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53 INTEREST TABLES AT !2% Constants  Function  Value   .120000  ,<2)  .116601  ,<4)  .114949  ,<12)  .113866   .113329   .107143   .110178   .111738   .112795   .113329   .892857  1/2  .944911   .972065  1/2  .990600   1.120000  (I + lV  1.058301  (1+/)!"  1.028737  (1+. 2  1.009489  . •0)  1.029150  j/,(4)  1.043938   1.053875   1.058867   1.089150   1.073938   1.063875   1.058867 
 v«  (1 + 1)"      .89286  1.12000  .8929  1.0000  1.000000   .79719  1.25440  1.6901  2.1200  .471698   .71178  1.40493  2.4018  3.3744  .296349   .63552  1.57352  3.0373  4.7793  .209234   .56743  1.76234  3.6048  6.3528  .157410   .50663  1.97382  4.1114  8.1152  .123226   .45235  2.21068  4.5638  10.0890  .099118   .40388  2.47596  4.9676  12.2997  .081303   .36061  2.77308  5.3282  14.7757  .067679   .32197  3.10585  5.6502  17.5487  .056984   .28748  3.47855  5.9377  20.6546  .048415   .25668  3.89598  6.1944  24.1331  .041437   .22917  4.36349  6.4235  28.0291  .035677   .20462  4.88711  6.6282  32.3926  .030871   .18270  5.47357  6.8109  37.2797  .026824   .16312  6.13039  6.9740  42.7533  .023390   .14564  6.86604  7.1196  48.8837  .020457   .13004  7.68997  7.2497  55.7497  .017937   .11611  8.61276  7.3658  63.4397  .015763   .10367  9.64629  7.4694  72.0524  .013879   .09256  10.80385  7.5620  81.6987  .012240   .08264  12.10031  7.6446  92.5026  .010811   .07379  13.55235  7.7184  104.6029  .009560   .06588  15.17863  7.7843  118.1552  .008463   .05882  17.00006  7.8431  133.3339  .007500   .05252  19.04007  7.8957  150.3339  .006652   .04689  21.32488  7.9426  169.3740  .005904   .04187  23.88387  7.9844  190.6989  .005244   .03738  26.74993  8.0218  214.5828  .004660   .03338  29.95992  8.0552  241.3327  .004144   .02980  33.55511  8.0850  271.2926  .003686   .02661  37.58173  8.1116  304.8477  .003280   .02376  42.09153  8.1354  342.4294  .002920   .02121  47.14252  8.1566  384.5210  .002601   .01894  52.79962  8.1755  431.6635  .002317   .01691  59.13557  8.1924  484.4631  .002064   .01510  66.23184  8.2075  543.5987  .001840   .01348  74.17966  8.2210  609.8305  .001640   .01204  83.08122  8.2330  684.0102  .001462   .01075  93.05097  8.2438  767.0914  .001304   .00960  104.21709  8.2534  860.1424  .001163   .00857  116.72314  8.2619  964.3595  .001037   .00765  130.72991  8.2696  1081.0826  .000925   .00683  146.41750  8.2764  1211.8125  .000825   .00610  163.98760  8.2825  1358.2300  .000736   .00544  183.66612  8.2880  1522.2176  .000657   .00486  205.70605  8.2928  1705.8838  .000586   .00434  230.39078  8.2972  1911.5898  .000523   .00388  258.03767  8.3010  2141.9806  .000467   .00346  289.00219  8.3045  2400.0182  .000417 
Appendix II Table numbering the days of the year For leap years the number of the day is one greater than the tabular number after February 28 » >< &     •c <    >> "3  <   1 u  E u >  &  «41                                                                                                                                                                                                                                                                                                                                                                                                                                                   
Appendix IV Statistical background A. Moments 1. The mean of a random variable X, denoted by , is die first moment about die origin m;. = E[X]. 2. The variance of a random variable X, denoted by or \ [ ], is die second moment about the mean ol = var[X] = E[X]  {E[X]f. 3. The standard deviation of a random variable X, denoted by tr, is the square root of the variance. 4. The covariance of two random variables X and y, denoted by otcov[X,Y], is defined as ay = cov[X, Y] = E[(X)(Y,y)] = E[XY\  E[X]EIY\ . 5. If die random variables X and are independent, dien EIXY] = E[X]E[Y] and ff  cov[X, F] = 0. 6. The mean of aX + bY, where a and b are constants, is given by + . 7. The variance of aX + bY, where a and b are constants, is given by aal + baj + labay . B. Distributions 1. Binomial distribution A discrete distribution defined by P4i pY Appendix IV < where n is a positive integer, 0 < p < 1, and x = 0, 1, 2, .... n. The moments are: mean = np variance = npq. 2. Uniform distribution A continuous distribution defined by Ax) = b  a where a < x < b. The moments are: a + b mean =   af variance = 3. Normal distribution A continuous distribution defined by where » < < oo, a> 0, and  » < x < ». The moments are: mean = variance = cr. Values of the cumulative distribution function are tabulated and used in finding probabiUties firom this distribution. 4. Lognormal distribution If = loggX and has a normal distiibution, dien X > 0 has a lognormal distribution. The moments are: mean = e variance = ˆ + \e!)• Note diat die mean and die variance a .
D. NewtonRaphson method 1. The iteration formula is given by 2. For this method g(r) = 0, which produces an extremely fast rate of convergence called "secondorder convergence." 3. This mediod does require diat f(x) can be computed and is nonzero. From the subject of numerical analysis, the first difference is defined by A/(0=/a+ 1)/(). Higher order differences can be developed by successively applying the above formula. A formula for summation by parts can be derived which is analogous to integration by parts i=n+l V 1 1 lb v1 (vlf (vlf t=n+l This formula will give practical results whenever higher order differences past a certain point can be safely ignored. In particular, if /(/) is a polynomial of degree m, dien (m + l)th and higher differences are all zero. Thus, this formula can be used to find the present value of varying annuities whose payments follow a polynomial. 5. This method is very simple to apply on a computer and convergence is guaranteed if ) is continuous. However, the rate of convergence is rather slow. C. Successive inverse interpolation 1. This method also requires two starting values, XQandxy, which have functional values of opposite sign. 2. The iteration formula is given by «/(« + i)>.+i/(„) +  ) 3. Two variations exist in applying this method after the first iteration. In the first variation the two most recently computed values of x for which the values of ) are of opposite sign are used. In the second variation die two most recendy computed values of x are used regardless of die sign of f[x). Further analysis of varying annuities
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