Mathematics Science with Economics

Question: Describe about the Mathematics Science with Economics or formulas. Answer: 1. The force of interest rate for the year 6 to 12 using the given formula is shown below. Year Interest Rate 6 0.028 7 0.031 8 0.034 9 0.037 10 0.04 11 0.043 12 0.046 From t=13, the interest rate applicable is 0.09 Accumulated amount after 22 years = 9600*1.028*1.031*1.034*1.037*1.04*1.043*1.046* 1.0910 = 29304.66 2. Initial loan borrowed can be computed in the following manner Installment Amount paid PV factor Present value 1 1480 0.988631 1463.174 2 1440 0.977391 1407.443 3 1400 0.966279 1352.79 4 1360 0.955293 1299.198 5 1320 0.944432 1246.65 6 1280 0.933694 1195.129 7 1240 0.923079 1144.618 8 1200 0.912584 1095.101 9 1160 0.902209 1046.562 10 1120 0.891951 998.9854 11 1080 0.88181 952.3553 12 1040 0.871785 906.6563 13 1000 0.861873 861.8734 14 960 0.852075 817.9915 15 920 0.842387 774.9961 16 880 0.83281 732.8726 17 840 0.823341 691.6067 18 800 0.813981 651.1844 19 760 0.804726 611.5919 20 720 0.795577 572.8155 21 680 0.786532 534.8417 22 640 0.77759 497.6574 23 600 0.768749 461.2494 24 560 0.760009 425.605 25 520 0.751368 390.7115 26 480 0.742826 356.5563 27 440 0.73438 323.1274 28 400 0.726031 290.4124 29 360 0.717777 258.3996 30 320 0.709616 227.0771 31 280 0.701548 196.4335 32 240 0.693572 166.4573 Total 23952.12 Loan borrowed is 23952 (ii) The loan outstanding at the starting of the 4th year was 11291 (iii)Capital component of 13th instalment is 870 Installment Starting Balance Amount Interest Principal Paid Ending Balance 1 23952.00 1480.00 275.45 1204.55 22747.45 2 22747.45 1440.00 261.60 1178.40 21569.04 3 21569.04 1400.00 248.04 1151.96 20417.09 4 20417.09 1360.00 234.80 1125.20 19291.88 5 19291.88 1320.00 221.86 1098.14 18193.74 6 18193.74 1280.00 209.23 1070.77 17122.97 7 17122.97 1240.00 196.91 1043.09 16079.88 8 16079.88 1200.00 184.92 1015.08 15064.80 9 15064.80 1160.00 173.25 986.75 14078.05 10 14078.05 1120.00 161.90 958.10 13119.94 11 13119.94 1080.00 150.88 929.12 12190.82 12 12190.82 1040.00 140.19 899.81 11291.02 13 11291.02 1000.00 129.85 870.15 10420.86 3. At time t=0, Amount deposited = 1,500 By t=15, the above 1500 at the given rate of the interest would amount to = 1500*(1.085)15 = 5099.614 At time t=3, Amount deposited = 1,000 By t=15, the above 1000 at the given rate of the interest would amount to = 1000*(1.085)12 = 2661.686 At time t=5, Amount deposited = 800 By t=15, the above 800 at the given rate of the interest would amount to = 800*(1.085)10 = 1808.787 Total amount in the fund at time 15 = 5099.614 + 2661.686 + 1808.787= 9570.09 4. For the initial four years, the nominal interest rate is 4.9 per annum convertible monthly Hence, annual effective interest rate (Till fourth year) = [1+ (4.9/1200)]12 -1 = 5.01156 For the remaining years, the nominal interest rate is 8.8 per annum convertible quarterly Hence, annual effective interest rate (From fifth year onwards) = [1+ (8.8/400)]4 -1 = 9.09468 An amount of 21,200 is receivable after 13 years. Value of the above amount at the beginning of the 5th year = (21200)/(1+(9.09468/100)9) = 9685.089 Present value of the amount = 9685.089/(1+(5.01156/100)4) = 7964.44 5. For the first 6 years, the payment is at the rate of 1536 per annum Hence, monthly contribution in the sinking fund = 1536/12 = 128 This contribution is done for 12*6 = 72 months Monthly interest rate applicable = 6.5/12 = 0.541667 FV of these payments at the end of 6 years = 128[(1+0.00541667)72 -1]/ 0.00541667 = 11234.71 FV of the above money by the 27th year end = 11234.71 *(1+0.00541667)21*12 = 43830.26 FV of the payments made from 7th year onwards to the end of 24th year = (1823/12)[(1+0.00541667)18*12 -1]/ 0.00541667 = 62033.48 FV of the above money by the 27th year end = 62033.48*(1+0.00541667)3*12 = 75350.31 Value of investment at the end of 27 years = 43830.26 + 75350.31 = 119180.60 6. It is known that at t=3 years an amount of 8700 is paid. PV of the above amount = 8700/(1.074)3 = 7022.737 It is known that at t=12 years an amount of 19400 is paid. Value of the above amount at t = 8 is 19400/(1.037)8 = 14506.8 Value of 19400 at t=0 is 14506.8/(1.074)8 = 8194.782 It is known that at t=28 years an amount of 14700 is paid. Value of the above amount at t = 8 is 14700/(1.037)40 = 3436.9 Value of 14700 at t=0 is 3436.9/(1.074)8 = 1941.48 PV of the fund = 7022.737+8194.782+ 1941.48 = 17159 7. The PV of the given annuity is calculated using table shown below. Year Amount PV factor Present Value 1 1280 1.00 1280.00 2 1280 0.97 1239.11 3 1280 0.94 1199.53 4 1280 0.91 1161.21 5 1280 0.88 1124.11 6 1280 0.85 1088.20 7 1280 0.82 1053.44 8 1280 0.80 1019.78 9 1397 0.77 1077.44 10 1397 0.75 1043.02 11 1397 0.72 1009.70 12 1397 0.70 977.45 13 1397 0.68 946.22 14 1397 0.66 915.99 15 1397 0.63 886.73 16 1397 0.61 858.40 17 1397 0.31 439.20 18 1397 0.29 408.56 19 1397 0.27 380.05 20 1397 0.25 353.54 21 1397 0.24 328.87 22 1397 0.22 305.93 23 1397 0.20 284.58 24 1397 0.19 264.73 25 1397 0.18 246.26 Total 19892.05 Hence, the total amount of reserve the insurer needs today is 19892.05.