# How to Calculate IRR by Hand

When it comes to investment returns, the bigger the number, the more profitable your investment has been. The best way to evaluate a project or an investment so you can decide whether to accept or reject it is through the internal rate of return. In investment jargon, **IRR is the interest rate that makes the net present value zero**. That needs some explaining, since you first must understand the concepts of present value and net present value, or the idea that money is more valuable now than it is later on.

## The Ins and Outs of Present Value

Imagine that you have $1,000 in your pocket right now. You could go to the store and blow the cash on gadgets, or you could use the money to make more money: invest it in a business project, buy some inventory to sell later at a higher price or simply put the money in the bank to earn interest.

Now, imagine that an investment could get you a guaranteed 10 percent return on your money. The $1,000 you have today will be worth $1,100 in 12 months' time because it has earned $1,000 times 10 percent, or $100. In 24 months' time, you'll have $1,210 due to compound interest.

What we're saying here is that $1,000 today is **worth exactly the same** as $1,100 next year, and both those amounts are **worth exactly the same** as $1,210 in two years' time when there's a 10 percent interest rate. If you turn the equation backward, $1,100 next year is only worth $1,000 now. In investing jargon, $1,100 next year has a present value of $1,000.

## From the Future Back to Now

Usually, when we talk about present value, we run the calculation backward. That's because we're interested in what money in the future is worth right now.

Suppose a business partner promises to pay you $1,000 next year. What is the present value? To reverse the calculation so you're taking the future payment back by one year, divide the dollar amount by 1.10. The $1,000 next year is worth $1,000/1.10, or $909.09 today.

If you were getting the money in three years, you'd divide the number by 1.10 three times:

$1,000/ 1.10 ÷ 1.10 ÷ 1.10 = $751.31 (to nearest cent).

This means that having $751.31 in your pocket today is worth exactly the same as having $1,000 in your pocket in three years' time.

## The Present Value With Exponents

While easy enough to perform, the present value calculation becomes unwieldy when you're projecting forward or working back over multiple years. Here, it's better to use exponents, or how many times to use the number in a multiplication.

For example, instead of calculating $1,000/ 1.10 ÷ 1.10 ÷ 1.10 to give the present value of $1,000 in three years' time, we can write the calculation as $1,000 ÷ 1.10^{3}= $751.31.

In fact, what we've just created here is the formula for present value (PV):

**PV= FV / (1+r) ^{n}**

Where:

- FV is future value
- r is the interest rate expressed as a decimal (0.10, not 10 percent)
- n is the number of years

Using this formula to calculate the PV of $1,000 in three years, you get:

PV = FV / (1+r)^{n}

PV = $1,000 / (1 + 0.10)^{3}

PV = $1,000 / 1.10^{3}

PV = $751.31

## The Ins and Outs of Net Present Value

So far, we've worked out the present value of money with a 10 percent rate of return. What about the net present value of money? Generally, when you make an investment, you have money going out (money you spend, invest or deposit) and money coming in (interest, dividends and other returns). When more comes in than goes out, the business is making a profit.

To get the net present value of an investment, you simply add what comes in and subtract what goes out. However, future values must be brought back to today's values to account for the **time-value of money.** The time-value of money is the concept that money in your pocket today (the present value) is worth more than the same sum in the future because of its earning potential.

So, what you're actually doing here is working out the present value of every deposit and receipt, and then adding or subtracting them to get the net present value.

## Example of Net Present Value

Suppose a business partner needs a $1,000 loan right now and will pay you back $1,250 in a year. You have the money, and it's currently earning 10 percent interest in a certificate of deposit. Is the loan a good investment when you can get 10 percent elsewhere?

The "money out" here is $1,000. Since you're making the loan right now, the PV is $1,000. The "money in" is $1,250, but you won't receive it until next year, so you first must work out the PV:

PV = FV / (1+r)^{n}

PV = $1,250 / (1 + 0.10)^{1}

PV = $1,250 / 1.10

PV = $1,136.36

The net present value here is $1,136.36 minus $1,000, or $136.36. With a 10 percent interest or **discount rate,** the loan has an NPV of $136.36. In other words, it is $136.36 better than a 10 percent deposit at the bank in today's money.

## Playing With the Numbers

Hopefully, you can see that a positive NPV is good (you're making money), and a negative NPV is bad (you're losing money). Beyond that, the discount rate you apply can change the situation – and sometimes quite dramatically.

