Net Present Value and Internal Rate of Return

Should a textile mill in Faisalabad buy a new loom? Should a startup in Lahore open a second branch? Every investment decision comes down to one question: are the future cash inflows worth more than the money spent today? Net present value (NPV) and internal rate of return (IRR) are the two most widely used tools for answering it, and every BBA, MBA, ACCA and CA student meets them in capital budgeting.

The Idea Behind NPV

A project usually costs money now and returns cash over several years. Because of the time value of money, we cannot simply add those amounts. Instead we discount each future cash flow back to today at the required rate of return (also called the cost of capital or hurdle rate) and subtract the initial investment.

If \( C_0 \) is the initial outlay and \( C_t \) is the cash flow at the end of year \( t \), then

$$ NPV = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+r)^t} $$

The decision rule is simple:

• NPV > 0: the project earns more than the required return, so it adds value. Accept.
• NPV < 0: the project destroys value. Reject.
• NPV = 0: the project earns exactly the required return.

NPV is measured in currency, so it tells you directly how much wealth (in today's rupees) the project creates for shareholders.

Worked Example 1: Evaluating a Machine

A bakery is considering an oven costing PKR 1,000,000. It expects extra cash flows of PKR 300,000, 400,000, 500,000 and 200,000 at the end of years 1 to 4. The required return is 12%. Find the NPV.

Step 1: Compute the discount factors \( 1/(1.12)^t \):

• Year 1: \( 1/1.12 = 0.892857 \)
• Year 2: \( 1/1.2544 = 0.797194 \)
• Year 3: \( 1/1.404928 = 0.711780 \)
• Year 4: \( 1/1.573519 = 0.635518 \)

Step 2: Multiply each cash flow by its factor.

Step 3: Add the present values and subtract the cost.

$$ NPV = 1{,}069{,}728.43 - 1{,}000{,}000 = \text{PKR } 69{,}728.43 $$

The NPV is positive, so the oven should be bought. It creates about PKR 69,728 of value in today's money, over and above the 12% return investors demand.

A related measure is the profitability index, the ratio of the present value of inflows to the initial cost:

$$ PI = \frac{1{,}069{,}728.43}{1{,}000{,}000} = 1.07 $$

A PI above 1 matches a positive NPV.

The Idea Behind IRR

The internal rate of return is the discount rate that makes the NPV exactly zero:

$$ 0 = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+IRR)^t} $$

You can think of it as the project's own built-in rate of return. The decision rule is:

• IRR > required return: accept.
• IRR < required return: reject.

Managers like IRR because a percentage is easy to communicate: "this project returns 15%" sounds more intuitive than "this project has an NPV of PKR 69,728".

Finding IRR by Trial and Interpolation

Except for very simple cases, there is no algebraic formula for IRR. By hand, we use trial and error followed by linear interpolation. Spreadsheets and calculators use iterative methods that do the same thing far more precisely.

Worked Example 2: IRR of the Bakery Project

We already know NPV at 12% is positive, so the IRR must be higher than 12%. Try higher rates.

Trial at 15%. Discount factors are 0.869565, 0.756144, 0.657516 and 0.571753:

$$ PV = 260{,}869.57 + 302{,}457.47 + 328{,}758.12 + 114{,}350.65 = 1{,}006{,}435.81 $$

$$ NPV_{15\%} = 1{,}006{,}435.81 - 1{,}000{,}000 = +6{,}435.81 $$

Still positive, but close to zero.

Trial at 18%. Discount factors are 0.847458, 0.718184, 0.608631 and 0.515789:

$$ PV = 254{,}237.29 + 287{,}273.77 + 304{,}315.44 + 103{,}157.78 = 948{,}984.28 $$

$$ NPV_{18\%} = 948{,}984.28 - 1{,}000{,}000 = -51{,}015.72 $$

Now negative, so the IRR lies between 15% and 18%.

Interpolate. Using the straight line between the two points:

$$ IRR \approx 15\% + \frac{6{,}435.81}{6{,}435.81 + 51{,}015.72} \times (18\% - 15\%) $$

$$ IRR \approx 15\% + 0.1120 \times 3\% = 15.34\% $$

An exact numerical solution gives 15.32%, so the interpolation is excellent. Interpolation slightly overstates the IRR because the NPV curve is convex, so always choose two trial rates as close together as you can.

Since 15.32% is above the 12% required return, IRR agrees with NPV: accept.

Worked Example 3: A Project to Reject

A second project costs PKR 500,000 and returns PKR 200,000 each year for 3 years. Because the cash flows are equal, we can use the annuity factor at 12%:

$$ \frac{1-(1.12)^{-3}}{0.12} = 2.401831 $$

$$ NPV = 200{,}000 \times 2.401831 - 500{,}000 = -\text{PKR } 19{,}633.75 $$

Solving for the rate that makes the NPV zero gives an IRR of about 9.70%, below the 12% hurdle. Both methods say reject. Notice that the project still makes a profit in plain rupee terms (PKR 600,000 back for PKR 500,000), but not enough to compensate for the time value of money.

When NPV and IRR Disagree

For a single independent project with a normal pattern (one outflow followed by inflows), NPV and IRR always give the same accept or reject decision. Problems appear in three situations:

1. Mutually exclusive projects of different sizes. A small project may have a higher IRR but a much smaller NPV. Because shareholders care about total wealth, choose the higher NPV.
2. Different timing of cash flows. IRR implicitly assumes cash flows are reinvested at the IRR itself, which may be unrealistic. NPV assumes reinvestment at the cost of capital, which is more sensible.
3. Non-conventional cash flows. If signs change more than once (for example, a mine with clean-up costs at the end), the NPV equation can have several IRRs, or none. The modified IRR (MIRR) fixes this by compounding inflows at the cost of capital.

The textbook verdict is clear: when in doubt, trust NPV.

Doing It in Software

In Excel, use =NPV(rate, values) + initial_cash_flow, remembering that Excel's NPV function assumes the first value occurs at the end of year 1. Use =IRR(range) with the outlay included as a negative number. Financial calculators have dedicated cash-flow keys, and Python's numpy_financial library provides npv and irr functions.

Did You Know?

• Surveys of chief financial officers have repeatedly found that IRR is used about as often as NPV in practice, even though theory prefers NPV.
• The Excel IRR function starts from a default guess of 10% and refines it by iteration; for unusual cash flows, a different guess can produce a different IRR.
• A project with a positive NPV at a 0% discount rate simply has total inflows greater than outflows; the NPV curve then falls as the rate rises and crosses zero at the IRR.

Key Takeaways

• NPV discounts all cash flows at the required return; accept projects with NPV above zero.
• IRR is the discount rate at which NPV equals zero; accept if it exceeds the required return.
• By hand, find IRR with two trial rates and linear interpolation.
• For independent, conventional projects both methods agree; for mutually exclusive projects, choose the higher NPV.
• Watch for multiple IRRs when cash flows change sign more than once.

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