**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.

i and others (very few, but some) look at this

example differently. most of my life, and most

of my projects, are constrained by cash. or

working capital. so i must pick between various

projects, based not on NPV, but getting my money

back the quickest. the concept of flush:

http://consultski.blogspot.com/2006/01/financial-flush-method-ffm.html

http://consultski.blogspot.com/2006/01/financial-flush-method-ffm-part-2.html

it is my hope to expand this example into a more

realistic scenario for part three in my series…

-ski

Great point Ski – I intend to write a definition of “payback period” soon. Another tie-breaker or decision-driver for investment decisions.

Thanks for the comment and links

How do you choose the discount rate for the projects you are comparing? This is always the puzzling part of NPV analysis.

Hi Ashley, thanks for reading and thanks for the comment/question!

I believe that there are two answers – one for companies, and one for individuals.

For individuals, I believe a conservative discount rate to use is one that can be achieved with relatively low risk investments (but not risk-free investments). Personally, I use 10%, as an approximation of the long term rate of return for investing in the stock market. This is my approximation for ‘cost of capital’. The efficient market theory implies that to maximize profits in the long run, the discount rate (or hurdle rate) should equal the cost of capital.

For companies, the interesting question is “Would this be dillutive or accretive to earnings?” In other words, what are all of my other projects returning – would this project have a higher or lower rate of return than that?

Unfortunately this is a bit of a circular reference, since all of those other projects need a discount rate for their calculations.

Choosing discount rates higher than the cost of capital tends to favor short term investments over long term investments (earlier cash flow is ‘overwieghted’).

Most companies (I believe) use the CAPM model (Capital asset pricing model) to determine the ideal internal hurdle rate for investment. Applying the CAPM model is such a complex effort, that it would be unreasonable for individual projects to do it. Having a single rate for the company simplifies this quite a bit, as long as the projects being considered have consistent “risk profiles” as other investments the company is making.

The corporations I’ve worked for have used numbers close to 20% as the hurdle rate (as communicated to project teams).

Sorry that doesn’t give a crisp answer, but there just isn’t a simple answer.

Dear Sir,

Would also like to Know can ou explain DPV( discounted present value) and IRR( internal rate return)with suitable examples as above which are really helpful and easy to understand.

hello

i am student in ph.d (watershed management) in malayzia.

my proposal in ph.d is (The evaluation of the implemented watershed management operation from both technical and socio-economical effect point of view).is it possible for you to help me,and send file about socio-economic and N.P.V AND IRR AND BENEFIT COST ANAYSIS.

THANK YOU

BAHRAM GOLRANG

Bahram, welcome to Tyner Blain. What is it you need help with, specifically?

Mahesh, thanks for the question, and welcome to Tyner Blain.

The DPV is the same as an NPV for a single cash flow. In the bond example, the DPV = NPV. In the series of payments example, you calculate the DPV of each payment, and sum them up to calculate the NPV. Wikipedia provides a good article on IRR, let me know if that doesn’t explain it well enough

Hi Sir,

I am a university student in Malaysia, I have read the above explaination about NPV and i understand very well. However, can you furnish me more on the assumptions of NPV (i.e., assumed that the inflow and outflow are certained every year) and also the limitations of NPV in relation to investment appraisal? The more assumptions and limitations the better. I am in the midst of preparing for final exam 2 days later, hope to get a fast reply from you. Thank you.

It looks like you compare wrong numbers at the end of the articles.

You say:

$10,000 one year from now has a net present value of $9,524

$500 per month for 20 months has a net present value of $9565

But $9,524 is received using the 5% effective interest rate and $9,565 is received using the 5.13% interest rate.

I guess the correct number for the first example has to be $9,512 = $10,000/(1 + 5.13%).

It does not dismiss the point but makes article hard to understand. I re-read the article multiple times before I got that this is a mistake but not my misunderstanding of the whole concept.

Anyway, thank you for article.

Thanks, Dennis!

Correction made. I really appreciate you taking the time to point this out – will definitely help other folks who read this in the future.