Before you start
You will be able to
- State the six principles of financial decision-making and explain the reasoning behind each.
- Apply the time value of money and the trade-off between risk and return to compare choices.
- Distinguish cash flow from accounting profit and explain how a profitable business can run short of cash.
- Identify the incremental cash flows of a decision, including opportunity costs, and exclude sunk costs and unchanged costs.
- Explain how diversification reduces risk and why it cannot remove risk that affects the whole economy.
- Explain the law of one price and calculate the profit from a simple arbitrage.
Review first
- What finance is: money, time and risk — You need future value, present value, expected value and the risk premium.
- Finance, accounting and economics — You need opportunity cost and the difference between accounting profit and cash flow.
Fictional case A fictional coffee business founded by Ana Lim. It starts as a catering and barista-training service, opens its own cafés and an online shop for beans, and plans a canned cold brew.
The idea
Fictional case
In the last week of February 2026, Ana has resigned from the café chain and has a week before Lantern opens. Each day brings a decision: whether to pay for an equipment warranty now or later, how much of her savings to put into the business, how much cash Lantern needs on day one, whether a workshop idea is worth pursuing. The decisions are small, but the reasoning behind them is the same reasoning a company uses for a new factory or a pension fund for its portfolio.
Finance condenses that reasoning into a handful of principles. Different textbooks list them differently and under different names; the six below are among the most widely used. This lesson takes them one at a time, each with a decision from Lantern’s first week, and then shows how they fit together. None needs more mathematics than the previous two lessons.
1. The time value of money
A dollar today is worth more than a dollar in the future, because today’s dollar can earn a return in the meantime. You met this principle in the first lesson. Its practical rule is that amounts arriving at different dates must be moved to the same date, by compounding or discounting, before they are compared or added.
Fictional case
The equipment supplier offers Ana a two-year extended warranty on the espresso machine for $600 paid now, or $650 paid at the end of the second year. Her spare cash earns 1.5% a year in her savings account. The present value of paying later is dollars. Paying $600 now is cheaper by about $30.93 in today’s money, so if she buys the warranty, paying now is the better choice. (Carried forward instead, $600 would have grown to only $618.14 by the time the $650 was due: the same answer.)
2. Risk and return go together
Higher expected returns are, on average, the reward for bearing more risk; there is no reliable way to earn more without accepting more uncertainty. Because investors are risk averse, a risky investment must offer a risk premium to attract their money. The flip side is a useful warning: an investment that promises a high return with little risk is either misunderstood or a fraud.
Fictional case
Ana’s savings account pays a certain 1.5%. Lantern, on the estimates in the first lesson, is expected to pay her $54,400 in its first year against $52,300 from her salary and interest, but with a one-in-five chance of only $12,000. The extra $2,100 she expects is the premium for that risk. Lantern is not a better investment than the bank because its expected return is higher; it is a riskier one that is expected to pay more. Whether the premium is enough is her judgement.
3. Cash flow, not accounting profit
Value comes from cash, not from accounting profit. Profit cannot pay wages or suppliers; cash can. Accounting profit is measured on the accrual basis, which records revenue when it is earned and spreads the cost of equipment over its life, so profit and cash can be far apart in any one period. Finance measures decisions in cash, and in when the cash arrives.
Fictional case
Suppose that in one month Lantern caters events invoiced at $6,000, payable in 30 days, and pays $3,800 in cash for supplies, wages and rent. Its espresso machine and grinder, which cost $12,000, are expected to last five years, so the accounts charge dollars of depreciation a month. The month’s accounting profit is dollars. Yet no customer has paid yet, so the business’s cash has fallen by $3,800. A profitable month, and Ana must have enough cash in the bank to survive it (all figures in this example are fictional).
4. Only incremental cash flows matter
Judge a decision by the cash flows it changes, and only those. An incremental cash flow is the difference between the cash flows with the decision and without it. Two kinds of cost need special care. An opportunity cost (the value of the best alternative given up) is incremental even though no money changes hands, so it must be counted. A sunk cost, money already spent that cannot be recovered whatever you decide now, is not incremental, so it must be ignored. So must costs that stay the same either way.
