# Investments

## Investments and Loans

When dealing with investments or loans, several cash flows are scheduled within a certain time frame, such as the

• present value (the value at the beginning of the scheduled time frame),

• future value (the value at the end of the scheduled time frame), and

• periodic payments during the scheduled time frame.

AIMMS provides several functions to calculate each of these cash flows (or the interest rate used) in the presence of all others.

## Constant Payments

Investments and loans with constant, periodic payments and a constant interest rate are special. When the payments are annual, such an investment is called an annuity. The constant payments of these investments consist of a principal and an interest payment. The principal payment will generally increase in time whereas the interest payment will decrease in time. Two different types of investments with constant payments and interest rates can be distinguished:

• The first type, also referred as type 0, has payments that are made at the end of each period.

• The second type, type 1, has payments that are made at the beginning of each period. This type has no interest payment at the beginning of the first period, but does have an extra period, after the last periodic payment, with an interest payment over the last period and an inverse principal payment.

## Equations

Cash flows can be either positive or negative, where a positive payment indicates that you are receiving this payment. Taking the interest into account, the total value of an investment must be equal to zero after all cash flows have occurred. For example, a positive present value and positive payments will lead to a negative future value: your debt has grown. The following equation expresses the relation between all the cash flows that take place

$v_p(1+r)^N + p\sum_{i=1}^N (1+r)^{i-1+T} + v_f = 0$

where $$v_p$$ is the present value, $$v_f$$ is the future value, $$p$$ is the constant periodic payment, $$r$$ is the constant interest rate and $$T$$ is the investment type as discussed above.

AIMMS supports the following investment functions with constant, periodic payments:

## Variable Payments

When the cash flows are variable (i.e. not constant), take place at irregular intervals, or when the interest rate varies over time, it still possible to compute present values, future values, and the internal rate of return, i.e. the rate received for an investment consisting of payments and income.

AIMMS supports the following investment functions for variable cash flows: