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Engineering Economics: free practice, theory and problems

Money has a time value: 100 NOK today is worth more than 100 NOK in a year, because the amount can earn interest in the meantime. This is the foundation of engineering economics: to compare two investments, a loan or savings spread over several years, you must convert every amount to the same point in time before comparing them directly. The tool for that is interest calculation – future value when you move forward in time, and present value (discounting) when you move backward.

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Interest and time value
Investment analysis
Costs and profitability

1. Interest and time value

What is it about?

Concepts and formulas

Future value of an amount $P$ after $n$ years at annual interest rate $r$ : $F = P(1+r)^n$ .
Present value of an amount $F$ received in $n$ years: $P = F/(1+r)^n$ . Converting a future amount to present value is called discounting, and $r$ is then called the discount rate.
Compound interest means that the interest in a period is also calculated on interest already earned, not just on the original amount. That gives exponential growth.
The nominal rate is the stated annual rate. The effective rate is the actual annual rate once you account for how often interest is added (compounding) and any fees: with monthly compounding of a nominal rate $r$ , the effective rate is $(1 + r/12)^{12} - 1$ .
The real interest rate is the rate adjusted for inflation: $(1 + r_{real}) = (1 + r_{nom})/(1 + i)$ , where $i$ is inflation. The real rate tells you how much your purchasing power actually grows.
Rule of 72: an amount doubles in about $72/r_{\%}$ years.

How to solve the problems

Decide whether you are moving forward in time (future value) or backward (present value).
Check whether the rate is nominal or effective, and whether it compounds more often than annually. Convert to an effective annual rate first if needed.
Insert into the right formula and calculate.
If the problem is about purchasing power over time, use the real rate instead of the nominal rate.

Example

You deposit 50,000 NOK at a nominal rate of 6% with monthly compounding. How much do you have after 4 years, in today's purchasing power, if inflation is 3% per year?

Future value (nominal NOK): $F = 50.000\cdot(1 + 0.06/12)^{12\cdot 4} \approx 63.524$ NOK.
The real value in today's money: $P = F/(1+i)^n = 63.524/1.03^{4} \approx 56.441$ NOK.

Answer: about 56,441 NOK in today's purchasing power, even though the account shows 63,524 NOK.

Common mistakes

Mixing nominal and effective rates without converting.
Using the nominal rate when the problem is really about purchasing power (then the real rate should be used).
Forgetting to keep months/years consistent in the exponent.
Thinking compound interest is only "a little extra" – over a long time the difference from simple interest is large.

Forward in time: multiply by

(1+r)^n

. Backward in time: divide by

(1+r)^n

. If the question asks about purchasing power, use the real rate.

Concepts in this part

Present value and discountingMoney today is worth more than the same amount in the future, because it can earn interest in the meantime. Th…Compound interestWith compound interest, interest is also calculated on previously earned interest, not just the original amoun…Effective and nominal interestNominal interest is the stated annual rate, while effective interest is the actual annual rate once compoundin…Real interest rateThe real interest rate is the rate adjusted for inflation, showing how much purchasing power actually grows. I…

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2. Investment analysis

What is it about?

Investment analysis is about deciding whether a project is profitable enough to carry out, and about comparing loan alternatives. Because payments and receipts occur at different times, you cannot simply add up the amounts – you must discount every cash flow to the same point in time (usually today) using a discount rate that reflects the required return and the risk. The three main methods are net present value (NPV), internal rate of return (IRR) and the payback period.

Concepts and formulas

Net present value: $NPV = -I_0 + \sum_{t=1}^{n} \dfrac{CF_t}{(1+r)^t}$ , where $I_0$ is the investment and $CF_t$ the cash flow in year $t$ . Profitable if $NPV > 0$ .
When the cash flow is the same amount $A$ every year, the sum can be written with the annuity factor: $NPV = -I_0 + A\cdot\dfrac{1-(1+r)^{-n}}{r}$ .
Internal rate of return (IRR): the discount rate that makes $NPV = 0$ . Profitable if the IRR is higher than the required return.
Simple payback period: $I_0$ divided by the annual saving/cash flow. Easy to understand, but ignores the time value of money and everything that happens after the payback period.
Annuity loan: equal payments each year, $A = P\cdot\dfrac{r}{1-(1+r)^{-n}}$ (interest + principal).
Serial loan: equal principal repayments, so the payment – and the total interest – ends up lower than for an annuity loan with the same rate and term.
The discount rate is usually set higher than the bank rate to account for risk and alternative returns (what the money could otherwise earn).

How to solve the problems

List the investment $I_0$ and the cash flows $CF_t$ year by year, with signs (payments negative, receipts positive).
Is the cash flow the same every year? Use the annuity factor. Is it different? Discount each year separately and add.
Compare $NPV$ with zero, or find the $IRR$ and compare it with the required return.
Remember that the payback period is only a rough extra measure, not a replacement for NPV/IRR.

Example

A machine costs 400,000 NOK and saves 90,000 NOK per year for 6 years. The discount rate is 8%. Is the investment profitable?

Annuity factor: $\dfrac{1-1.08^{-6}}{0.08} \approx 4.623$ .
$NPV = -400.000 + 90.000\cdot 4.623 \approx 16.059$ NOK.

Answer: $NPV > 0$ , so the investment is profitable.