Fictional case
For Ana, opening Lantern has two opportunity costs: the $52,000 salary she gives up and the $300 of interest a year that her $20,000 would have earned at 1.5%. Both count. In January she paid $900 from her January pay for a logo and a market-research report (a fictional detail of this lesson). That money is gone whether or not Lantern opens: it is sunk, and it should play no part in her decision. Her flat’s rent is the same either way, so it is not incremental either. The comparison that matters is the one from the first lesson: $54,400 expected from Lantern against $52,300 given up.
5. Diversification reduces risk
Combining investments whose results do not move in lockstep reduces the risk of the whole without a matching fall in expected return: the expected return is the weighted average of the parts, but the risk is lower than the weighted average of theirs. When one investment does badly, another may do well, so the ups and downs partly cancel. This is diversification, summed up in the saying “don’t put all your eggs in one basket”. It cannot remove every risk: when the whole economy turns down, most investments fall together.
Fictional case
Ana’s finances are about to be concentrated in one basket. Her income will come from Lantern, and $20,000 of her $26,000 of savings will be invested in it, so a bad year for Lantern would hit both at once. Keeping $6,000 as an emergency fund in an insured bank account, outside the business, is a small act of diversification: that money does not depend on Lantern’s fortunes. Inside the business, Lantern’s two activities, catering for offices and events and running barista workshops, give a little diversification too, though both depend on the same local economy.
6. No arbitrage: the law of one price
Two things that produce identical cash flows should sell for the same price. This is the law of one price. If they did not, anyone could buy the cheaper one and sell the dearer one at once, locking in a profit with no risk and none of their own money. Such a trade is an arbitrage. Traders who spot one rush to exploit it, and their buying and selling pushes the two prices together, so in well-functioning markets arbitrage opportunities are rare and short-lived.
Fictional case
Ana finds the same model of coffee grinder listed at $1,150 by one supplier and $1,000 by another. Before assuming she has found a bargain, she checks what the price includes. The dearer listing includes delivery, installation and a two-year warranty; the cheaper one includes none of them. Once she prices those extras, the two offers cost about the same. The law of one price did not fail: the two offers were not identical. When prices for apparently identical things differ, the first question is what makes them different.
In depth
How the principles fit together
The six principles are not a list of separate tips. Together they describe how to value any decision. Start from the cash flows the decision changes (principles 3 and 4). Each arrives at a particular date and is uncertain, so estimate its expected value and move it to today by discounting (principle 1) at a rate that includes a premium for its risk (principle 2). The premium should reflect only the risk that cannot be diversified away (principle 5). And the result should be consistent with the prices of similar things already traded in markets (principle 6). F05 Corporate Finance turns this into a full method, called net present value; the lessons so far already contain its logic.
| Principle | In one line | The mistake it prevents |
|---|---|---|
| Time value of money | Move amounts to the same date before comparing them | Treating $650 in two years as more than $630.93 today |
| Risk and return | Higher expected return is the reward for higher risk | Comparing a risky return with a safe one as if they were alike |
| Cash, not profit | Decisions are measured in cash and its timing | Running out of cash in a profitable month |
| Incremental cash flows | Count only what the decision changes, including opportunity costs | Letting sunk costs or unchanged costs sway a decision |
| Diversification | Combine investments that do not move in lockstep | Holding everything in one basket and calling it bad luck when it breaks |
| No arbitrage | Identical cash flows have one price | Believing in a risk-free profit that is really a hidden difference |
Cash and profit: why they differ
Profit and cash flow differ for three main reasons, each of which you will study in F01. Timing of revenue: revenue is recorded when the service is delivered, but customers on credit pay later, and deposits arrive before the work is done. Timing of expenses: supplies bought in one month may be used in the next, and bills may be paid before or after the cost is recorded. Long-lived assets: equipment is paid for at once but expensed gradually through depreciation. Over the whole life of a business the totals come together, but within a month or a year the gap can be large, and a business that runs out of cash can fail even when its accounts show a profit. That is why lenders and finance managers watch cash forecasts as closely as profit.
