# Inflation and investment appraisal

This article discusses the nominal terms and real terms approaches to investment appraisal using the net present value method, and also considers the impact of taxation in the context of these approaches. This is an area of the syllabus where mistakes are often made by unprepared candidates.

#### The effect of inflation on cash flows

In a business environment with inflation, future cash flows will have decreasing purchasing power in current value terms as time passes. For example, if inflation is expected to be 5% per year and a cash amount of $100.00 is received at the end of each year for three years, the deflated values of these future cash receipts are as follows:

Year | Cash received | Deflation factor | Deflated value |
---|---|---|---|

1 | $100.00 | 1/1.05 = 0.952 | $95.20 |

2 | $100.00 | 1/1.05^{2} = 0.907 |
$90.70 |

3 | $100.00 | 1/1.05^{3} = 0.864 |
$86.40 |

In order to maintain the purchasing power of future cash receipts, the cash received must be inflated. Using the earlier example and maintaining the purchasing power of $100.00 gives:

Year | Cash received | Inflation factor | Inflated value |
---|---|---|---|

1 | $100.00 | 1.05 | $105.00 |

2 | $100.00 | 1.05^{2} = 1.1025 |
$110.25 |

3 | $100.00 | 1/1.05^{3} = 1.1576 |
$115.76 |

The inflated values in this table are also called nominal values.

#### General inflation and specific inflation

It is important to grasp the difference between general inflation and specific inflation. General inflation is measured by a published measure, such as the eurozone Harmonised Index of Consumer Prices (HICP). Specific inflation means that specific project variables such as selling price, variable costs and fixed costs inflate at different rates, such as 5% for selling price, 4% for variable costs and 6% for fixed costs.

#### Real and nominal costs of capital

The real cost of capital (r) and the nominal cost of capital (i) are related by general inflation (h) in the Fisher formula, provided in the examination formulae sheet:

(1 + i) = (1 + r)(1 + h)

If the real cost of capital is 4.0% and the general rate of inflation is 4.8%, the nominal cost of capital is 9.0%:

(1 + 0.040) (1 + 0.048) = 1.08992 or 9.0%

Since costs of capital are normally given in nominal terms, it is more usual to calculate the real cost of capital by deflating the nominal cost of capital by the general rate of inflation:

(1 + 0.090) / (1 + 0.048) = 1.04 or 4%

The calculated real cost of capital is 4%.

#### Nominal cash flows

Nominal cash flows are current price terms cash flows that have been inflated into future values, as illustrated above, using either general or specific inflation.

**Example of calculating nominal cash flows using specific inflation**

Selling price (current price terms) | $5.30 per unit |

Variable cost (current price terms) | $3.15 per unit |

Selling price inflation | 5% per year |

Variable cost inflation | 4% per year |

Forecast sales volume is 300,000 units per year, increasing by 50,000 units per year, and the investment project is expected to last for four years.

Inflated selling prices

Year 1: 5.30 x 1.05 = $5.57 per unit

Year 2: 5.30 x 1.05^{2} = $5.84 per unit

Year 3: 5.30 x 1.05^{3} = $6.14 per unit

Year 4: 5.30 x 1.05^{4} = $6.44 per unit

Inflated sales revenue

Year 1: 5.57 x 300,000 = $1,671,000

Year 2: 5.84 x 350,000 = $2,044,000

Year 3: 6.14 x 400,000 = $2,456,000

Year 4: 6.44 x 450,000 = $2,898,000

Inflated variable costs

Year 1: 3.15 x 1.04 = $3.28 per unit

Year 2: 3.15 x 1.04^{2} = $3.41 per unit

Year 3: 3.15 x 1.04^{3} = $3.54 per unit

Year 4: 3.15 x 1.04^{4} = $3.69 per unit

Using year 2 inflated costs as an example, when performing these calculations in a spreadsheet the following methods can be used

=3.15*1.04^2

=3.15*POWER(1.04,2)

There are other methods of calculating these figures and any approach which gives the correct figures will be given credit in the exam.