Let's try the same loan investment, but say we require a 15 percent return.

The money out is still $1,000 PV. This time, though, the money in has the following calculation:

PV = FV / (1+r)^{n}

PV = $1,250 / (1 + 0.15)^{1}

PV = $1,250 / 1.15

PV = $1,086.96

So, at 15 percent interest, the same investment is worth only $86.96. Generally, you'll find that the lower the interest rate, the easier it is to get a decent NPV. High interest rates are tough to achieve. When the rate seems too good to be true, your NPV might not look so good.

## What's the Significance?

The net present value is a mathematical way of figuring out **today's equivalent of a return that you're going to receive on a future date,** whether that date is 12, 36 or 120 months in the future. Its main benefit is to help you establish a specific interest rate as a benchmark for comparing your projects and investments.

Suppose, for example, your company is considering two projects. Project A will cost $100,000 and is expected to generate revenues of $2,000 a month for five years. Project B will cost more – $250,000 – but the returns are projected to be $4,000 per month for 10 years. Which project should the company pursue?

Let's assume the company wants to achieve 10 percent as the minimum acceptable return percentage that the project must earn in order to be worthwhile. At this rate, Project A will return an NPV of **minus $9,021.12.** In other words, the company would lose money. Project B, on the other hand, has an NPV of **$44,939.22**. Assuming the two projects are similar risks, the company should green light project B.

When comparing projects by NPV, it's critical to use the same interest rate for each, or you're not comparing apples with apples, and your calculations will have little practical value. You can use an online NPV calculator to quickly run the calculations at various interest or discount rates.

## Ins and Outs of the Internal Rate of Return

The **interest rate that makes the NPV zero** is called the internal rate of return.

Calculating the IRR is desirable because it lets you see at a glance the rate of return you can anticipate from a specific investment, even if the returns won't land in your account for many years. This allows you to benchmark the project or investment against another you might have made or against an industry average rate of return.

If your stock investments are achieving an IRR of 14 percent, for example, and the stock market is averaging returns of only 10 percent over the same period, then you clearly made some good investment decisions. You may wish to channel more cash into that particular stock portfolio since you're outperforming the usual benchmarks.

## How Do You Calculate IRR?

To calculate IRR manually without the use of software or a complicated IRR formula, you must use the trial and error method. As the name implies, you're going to guess the rate of return that will give an NPV of zero, check it by running the calculation with the rate you've guessed, and then adjust the percentage up or down until you get as close to zero as you possibly can.

It's not scientific, but it is effective and you can usually find the IRR after a couple of tries.

## IRR Trial and Error Method Example

Suppose that you have an opportunity to invest $5,000 for three years and receive:

- $200 in the first year
- $200 in the second year
- $5,200 when the investment is closed in year three

What is the NPV at 10 percent interest?

Here, we have money out of $5,000. To calculate the PV of the future returns, we run the following calculation:

PV = FV / (1+r)n

So:

Year 1: $200 / 1.10 = $181.82

Year 2: $200 / 1.10^{2} = $165.29

Year 3: $5,200 / 1.10^{3} = $3,906.84

Adding those up gets:

NPV = ($181.82 + $165.29 +$3,906.84) - $5,000

NPV = minus $746.05

The objective, remember, is to find the interest rate than makes the NPV zero. Ten percent is way off, so let's try another guess, say 5 percent.

Year 1: PV = $200 / 1.05 = $190.48

Year 2: PV = $200 / 1.05^{2} = $181.41

Year 3: PV = $5,200 / 1.05^{3} = $4,491.96

Adding these figures up gets:

NPV = ($190.48 + $181.41 + $4,491.96) - $5,000

NPV = minus $136.15

We now know that, for this calculation, the required IRR is less than 5 percent. Let's adjust again, this time to 4 percent:

Year 1: PV = $200 / 1.04 = $192.31

Year 2: PV = $200 / 1.04^{2} = $184.91

Year 3: PV = $5,200 / 1.04^{3} = $4,622.78

Now, the NPV is:

NPV = ($192.31+ $184.91 + $4,622.78) - $5,000

NPV = $0

Using the trial and error method, we've found the IRR that returns an NPV of zero, and the answer is 4 percent. In other words, this particular investment should earn a 4 percent return assuming all goes according to plan.