INTEREST TABLES AT !2%

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

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. 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