Inflated total variable cost

Year 1: 3.28 x 300,000 = $984,000

Year 2: 3.41 x 350,000 = $1,193,500

Year 3: 3.54 x 400,000 = $1,416,000

Year 4: 3.69 x 450,000 = $1,660,500

Nominal terms total contribution

Year 1: 1,671,000 – 984,000 = $687,000

Year 2: 2,044,000 – 1,193,500 = $850,500

Year 3: 2,456,000 – 1,416,000 = $1,040,000

Year 4: 2,898,000 – 1,660,500 = $1,237,500

#### Real cash flows

Real cash flows are found by deflating nominal cash flows by the general rate of inflation.

**Example of calculating real cash flows by deflating nominal cash flows**Using the nominal cash flows calculated above and a general rate of inflation of 4.8%:

Real terms total contribution

Year 1: 687,000/ 1.048 = $655,534

Year 2: 850,500/ 1.048^{2} = $774,376

Year 3: 1,040,000/ 1.048^{3} =$903,544

Year 4: 1,237,500/ 1.048^{4} = $1,025,888

#### Nominal terms approach to investment appraisal

The nominal terms approach to investment appraisal involves discounting nominal cash flows with a nominal cost of capital in calculating the NPV of an investment project.

**Example of calculating nominal terms NPV**Using the nominal contributions calculated earlier, a nominal discount rate of 9.0% and an assumed initial investment of $1,000,000:

Year 1: 687,000/ 1.09 = | $630,275 |

Year 2: 850,500/ 1.09^{2} = |
$715,849 |

Year 3: 1,040,000/ 1.09^{3} = |
$803,071 |

Year 4: 1,237,500/ 1.09^{4} = |
$876,676 |

$3,025,871 | |

Initial investment | $1,000,000 |

Nominal NPV | $2,025,871 |

#### Real terms approach to investment appraisal

The real terms approach to investment appraisal involves discounting real cash flows with a real cost of capital in calculating the NPV of an investment project.

**Example of calculating real terms NPV**Using the real contributions calculated earlier, a real discount rate of 4.0% and the initial investment of $1,000,000:

Year 1: 655,534/ 1.04 = | $630,321 |

Year 2: 774,376/ 1.04^{2} = |
$715,954 |

Year 3: 903,544/ 1.04^{3} = |
$803,247 |

Year 4: 1,025,888/ 1.04^{4} = |
$876,933 |

$3,026,455 | |

Initial investment | $1,000,000 |

Real NPV | $2,026,455 |

Allowing for rounding, the nominal NPV and the real NPV are identical, as can be seen by conducting these calculations with a spreadsheet.

#### The effect of taxation

What is the effect on the NPV calculations of including taxation? Assume corporation tax of 25% and straight-line tax-allowable depreciation (TAD) over four years with zero residual value.

#### Nominal terms NPV calculation where tax is not deferred

Annual TAD will be 1,000,000/ 4 = $250,000 per year

Annual TAD tax benefit will be 250,000 x 0.25 = $62,500 per year

Nominal terms after-tax cash flows

Year 1: 687,000 – (687,000 x 0.25) + 62,500 = $577,750

Year 2: 850,500 – (850,500 x 0.25) + 62,500 = $700,375

Year 3: 1,040,000 – (1,040,000 x 0.25) + 62,500 = $842,500

Year 4: 1,237,500 – (1,237,500 x 0.25) + 62,500 = $990,625

The nominal after-tax cost of capital is approximately 9 x (1 – 0.25) = 6.75%

Discounting to find the nominal terms after-tax NPV:

Year 1: 577,750/ 1.0675 = | $541,218 |

Year 2: 700,375/ 1.0675^{2} = |
$614,603 |

Year 3: 842,500/ 1.0675^{3} = |
$692,574 |

Year 4: 990,625/ 1.0675^{4} = |
$762,848 |

$2,611,243 | |

Initial investment | $1,000,000 |

Nominal NPV | $1,611,243 |

#### Real terms NPV calculation where tax is not deferred

Real terms after-tax cash flows

Year 1: 577,750/ 1.048 = $551,288

Year 2: 700,375/ 1.048^{2} = $637,688

Year 3: 842,500/ 1.048^{3} = $731,958

Year 4: 990,625/ 1.048^{4} = $821,229

The real after-tax cost of capital is related to the nominal after-tax cost of capital by the Fisher equation, so the real after-tax cost of capital is approximately (1.0675/ 1.048) = 1.0186 or 1.86%

