# Time Value of Uneven Cash Flows

When a cash flow stream is uneven, the present value (PV) and/or future value (FV) of the stream are calculated by finding the PV or FV of each individual cash flow and adding them up.

A stream of cash flows is uneven when:

• All amounts in the series of cash flows are not equal, and/or
• There is unequal time between any two cash flows.

For example, coupon payments of a conventional bond constitute a series of even cash flows. It is because in case of a conventional bond, coupon payments are a fixed percentage of the face value of the bond. In other words, they are the same for each coupon period. However, in case of a floating-rate bond, the coupon payments are uneven. It is because in case of a floating-rate bond, coupon payments move up or down depending on the reference interest rate.

When cash flows are unequal and irregular, we cannot use the standard formulas for present value or future value of an annuity or present value of annuity factors tables. What we need to do is to calculate the present value or future value of each individual cash flow after considering the time period between the cash flow date and the valuation date i.e. the reference date, the date on which we want to calculate the PV or FV.

## Present Value of Uneven Cash Flows

We need to calculate present value of each cash flow using the present value of a single sum of money formula and then add together all the present values.

Where the cash flows are unequal but regular, we can use the following formula when CF1, CF2, CF3 and CFn are the uneven cash flows:

$$\text{PV}={\rm \text{CF}} _ \text{0}+\frac{{\rm \text{CF}} _ \text{1}}{{(\text{1}+\text{r})}^\text{1}}+\frac{{\rm \text{CF}} _ \text{2}}{{(\text{1}+\text{r})}^\text{2}}+\frac{{\rm \text{CF}} _ \text{3}}{{(\text{1}+\text{r})}^\text{4}}+\text{...}+\frac{{\rm \text{CF}} _ \text{n}}{{(\text{1}+\text{r})}^\text{n}}$$

Since CF0 occurs at time 0, its present value factor is 1.

Where the cash flows are both unequal and irregular, we need to manually calculate the total number of periods between the reference date and the cash flow date and use the following formula to calculate the component present value:

$${\rm \text{PV}} _ \text{n}=\frac{{\rm \text{CF}} _ \text{n}}{{(\text{1}+\text{r})}^\text{n}}$$

The present value of the whole stream of cash flows is the sum of all component present values.

## Future Value of Uneven Cash Flows

The procedure for calculating future value of uneven cash flows is similar. We just need to find future value of each individual cash flow and sum them up.

Where n is the total number of periods from time 0 to the reference date for future value, we can use the following formula to calculate future value:

$$\text{FV}={\rm \text{CF}} _ \text{0}\times{(\text{1}+\text{r})}^\text{n}+{\rm \text{CF}} _ \text{1}\times{(\text{1}+\text{r})}^{\text{n}-\text{1}}+{{\rm \text{CF}} _ \text{2}\times(\text{1}+\text{r})}^{\text{n}-\text{2}}+\text{...}+\ {\rm \text{CF}} _ \text{n}$$

Please note that CF0 is compounded for the whole n periods and CFn is compounded for 0 periods, hence it has future value factor of 1.