Editing Actuarial notation (section)

===Interest rates===
<math>\,i</math> is the annual [[effective interest rate]], which is the "true" rate of interest over ''a year''. Thus if the annual interest rate is 12% then <math>\,i = 0.12</math>.

<math>\,i^{(m)}</math> (pronounced "i ''upper'' m") is the [[nominal interest rate]] convertible <math>m</math> times a year, and is numerically equal to <math>m</math> times the effective rate of interest over one <math>m</math><sup>''th''</sup> of a year. For example, <math>\,i^{(2)}</math> is the nominal rate of interest convertible semiannually. If the effective annual rate of interest is 12%, then <math>\,i^{(2)}/2</math> represents the effective interest rate every six months. Since <math>\,(1.0583)^{2}=1.12</math>, we have <math>\,i^{(2)}/2=0.0583</math> and hence <math>\,i^{(2)}=0.1166</math>. The <sup>"(m)"</sup> appearing in the symbol <math>\,i^{(m)}</math> is not an "[[exponent]]." It merely represents the number of interest conversions, or compounding times, per year. Semi-annual compounding, (or converting interest every six months), is frequently used in valuing [[Bond (finance)|bonds]] (see also [[fixed income securities]]) and similar [[monetary financial liability]] instruments, whereas home [[mortgages]] frequently convert interest monthly. Following the above example again where <math>\,i=0.12</math>, we have <math>\,i^{(12)}=0.1139</math> since <math>\,\left(1+\frac{0.1139}{12}\right)^{12}=1.12</math>.

Effective and nominal rates of interest are not the same because interest paid in earlier measurement periods "earns" interest in later measurement periods; this is called [[compound interest]]. That is, nominal rates of interest credit interest to an investor, (alternatively charge, or [[debit]], interest to a debtor), more frequently than do effective rates. The result is more frequent compounding of interest income to the investor, (or interest expense to the debtor), when nominal rates are used.

The symbol <math>\,v</math> represents the [[present value]] of 1 to be paid one year from now:
:<math>\,v = {(1+i)}^{-1}\approx 1-i+i^2</math>
This present value factor, or discount factor, is used to determine the amount of money that must be invested now in order to have a given amount of money in the future. For example, if you need 1 in one year, then the amount of money you should invest now is: <math>\,1 \times v</math>. If you need 25 in 5 years the amount of money you should invest now is: <math>\,25 \times v^5</math>.

<math>\,d</math> is the [[annual effective discount rate]]:
:<math>d = \frac{i}{1+i}\approx i-i^2</math>
The value of <math>\,d</math> can also be calculated from the following relationships: <math>\,(1-d) = v = {(1+i)}^{-1}</math>
The rate of discount equals the amount of interest earned during a one-year period, divided by the balance of money at the end of that period. By contrast, an annual effective rate of interest is calculated by dividing the amount of interest earned during a one-year period by the balance of money at the beginning of the year. The present value (today) of a payment of 1 that is to be made <math>\,n</math> years in the future is <math>\,{(1-d)}^{n}</math>. This is analogous to the formula <math>\,{(1+i)}^{n}</math> for the future (or accumulated) value <math>\,n</math> years in the future of an amount of 1 invested today.

<math>\,d^{(m)}</math>, the nominal rate of discount convertible <math>\,m</math> times a year, is analogous to <math>\,i^{(m)}</math>. Discount is converted on an <math>m</math><sup>''th''</sup>-ly basis.

<math>\,\delta</math>, the [[force of interest]], is the limiting value of the nominal rate of interest when <math>m</math> increases without bound:

:<math>\,\delta = \lim_{m\to\infty}i^{(m)}</math>

In this case, interest is [[continuously compounded interest|convertible continuously]].

The general relationship between <math>\,i</math>, <math>\,\delta</math> and <math>\,d</math> is:

:<math>\,(1+i) = \left(1+\frac{i^{(m)}}{m}\right)^{m} = e^{\delta} = \left(1-\frac{d^{(m)}}{m}\right)^{-m} = (1-d)^{-1}</math>

Their numerical value can be compared as follows:

:<math>\, i > i^{(2)} > i^{(3)} > \cdots > \delta > \cdots > d^{(3)} > d^{(2)} > d</math>