An annuity is a series of identical payments occurring at equal time intervals. When the payments appear at the end of each time period, the annuity is said to be an ordinary annuity. Ordinary annuity is thus a series of payments paid to cover some sort of expense. Time horizon typically refers to the time period associated with accomplishing an investment objective. Time horizons can be short term (1 to 4 years), medium term (5 to 10 years) or long term (more than 10 years) . The following timeline represent an ordinary annuity comprised of five payments of $100 each:

$100 $100 $100 $100 $100

0 1 2 3 4 5 t=5 and i=10%

The equal periods of time (represented by t) between the identical payments of $100 could be a year, a 6-month period, a quarter of a year, a month, etc. In the above example, t = 5 periods of one year each. The interest rate (represented by i) is used to discount the $100 payments to time period 0. The interest rate might be the company's required rate, its target rate, its cost of capital, etc. The $100 amounts are often represented by either the letters "PMT" (for "payment" or receipt).

Immediate annuity is the cash flows start immediately, as in a saving plan or a lease. It is an investment that provides a regular income over an agreed period in exchange for a lump sum payment and also can be purchased with ordinary money or with superannuation money. The graph below shows the time horizon for an immediate annuity:

$100 $100 $100 $100 $100

0 1 2 3 4 5

FV=PMT [(1+i) ^n – 1 / i]

An immediate annuity can be described a short-term annuities or a single-premium immediate annuities that pays out for a specific period of time, or it may reference a multi-year guaranteed annuity .

Furthermore, there are three types of immediate annuities :1) Lifetime; 2)Term Certain;3)Complying Annuity. Lifetime is the payment continues for the investors’ entire life regardless of how long they live. Time certain is the payment continues for a pre-determined period. And complying Annuity must be payable for a lifetime and it cannot be withdrawn within six months.

An annuity is a series of equal cash flows, equally distributed over time . A regular annuity is simply an annuity where the first payment is made at the end of the period. We can calculate the regular payment of annuities by the following formula:

Annuities involve a set of identical payments over a given number of periods, many financial decision involve cash flows that are irregular. These irregular cash flows consist of a series of annuity payment plus an additional final lump sum in Year N, and all other irregular streams .

The first type of irregular is the bonds payment of annuities (shown in the graph below). For example, it is a 5-year, 12%, ordinary annuity plus a final payment of $1000, then we could use Excel’s PV function to find PV of the first type irregular payment, =PV(I,N,PMT,FV)=PV(0.12,5,100,1000)=-$927.90

The second type of irregular payment is stocks and capital investment (illustrates below). In order to find the present value of the second type of irregular payment, we can calculate the PVs of the cash flow stream which equals the sum of the individual PVs or PVs of the irregular cash flows stream. Also we can calculate the FV of the cash flow stream which equals the sum of the cash flows’ FVs or the FV of the stream .

The PV of irregular payment: $89.29+$239.16+$213.53+$190.66+$283.71= $1,016.35

The FV of irregular payment: $0+$157.35+$421.48+$376.32+$336.00+$500.00=$1,791.15

PVs of the CFs: $89.29 $239.16 $213.53 $190.66 $283.71 FV of each CF: $0 $157.35 $421.48 $376.32 $336.00 $500.00…