Discounting to find the real terms after-tax NPV:

Year 1: 551,288/ 1.0186 = | $541,221 |

Year 2: 637,688/ 1.0186^{2} = |
$614,612 |

Year 3: 731,958/ 1.0186^{3} = |
$692,588 |

Year 4: 821,229/ 1.0186^{4} = |
$762,868 |

$2,611,289 | |

Initial investment | $1,000,000 |

Real NPV | $1,611,289 |

Once again, considering rounding, the nominal terms and real terms after-tax NPVs are the same.

#### Nominal terms NPV calculation where tax is deferred

Annual TAD will be 1,000,000/ 4 = $250,000 per year

Annual TAD tax benefit will be 250,000 x 0.25 = $62,500 per year

Nominal terms after-tax cash flows

Year 1: $687,000

Year 2: 850,500 – (687,000 x 0.25) + 62,500 = $741,250

Year 3: 1,040,000 – (850,500 x 0.25) + 62,500 = $889,875

Year 4: 1,237,500 – (1,040,000 x 0.25) + 62,500 = $1,040,000

Year 5: 62,500 – (1,237,500 x 0.25) = -$246,875

The nominal after-tax cost of capital is again approximately 9 x (1 – 0.25) = 6.75%

Discounting to find the nominal terms after-tax NPV:

Year 1: 687,000/ 1.0675 = | $643,560 |

Year 2: 741,250/ 1.0675^{2} = |
$650,473 |

Year 3: 889,875/ 1.0675^{3} = |
$731,519 |

Year 4: 1,040,000/ 1.0675^{4} = |
$800,870 |

Year 5: -246,875/ 1.0675^{5} = |
-$178,089 |

$2,648,333 | |

Initial investment | $1,000,000 |

Nominal terms NPV | $1,648,333 |

#### Real terms NPV calculation where tax is deferred

Real terms after-tax cash flows

Year 1: 687,000/ 1.048 = $655,534

Year 2: 741,250/ 1.048^{2} = $674,904

Year 3: 889,875/ 1.048^{3} = $773,117

Year 4: 1,040,000/ 1.048^{4} = $862,161

Year 5: -246,875/ 1.048^{5} = -$195,286

The real after-tax cost of capital was calculated above to be 1.86%

Discounting to find the real terms after-tax NPV:

Year 1: 655,534/ 1.0186 = | $643,564 |

Year 2: 674,904/ 1.0186^{2} = |
$650,481 |

Year 3: 773,117/ 1.0186^{3} = |
$731,534 |

Year 4: 862,161/ 1.0186^{4} = |
$800,892 |

Year 5: -195,286/ 1.0186^{5} = |
-$178,095 |

$2,648,376 | |

Initial investment | $1,000,000 |

Real NPV | $1,648,376 |

Once again, considering rounding, the nominal terms and real terms after-tax NPVs are the same.

**Nominal terms or real terms approach to investment appraisal?**If an exam question contains specific inflation rates, but does not provide a general rate of inflation, the nominal terms approach must be used.

If an exam question contains specific inflation rates and also provides a general rate of inflation, the nominal terms approach is be quicker and is recommended, since nominal cash flows must be calculated using specific inflation before the deflating these by the general rate of inflation to give real cash flows for use in a real terms approach. Note that if a real terms approach is adopted, the specific inflation rates cannot be ignored.

Of course, a question may explicitly require a nominal terms approach to be adopted, or a real terms approach, or both.

#### Conclusion

If care is taken to understand the differences between the nominal terms approach and the real terms approach to investment appraisal, and if care is taken to understand the requirements of an exam question in the area of investment appraisal that incorporates inflation and taxation, candidates are likely to do well in this part of the syllabus.