# Present value of increasing payments

Find the present value of installment payments of P1,000 now, P2,000 at the end of the first year, P3,000 at the end of the second year, P4,000 at the end of the third year, P5,000 at the end of the fourth year, if money is worth 10% compounded annually.

### The present value can be

The present value can be thought of as the equivalent amount of money that would be paid up front as a lump sum, which has the same time value of money as the cash flow in question.

For example, if I say that $2000$ will be paid to you one year from now, and the effective annual rate of interest is $i=0.1$, the present value of this payment is the amount which, if held by you over the same amount of time, would equal $2000$ at the end of one year. That is to say....

$$Present \space value (1+i)= 2000$$
or
$$Present \space value = \frac{2000}{1+0.1} = 1818.18$$

So...it means that receiving $1818.18$ now and receiving $2000$ after a year, assuming its annual interest rate is $0.1$, means the same thing.

With that in mind, the present value of cash-flow would be:

I was paid $1000$, $2000$, $3000$, $4000$ and $5000$ after now, $1$, $2$, $3$, and $4$ years, respectively. Translating into an equation, it becomes...

$$1000 + \frac{2000}{(1+0.1)^1} + \frac{3000}{(1+0.1)^2} + \frac{4000}{(1+0.1)^3} + \frac{5000}{(1+0.1)^4}$$

which equals $11717.85$

Therefore, the present value of the cash flow would be $\color{green}{11717.85}$ pesos.

Alternate solutions are highly encouraged....