Discounting

Calculations done without Risks.

The sum of discounted future cash flows $C{F}_t$ more than the investment $I$? $NPV=-I+\frac{C{F}_1}{1+r}+\frac{C{F}_2}{(1+r{)}^2}+\frac{C{F}_3}{(1+r{)}^3}+...+\frac{C{F}_t}{(1+r{)}^t}$; $NPV>0$ → $(-I,C{F}_1,C{F}_2,C{F}_3,...,C{F}_t)$ is a profitable investment $r$ … discount rate $NPV$ … Net Present Value

Example Taxi

Assumption: the taxi business is closed after 5 years

$-40,000+\frac{30,000}{1+r}+\frac{30,000}{(1+r{)}^2}+\frac{30,000}{(1+r{)}^3}+\frac{30,000}{(1+r{)}^4}+\frac{45,000}{(1+r{)}^5}=1001,637$ for $r=0.05$

The Net Present Value is positive, it would be a worthwhile investment

Careful: This price also includes the driving of the taxi (done by entrepreneur)

After hiring a taxi driver for 24k per year: $-40,000+\frac{6,000}{1+r}+\frac{6,000}{(1+r{)}^2}+\frac{6,000}{(1+r{)}^3}+\frac{6,000}{(1+r{)}^4}+\frac{21,000}{(1+r{)}^5}=-2.270$ for $r=0.05$

The Net Present Value now is negative. This would not be a good investment. You would get more money just saving with the bank at 5%

0 Discount Rate

when solving the $NPV=0$ for $r$ we can find the Internal Rate of Return.

Invest 200 now, get 220 in a year: $\frac{220}{200}-1=10\%$

But how can we solve for $r$ in more years than just 2? → Newtons Approximation Basically brute force the values

After brute forcing we get $0.032<r<0.033$ for this specific problem.