Common mistakes

Using the annuity factor when the cash flows actually differ from year to year.
Mixing up the discount rate and the bank rate – the discount rate should reflect the risk of this specific project.
Concluding from the payback period alone and ignoring cash flows after it.
Forgetting that the investment $I_0$ must be negative (a payment) in the NPV formula.

NPV > 0

IRR >

required return means profitable. Different cash flows: discount each year separately – never use the annuity factor then.

Concepts in this part

Net present value (NPV)Net present value sums the present value of all future cash flows and subtracts the investment. If NPV is posi…Internal rate of return (IRR)The internal rate of return is the rate that makes net present value equal to zero — the point where the proje…Payback periodSimple payback period is how long it takes before the annual savings or income has covered the investment, wit…Annuity and serial loansAn annuity loan has equal instalments through the whole term, where the interest portion decreases and the pri…

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3. Costs and profitability

What is it about?

Before you can say whether a product or project is profitable, you need to understand how costs behave. Some costs are fixed no matter how much you produce, others are variable and follow the production volume. This distinction underlies break-even analysis, pricing decisions and choices such as picking one production method over another.

Concepts and formulas

Fixed costs $FC$ : do not change with the quantity produced (rent, fixed salaries, insurance).
Variable cost $v$ per unit: increases in proportion to the number of units produced (raw materials, electricity per unit).
Contribution margin per unit: $cm = p - v$ , where $p$ is the selling price. The contribution margin ratio is $cm/p$ (the share of the price left to cover fixed costs and provide profit).
Break-even in units: $Q_0 = FC/(p - v)$ . In revenue: $Q_0\cdot p$ .
Target profit $Z$ : the number that must be sold is $Q = (FC + Z)/(p - v)$ – the same formula as break-even, just with $Z$ added to the fixed costs.
Straight-line depreciation: $(I - \text{salvage value})/n$ per year, the same amount every year.
Declining-balance depreciation: the book value is reduced by a fixed percentage $d$ each year: $V_n = I(1-d)^n$ . It gives higher depreciation early and lower later, unlike straight-line depreciation.
Opportunity cost: the value of the best alternative you give up. Sunk cost: costs already incurred that cannot be recovered – they should not affect new decisions.
Economies of scale: the cost per unit falls as volume increases, because the fixed costs are spread over more units.

How to solve the problems

Sort the costs into fixed and variable.
Set up the result as (price − variable cost) × quantity − fixed costs, and set it equal to zero (or to the target profit) to find the break-even point.
For depreciation: decide whether the method is straight-line or declining-balance, and use the right formula.
Ignore sunk costs and include the correct opportunity cost when comparing alternatives.

Example

A company has fixed costs of 180,000 NOK per year. The product sells for 320 NOK and costs 200 NOK in variable cost per unit. How many must be sold to achieve a profit of 60,000 NOK?

Contribution margin: $cm = 320 - 200 = 120$ NOK per unit.
$Q = (FC + Z)/cm = (180.000 + 60.000)/120 = 2.000$ units.

Answer: 2,000 units.

Common mistakes

Computing the break-even point with the full selling price instead of the contribution margin.
Using the straight-line formula when the problem describes a fixed percentage rate per year (that is declining-balance depreciation).
Letting a sunk cost affect a new decision ("we've already spent so much on this").
Forgetting that the fixed cost per unit changes with volume, even though the total fixed cost does not.

Break-even:

Q_0 = FC/cm

. Add the target profit to the numerator to find a sales goal. Declining-balance multiplies by

(1-d)

each year; straight-line subtracts the same amount every year.

Concepts in this part

Fixed and variable costsFixed costs, like rent, do not change with the quantity produced. Variable costs, like raw materials, increase…Contribution margin and break-evenThe contribution margin per unit is the price minus variable cost per unit — what is left to cover fixed costs…Straight-line depreciationStraight-line depreciation spreads the loss in value of an asset evenly over its lifetime, so the same amount …Opportunity cost and sunk costOpportunity cost is the value of the best alternative given up by choosing something else, and should count in…

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Example problems with solutions

Here are some of the problems in engineering Economics. In the app, calculation problems get new numbers every time, so you can practise until it sticks – and take a graded practice exam before the real one.

Interest and time value: Why is 100 NOK today worth more than 100 NOK in one year?

Answer: The money can earn interest in the meantime (plus inflation and risk)

This is the basis for discounting.

Investment analysis: When is an investment profitable according to the net present value method?

Answer: When the net present value (NPV) is greater than zero

NPV > 0 means the return is better than the discount rate.

Costs and profitability: What is the difference between fixed and variable costs?

Answer: Fixed costs do not change with the quantity produced, variable costs do

Rent is fixed, and raw materials are variable.

Interest and time value: What is the present value of an amount

F

n

years at interest rate

r

Answer: $F/(1+r)^n$

This is called discounting.

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Matches these university courses

The content covers the syllabus found in engineering degrees, for example:

INN200 (NMBU)

Engineering Economics: free practice, theory and problems

Contents

1. Interest and time value

What is it about?

Concepts and formulas

How to solve the problems

Example

Common mistakes

Concepts in this part

2. Investment analysis

What is it about?

Concepts and formulas

How to solve the problems

Example

Common mistakes

Concepts in this part

3. Costs and profitability

What is it about?

Concepts and formulas

How to solve the problems

Example

Common mistakes

Concepts in this part

Example problems with solutions

Matches these university courses