キャッシュフロー計算機

Net Present Value (NPV)

Net Present Value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time.

NPV answers the question: “Is this investment worth making in today’s dollars?” or “What is the current value of my future cash flows?”

The formula for calculating NPV is:

$$NPV=\sum _{t=0}^n\frac{C{F}_t}{(1+r{)}^t}$$

Where:

• NPV = Net Present Value
• CF_t = Cash flow at time t
• r = Discount rate (required rate of return)
• n = Total number of periods
• t = Time period

A positive NPV indicates a potentially profitable investment, while a negative NPV suggests the investment may result in a net loss.

Example:

Consider an investment that requires an initial outlay of $10,000 (negative cash flow) and is expected to generate $4,000, $4,000, and $5,000 in the next three years. With a discount rate of 10%:

$$NPV=-10,000+\frac{4,000}{(1+0.1{)}^1}+\frac{4,000}{(1+0.1{)}^2}+\frac{5,000}{(1+0.1{)}^3}=679.12$$

Since the NPV is positive, this investment is projected to be profitable.

Internal Rate of Return (IRR)

Internal Rate of Return (IRR) is the discount rate that makes the net present value of all cash flows equal to zero.

IRR answers the question: “What is the rate of return on my investment?” or “What interest rate am I effectively earning?”

The formula for calculating IRR involves finding the rate r that satisfies the equation:

$$\sum _{t=0}^n\frac{C{F}_t}{(1+r{)}^t}=0$$

Where:

• CF_t = Cash flow at time t
• r = Internal Rate of Return
• n = Total number of periods
• t = Time period

IRR can be used to rank several prospective investments or projects. Higher IRR values indicate more desirable investments, assuming all other factors are equal.

Example:

Using the same investment from the NPV example: initial outlay of $10,000 with returns of $4,000, $4,000, and $5,000 over three years. The IRR would be the rate r that satisfies:

$$-10,000+\frac{4,000}{(1+r{)}^1}+\frac{4,000}{(1+r{)}^2}+\frac{5,000}{(1+r{)}^3}=0$$

Solving this equation yields an IRR of approximately 14.3%, meaning the investment provides a 14.3% annual return.

Modified Internal Rate of Return (MIRR)

Modified Internal Rate of Return (MIRR) improves upon the IRR by addressing the reinvestment rate assumption and the possibility of multiple IRRs.

MIRR answers the question: “What is my rate of return if negative cash flows are financed at a financing rate and positive cash flows are reinvested at a reinvestment rate?”

The formula for calculating MIRR is:

$$MIRR=\sqrt[n]{\frac{FV(\text{positive cash flows, reinvestment rate})}{PV(\text{negative cash flows, financing rate})}}-1$$

Or expressed differently:

$$MIRR={\left(\frac{{\sum }_{t=0}^{n}C{F}_t^{+}\times (1+r_r{)}^{n-t}}{-{\sum }_{t=0}^n\frac{C{F}_t^{-}}{(1+r_f{)}^t}}\right)}^{\frac{1}{n}}-1$$

Where:

• MIRR = Modified Internal Rate of Return
• CF_t^+ = Positive cash flows at time t
• CF_t^- = Negative cash flows at time t
• r_r = Reinvestment rate
• r_f = Financing rate
• n = Total number of periods
• t = Time period

Example:

Continuing with our investment scenario, if the reinvestment rate is 8% and the financing rate is 10%:

Future value of positive cash flows: $4,000(1+0.08)^2 + $4,000(1+0.08)^1 + $5,000 = $13,648

Present value of negative cash flows: $10,000

$$MIRR={\left(\frac{13,648}{10,000}\right)}^{\frac{1}{3}}-1=10.9\%$$

This MIRR of 10.9% is lower than the IRR of 14.3%, reflecting the more realistic assumption that returns are reinvested at 8% rather than at the IRR.

Net Future Value (NFV)

Net Future Value (NFV) represents the value of all cash flows at the end of the investment period, assuming they are compounded at a given interest rate.

