Business Economics: free practice, theory and problems

1. Contribution margin and break-even

What is it about?

How many units must a company sell to break even? The answer comes from the contribution margin: what each unit sold contributes towards covering the fixed costs.

Concepts and formulas

• Fixed costs (FC) are the same however much you sell: rent, fixed salaries.
• Variable costs (VC per unit) grow with volume: materials, shipping.
• Contribution margin per unit: $\text{CM} = \text{price} - \text{VC}$.
• Contribution margin ratio: $\dfrac{\text{CM}}{\text{price}} \cdot 100\,\%$.
• Break-even point (units): $x_0 = \dfrac{\text{FC}}{\text{CM}}$. Break-even revenue: $\dfrac{\text{FC}}{\text{CM ratio}}$.
• Profit: $\text{CM} \cdot x - \text{FC}$.
• Margin of safety: how much sales can fall before you make a loss.

Example

Price 200 NOK, VC 120 NOK, FC 400,000 NOK. CM = 80 NOK and the ratio is 40 %. Break-even: $400\,000/80 = 5000$ units.

Practise contribution margin and break-even in the app →

2. Costs and profitability

🔗 Shared curriculum with Engineering economics. The questions are the same, but your progress counts separately for each subject.

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

1. Sort the costs into fixed and variable.
2. 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.
3. For depreciation: decide whether the method is straight-line or declining-balance, and use the right formula.
4. 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?

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

Practise costs and profitability in the app →

3. Interest and time value

🔗 Shared curriculum with Engineering economics. The questions are the same, but your progress counts separately for each subject.

What is it about?

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.

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

1. Decide whether you are moving forward in time (future value) or backward (present value).
2. 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.
3. Insert into the right formula and calculate.
4. 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?

1. Future value (nominal NOK): $F = 50.000\cdot(1 + 0.06/12)^{12\cdot 4} \approx 63.524$ NOK.
2. 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.

Practise interest and time value in the app →

4. Investment analysis

🔗 Shared curriculum with Engineering economics. The questions are the same, but your progress counts separately for each subject.

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

1. List the investment $I_0$ and the cash flows $CF_t$ year by year, with signs (payments negative, receipts positive).
2. Is the cash flow the same every year? Use the annuity factor. Is it different? Discount each year separately and add.
3. Compare $NPV$ with zero, or find the $IRR$ and compare it with the required return.
4. 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?

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

Practise investment analysis in the app →

5. Budgets and liquidity

What is it about?

A budget is a plan for the future in money. The income budget shows whether the company makes a profit. The cash budget shows whether it has money in the bank to pay the bills. A profitable company can still go bankrupt if liquidity fails.

Concepts and formulas

• Revenue and expense (profit) are recorded when the sale/consumption happens. Receipts and payments (liquidity) when the money actually moves.
• Sales on credit: the revenue comes now, the receipt later (for example 30 days' credit).
• Cash budget: closing balance = opening balance + receipts − payments.
• Depreciation is an expense, but not a payment.
• Buying a machine is a payment straight away, but is expensed gradually via depreciation.

Example

Opening bank balance 50,000 NOK. Receipts 120,000 NOK, payments 150,000 NOK. Closing $= 50\,000 + 120\,000 - 150\,000 = 20\,000$ NOK.

Practise budgets and liquidity in the app →