Page 2 of 3
A better investment?
If you are still not convinced about the importance of the NPV consider the result of making an alternative investment at the safe interest rate of I% using just the negative cashflows (recall that the negative cashflows are what you would have invested to receive the positive element of the cashflow). This represents what you could have earned on the money you have invested in the "risky" project if you had invested it into a "safe" project at the current safe interest rate. Of course if you had invested in the safe project then you wouldn't recieve any of the postive values generated by the risky project - just the income from the investments at the safe interest rate.
Each such cash sum invested at I% will eventually grow to:
where m is the number of time periods left to the end of the investment. The negative sign is needed simply because of our cashflow convention that an investment of $S is a cashflow of -S.
This is simply the standard compound interest formula applied to each of the negative cashflows. The value of the entire "safe" investment is just the sum of all such values. It really is just the total you would receive from an alternative safe investment with the same cash outflows.
In other words the future value of the safe investment FVsafe is just the sum of each negative cashflow compounded (at I%) forwards to the end of the investment.
Now the NFV of the original risky investment is the sum of the NFV of the negative and positive cashflow calculated separately so:
where NFV(risk+) is the NFV of the positive part of the cashflow and NFV(-risk) is the NFV of just the negative part of the cashflow. Again we assume that the interest rate used is the safe rate.
You can also see that theFuture Value of the "safe" investment is i.e. the alternative were you simply invest your inputs, i.e. the negative cash flow items at the safe rate is:
FV(safe) = -NFV(risk-)
that is just the NFV of the negative components of the cash flow.
So putting these two together we can see that if the NFV of the original risky investment is just the NFV of the positive cashflows minus the NFV of the alternative safe investment, that is
This gives us a basic relationship between the NFV of the entire cashflow and the NFV of the alternative safe investment. Notice that this is what the entire risky investment is worth at the end of its term assuming that each item in the cash flow was invested at the safe rate.
Consider what this formula means if the NFV for the original ‘risky’ investment is zero. Then the NFV of the positve cashflow is equal to the total future value of the alternative safe investment. That is, as:
Thus you can see that when the NFV, and hence the NPV is zero, then the future value of the positive part of the cashflow is equal to the final value of the safe investment. However you have to be very careful that you understand exactly what is being compared against what.
The safe investment involves depositing the negative cashflows at I% at the time that they would occur in the original investment. The future value of this investement is equal to the value of the positive cashflows in the original ‘risky’ investment but notice that for this to be true these also have to be invested as they are received at the ‘safe’ interest rate of I%. Thus this comparison of values still takes into account the time when each sum of money is received. But as long as you do reinvest each sum of money at I% as it is received NFV(risk+) really does represent the amount of money you have received at the end of the risky investment and of course FVsafe is the amount you receive at the end of the safe investment. When these two quantities are equal the worth of both investments are the same.
When the NFV and hence the NPV of the risky investment is positive then:
and when it is negative then:
In other words a positive NPV does imply that you make more from the risky investment than a safe investment at I% and a negative NPV indicates that you would be better off making the safe investment.