NFV answers the question: “What will be the future value of all my cash flows at the end of the investment horizon?”

The formula for calculating NFV is:

$$NFV=\sum _{t=0}^{n}C{F}_t\times (1+r{)}^{n-t}$$

Where:

• NFV = Net Future Value
• CF_t = Cash flow at time t
• r = Interest rate
• n = Total number of periods
• t = Time period

Example:

Using our investment scenario with a 10% interest rate:

$$NFV=-10,000\times (1+0.1{)}^3+4,000\times (1+0.1{)}^2+4,000\times (1+0.1{)}^1+5,000=900.40$$

This means that at the end of the 3-year period, the net value of all cash flows, including the initial investment and all returns compounded at 10%, would be $900.40.

Payback Period

Payback Period is the time required to recover the initial investment in a project.

Payback Period answers the question: “How long will it take to get my money back?”

For uneven cash flows, the formula is:

$$\text{Payback Period}=A+\frac{B}{C}$$

Where:

• A = Last period with a negative cumulative cash flow
• B = Absolute value of cumulative cash flow at the end of period A
• C = Cash flow during the period after A

A shorter payback period means the investment is more liquid and less risky.

Example:

For our investment with an initial outlay of $10,000 and annual returns of $4,000, $4,000, and $5,000:

Year 0: -$10,000
Year 1: -$6,000 ($10,000 - $4,000)
Year 2: -$2,000 ($6,000 - $4,000)
Year 3: +$3,000 ($2,000 + $5,000)

The last period with a negative cumulative cash flow is Year 2, with a remaining deficit of $2,000. The cash flow in the next period is $5,000.

$$\text{Payback Period}=2+\frac{2,000}{5,000}=2.4\text{ years}$$

So it would take 2.4 years to recover the initial investment.

Discounted Payback Period

Discounted Payback Period is similar to the regular payback period but uses discounted cash flows instead of nominal cash flows.

Discounted Payback Period answers the question: “How long will it take to recover my investment in present value terms?”

The calculation involves finding the point at which the cumulative discounted cash flows become positive:

$$\sum _{t=0}^{DPP}\frac{C{F}_t}{(1+r{)}^t}=0$$

Where:

• DPP = Discounted Payback Period
• CF_t = Cash flow at time t
• r = Discount rate
• t = Time period

Example:

For our example with a 10% discount rate:

Year 0: -$10,000
Year 1: $4,000 ÷ 1.10 = $3,636.36
Year 2: $4,000 ÷ 1.21 = $3,305.79
Year 3: $5,000 ÷ 1.33 = $3,759.40

Cumulative discounted cash flows:
Year 0: -$10,000
Year 1: -$6,363.64
Year 2: -$3,057.85
Year 3: +$701.55

The last period with a negative cumulative discounted cash flow is Year 2, with a remaining deficit of $3,057.85. The discounted cash flow in the next period is $3,759.40.

$$\text{Discounted Payback Period}=2+\frac{3,057.85}{3,759.40}=2.81\text{ years}$$

So it would take 2.81 years to recover the initial investment in present value terms, which is longer than the regular payback period of 2.4 years.

Frequency in Cash Flows

In cash flow analysis, frequency refers to how many times a particular cash flow repeats at a given period. This is particularly useful when the same payment occurs multiple times in succession, allowing for more concise representation of recurring cash flows.

When a cash flow has a frequency greater than 1, the calculation formulas account for this repetition by:

$$C{F}_{\text{total}}=C{F}_{\text{amount}}\times \text{Frequency}$$

This frequency factor is integrated into all the cash flow calculations discussed above.

Example:

Instead of entering three identical annual payments of $5,000 at periods 1, 2, and 3, you could enter a single cash flow of $5,000 with a frequency of 3 at period 1. The calculator would then treat this as:

Period 1: $5,000
Period 2: $5,000
Period 3: $5,000

This simplifies data entry and makes the cash flow timeline more readable, especially for complex investment scenarios with multiple recurring payments.