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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

(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 >

&

«4-1



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 = + \e-!)

Note diat die mean and die variance a .



D. Newton-Raphson method

1. The iteration formula is given by

2. For this method g(r) = 0, which produces an extremely fast rate of convergence called "second-order convergence."

3. This mediod does require diat f(x) can be computed and is non-zero.

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

l---b

v-1 (v-lf (v-lf

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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