The formular for NPV is: Net Present value formula
$$\sum_{t=1}^n= \frac{Rt}{(1+i)^t}$$
where:
Rt =Net cash inflow-outflows during a single period t
i=Discount rate or return that could be earned in alternative investments
t=Number of timer periods
Which means taking projected cash flow for each year and dividing it by (1 + discount rate). For example this is the formula for a 5 year project and 3% discount rate:
$$\sum_{t=1}^n= \frac{Rt}{(1+0.03)^5}$$
This formula has one serious problem: considers that the discount rate does not change throughout the project. Which means for example that the risk free rate dont change in 5 years ( premise totally out of reality ). My question is very simple: is possible to consider differents rates per year for Net Present Value calculation? for example for the first year 3% for the next year 3.4% etc. If i calculate the npv for the first year with 3% of discount rate and sum by the npv for the second year with 3.4% discount rate??
To solve this doubt, I created a cashflow, separate it into two cash flows, calculated the NPV for each separately and added the same discount rate.
# R CODE
library(FinCal)
total_cashflow=c(10,10,10,20,60)
split1_cashflow=c(10,10)
split2_cashflow=c(10,20,60)
#Net present Value for the complete cashflow 3% discout rate
npv(0.03,total_cashflow)
> 100.7468
#Sum of Net present Value for the two split cashflows 3% discout rate
npv(0.03,split1_cashflow)+npv(0.03,split2_cashflow)
> 105.682
Why are the results different?
Best Answer
If we assume that the first payment is made immediately then your first result is right. It starts at $t=0$ and ends at $t=4$
$$PV_1=\frac{10}{1.03^0}+\frac{10}{1.03^1}+\frac{10}{1.03^2}+\frac{20}{1.03^3}+\frac{60}{1.03^4}=100.7468 \ \ \color{\limegreen}{\checkmark}$$
The split approach begins for the first two payments at $t=0$ and ends at $t=1$. For the next three payments it begins once again at $t=0$ and ends at $t=2$.
$$PV_2=\underbrace{\frac{10}{1.03^0}+\frac{10}{1.03^1}}_{\textrm{split 1}}+\underbrace{\frac{10}{1.03^0}+\frac{20}{1.03^1}+\frac{60}{1.03^2}}_{\textrm{split 2}}=105.682 \ \ \color{red}{\times}$$
This is not the intention of the first approach.