers should allocate their asset according to their liability duration in order to maintain their solvency.
Such consequence is not observed in equity. The change of the equity is affected by the reserve method and the discount rate. Different strategy in different regulation shows little effect on the shareholder's equity. This may be resulted from the
simulation assumptions since we only consider one product line of the insurer and the simulation length is limited. We also do not consider the new booking policies of the insurer. This leaves an unfinished issue for further research which aims to test the impact of IFRS4 adoption in Taiwan.
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References
Ahlgrim, K. C., D'Arcy, S. P., and Gorvett, R.W., 2004, The Effective Duration and Convexity of Liabilities of Property-Liability Insurers under Stochastic Interest Rates.
Geneva Papers on Risk and Insurance Theory 29, 75-108.
Ballota, L., Esposito, G., and Haberman, S., 2006, The IASB Insurance Project for Life Insurance Contracts: Impact on Reserving Methods and Solvency Requirements.
Mathematics and Economics 39, 356–375.
Chan, F. S., 2009, Asset and liability management for life insurers: effective duration and effective convexity analysis and simulation optimization, Ph.D. dissertation from
National Chengchi University (Taiwan).
Cox, J. C., Ingersoll, J. E., and Ross, S. A., 1985, A Theory of the Term Structure of Interest Rate. Econometrica, 53 (2) 385-407.
Honegger, R., and C. Mathis, 1993, Duration of Life Insurance Liabilities and Asset Liability Management, Working paper presented at the AFIR Colloquium in Rome.
Leibowitz, M. L., 1995, Total Portfolio Duration: A New Perspective on Asset Allocation, Financial Analysts Journal 51 (1), 139-144.
Markowitz, H. M., 1952, Portfolio Selection. The Journal of Finance 7 (1), 77–91.
Sharpe, W. F., and Lawrence, G. T., 1990, Liabilities: A New Approach. The Journal
of Portfolio Management, 5-10.
Tsai, C., 2009, The Term Structure of Reserve Durations and the Duration of Aggregate Reserves. The Journal of Risk and Insurance 72 (2), 419-441.
Waring, M. B., 2004, Liability-Relative Investing. The Journal of Portfolio
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
58
Management 30 (4), 8-20.
Waring, M. B., 2004, Liability-Relative Investing II. The Journal of Portfolio
Management 31 (1), 40-53.
Yu, T. Y., Tsai, C., and Huang. H. T., 2010, Applying Simulation Optimization to the Asset Allocation of a Property-Casualty Insurer. European Journal of Operational
Research 207, 499-507.
Yu, T. Y., Tsai, C., Huang. H. T., and Chen. C. L., 2011, Applying Simulation
Optimization to Dynamic Financial Analysis for the Asset–Liability Management of a Property–Casualty Insurer. Applied Financial Economics 21, 505-518.