Net present value, or NPV is the great equalizer of financial analysis.
NPV allows us to compare any two investments and determine which is the better investment.
NPV tells us how many dollars, today, we would be willing to spend to receive money in the future. NPV lets us compare investments that pay back money in very different ways – we can decide if we would rather have $10,000 in one year, or $500 per month for 20 months. Without NPV, the two investments appear to be the same (they both return $10,000), but one of them is better than the other.
NPV calculation for single payments
NPV compares these values based upon a simple assumption – any money sitting in our bank account would be placed in a risk-free, guaranteed investment at a fixed interest rate. Imagine that our bank is offering savings bonds with a 5% interest rate. We purchase a bond worth $10,000 one year from now. $10,000 is the expected value of the bond. We will have to pay the bank $10,000/(1.05) = $9,524.
The NPV, or net present value of the $10,000 bond is $9,524.
Simple and compound interest
The example above used simple interest, to make the idea easy to grasp. In reality, most investments will return what is known as compound interest. Simple interest is calculated against the original investment amount. With a simple interest rate of 5% per year, and an investment (or loan) of $100, the annual interest payment would be $5 (5% of $100). This is the way most people think about interest rates, but it isn’t the way that most companies use interest rates.
Most companies and banks use what is called a compound interest rate – specifically with daily compounding. Compound interest can be thought of as a series of simple interest calculations. The period of investment (1 year) is divided into smaller periods and the interest is recalculated at the end of each mini-period. Daily compounding means that this recalculation happens every day.
The interest for each mini-period (a day) is calculated using simple interest. For an annual interest rate of 5%, compounded daily, the daily interest rate is 5%/365 = 0.0137%. With our investment of $9524 above, a simple interest rate calculation at 5% would yield an annual interest payment of $9524 * 0.05 = $476 ($9524+ $465 = $10,000). With daily calculation, we would earn 0.0137% * 9524 = $1.30. Simple interest just takes that $1.30 per day and multiplies it by 365 days in the year to get $476 per year.
Compound interest calculation takes that $1.30 and adds it to the principal before recalculating for the next day.
- Day 1: $9524 loaned + $1.30 interest due = $9525.30 owed.
- Day 2: $9525.30 owed + $1.30 interest due = $9526.60 owed.
- Day 3: $9526.60 owed + $1.31 interest due = $9527.91 owed.
- […]
At the end of the year, the total amount due is $10,011. This is the result of compound interest. The effect is small in our example, but increases for higher interest rates and longer loan periods.
When an investment offers an annual interest rate X% compounded daily, we can convert that to an effective annual rate Y% with the following equation: Y% = (1 + (X%/365))^365 – 1. The effective interest rate (also called the effective yield) for a 5% annual rate compounded daily is (1 + (.05/365))^365 = 5.13%
We always perform NPV calculations in terms of the effective yield.
The reason we explained this is to allow us to evaluate investments that don’t pay in a lump sum at the end of the period.
NPV calculation for a series of payments
We just calculated NPV for a single, lump-sum payment. We can also calculate NPV either for a series of repeating payments, or for a collection of arbitrary payments. The net present value of multiple payments is the sum of the net present value of each of the payments.
NPV for a series of payments
We are evaluating a series of $500 payments over twenty months. With a 5.13% effective annual interest rate, we can compute the NPV for each of the payments individually. The effective monthly interest rate is 1/12 of the annual rate, or 0.43%
Assume we get paid at the end of each month (this is the common case).
- Payment 1: $500 future value = $500/(1 + 0.0043) = $497.87
- Payment 2: $500 future value = $500/(1 + 0.0043)^2 = $495.75
- Payment 3: $500 future value = $500/(1 + 0.0043)^3 = $493.65
- […]
- Payment 20: $500 future value = $500/(1+ 0.0043)^21 = $459.14
The total of all the NPV calculations is $9,565
Conclusion
We were able to compare the two investments and determine which one is worth more to us in the present:
- $10,000 one year from now has a net present value of $9,512 [updated 2010.01.10]
- $500 per month for 20 months has a net present value of $9565
The monthly payments are worth more to us today than the lump sum payment.
Some companies use a payback period analysis to compare investment alternatives – this is usually a bad idea, as it doesn’t identify the investment with the highest ROI, just the one where we get our money back the fastest.
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