i am new in finance and want to know not only how to calculate financial parameters using formulas,but underlined idea as well, for instance , let us consider following problem :
A has invested US $100 in 2016. The payment has been made yearly. The interest rate is 10% p.a. What would be the FV in 2019?
i know that formula for future value compounded yearly given present value and interest rate after $n$ years is given by
$ F=p*(1+r)^n $
in our case
p=-100
r=10%
n=3
formulas says that
but it should be =FV(10%,3,0,-100,0) yes?what does payment yearly means? first of all let us consider mathematically
so i have invested 100 dollar for three year yes at 10% interest rate, below there is table which shows years and given interest which i will get
formula is given by
so where i am making mistake? what does means payment has been made yearly ?
Imagine that you have $P_0=\$\, 100$ on a bank account at $i=10\%$ interest.
After one year you will have $P_1=P_0(1+i)=\$\,110$, after two year $P_2=P_1(1+i)=P_0(1+i)^2$ and so on. So for $n$ years you will have $$P_n=P_0(1+i)^n$$
If you withdraw money every year, for example $W$, you will have \begin{align*} n=0 &\qquad P_0=100\\ n=1 &\qquad P_1=(P_0-W)(1+i)=P_0(1+i)-W(1+i)\\ n=2 &\qquad P_2=(P_1-W)(1+i)=P_1(1+i)-W(1+i)=P_0(1+i)-W(1+i)^2-W(1+i)\\ \ldots\\ \end{align*} So at year $n$ you will have $$ P_n=P_0(1+i)^n-W\sum_{k=1}^n(1+i)^n=P_0(1+i)^n-W\frac{(1+i)^n-1}{i} $$ that is the total future vale is the future value of $P_0$ minus the future value of the stream of withdrawals.
If you instead deposit the value $D$ every year you will have, putting $D=-W$ $$ P_n=P_0(1+i)^n+D\sum_{k=1}^n(1+i)^n=P_0(1+i)^n+D\frac{(1+i)^n-1}{i} $$ that is the total future vale is the future value of $P_0$ plus the future value of the stream of deposits.
In excel funtion $\mathtt{FV(rate,nper,pmt,[pv],[type])}$, we have
Pay attention on the sign in excel: a deposit has the minus sign, whereas a withdrawal has the plus sign.