The Accounting Rate of Return (ARR) is a simple method of calculation based on net profit or book value. It is little used now as it does not take into account the time value of money.

ARR = average net profit / average book value of investment * 100
or
ARR = average net profit / total initial investment value * 100

The average book value, method alpha, will give a higher result.

Payback is the peroid of time it takes the cost of an initial investment to be recouped. It also provides a low-level assessment of risk (high payback time is higher reisk). It also ignores the time value of money, however by applying a discounted payback period where cash flows are are discounted to reflect time value, this value can be improved.

Payback (in years) = Initial Investment / Net Cash Inflow

The Internal Rate of Return (IRR) method uses the present value of future dollars received against the present value of cash outflows. The IRR method projects against a rate of return and discounts the values of cash flows in an out of a business. It is possible, and problematic, to have more than one IRR in a given project and it can rate investments different to the NPV.

To calculate the IRR the following formula is used with R to be solved

OC (original cost) = NCF1/(1+R)^1 + NCF2/(1+R)^2 + ... NCFn/(1+R)^n
NCF = net cash flow, R = IRR/100, n = number of periods

Where net cash flow is the same.

OC = NCF * PVF(IRR,n)

PVF = Present Value Factor

e.g., A $17946 investment productes a benefit of $6000 for 5 years. OC = 17946, NCF = 6000 n = 5, R is unknown.

OC = NCF * PVF(IRR,n)
17946 = 6000 * PVF(IRR,5)
17946/6000 = PVF(IRR,5)
= 2.991 or IRR equals roughly 20%.

To calculate where NCF is not the same each year either a computerised solution or trial and error is usually required.
The Net Present Value (NPV) uses the present value of future dollars against the the future value of cash outflows. Discounted flows can be added and, for mutually exclusive projects, the NPV method is considered superior.

Unlike IRR, NPV expresses results in dollar amounts. The NPV is determined by calculating the present value of all cash inflows and outflows at a certain rate and then adding the two together to arrive at a positive or negative result. A positive NPV suggests a project should be accepted and a negative rejected.

NPV = -OC + NCF1/(1+K)^1 + NCF2/(1+K)^2... NCF3/(1+K)^N

K = cost of capital/100

When the net cash flow is the same each year the formula becomes NPV = -OC + (NCF * PVF(K,n))

Problems arise when mutually exclusive invesments with unequal lives are being ranked. This is mainly due to the different reinvestment assumptions. The NPV method is generally preferred.