Incremental thinking in practice
Finding the incremental cash flows means asking, for every item, “what would be different if we did not do this?”. Four kinds of item need care.
- Opportunity costs count, even though no cash changes hands. A building the business already owns is not free if it could otherwise be rented out: the rent forgone is a cost of using it.
- Sunk costs do not count. Money spent on research, a design or a failed launch is gone whichever choice is made. Letting it influence the choice, often by continuing a failing project “so the money isn’t wasted”, is called the sunk cost fallacy.
- Allocated costs count only if they change. Many organisations share head-office costs among their departments. If closing a department would not reduce those costs, its share of them is not incremental, however it is labelled in the accounts.
- Side effects count. A new product that takes sales from an existing one, or a new service that brings customers to an old one, changes cash flows elsewhere in the business, and those changes belong to the decision.
Diversification and its limits
How much diversification helps depends on how closely investments move together, which statisticians measure with the correlation coefficient: +1 for investments that always move together, −1 for investments that always move in opposite directions, and 0 for no linear relationship. The more closely two investments move together, the less combining them reduces risk. Owning ten airline shares is little safer than owning one, because fuel prices and travel demand hit them all; owning an airline and an oil company helps more, because higher fuel prices hurt one and help the other.
Risk therefore comes in two kinds. Unsystematic risk (also called specific or diversifiable risk) belongs to one company or industry: a product recall, a lawsuit, a key customer lost. It can be largely diversified away by holding many investments. Systematic risk (market or non-diversifiable risk) affects nearly everything at once: recessions, changes in interest rates, pandemics. No amount of diversification removes it. Because investors can remove unsystematic risk cheaply by diversifying, markets do not pay a premium for bearing it; the risk premium rewards systematic risk. That is a statement about market prices, where investors can diversify cheaply. An owner like Ana, with her income and most of her savings in one business, cannot diversify Lantern’s specific risk away, so she may reasonably want a larger premium. F06 Investment Analysis develops this into the capital asset pricing model.
Arbitrage and why prices line up
An arbitrage, in the strict sense, needs no money of your own and carries no risk, yet produces a profit. The law of one price follows from the assumption that such opportunities do not last. It also explains why present value is more than a calculation. Suppose a safe deposit pays 3% a year and a safe promise to pay $1,030 in one year is for sale. Its price must be dollars. If it sold for $990, you could borrow $990 at 3%, buy the promise, collect $1,030 in a year, repay dollars, and keep dollars, with no risk and none of your own money. Traders would keep doing this until the price rose to $1,000. Present value is the price at which no arbitrage is possible.
In real markets the law holds only approximately. Transaction costs, transport, taxes and limits on borrowing or on short selling (selling something borrowed, to buy it back later) can let prices differ by up to the cost of exploiting the gap; studies of historical commodity markets describe the law as a pull towards one price rather than a state that holds at every moment. Arbitrage is also often confused with ordinary speculation. Buying shares you expect to rise is not arbitrage, because it carries risk. Later courses use no-arbitrage reasoning to price bonds (F06 Investment Analysis) and derivatives (F08 Advanced Financial Decision-Making).
The limits of the principles
The principles describe how to reason, not what the answer is. Every one of them works on estimates: of cash flows, of probabilities, of the right discount rate. They also assume people decide calmly and consistently, and behavioural finance has documented many ways in which real people do not, such as weighting losses more heavily than equal gains and clinging to sunk costs. Used well, the principles make those estimates and biases visible, so that a decision can be questioned before it is made.
Formulas
Incremental cash flow of a decision
- Cash flows of the business (or person) if the decision is taken, in a given period$
- Cash flows in the same period if it is not taken$
Assumes. Opportunity costs are included as cash given up; sunk costs, which are the same in both cases, cancel out.
Work it out period by period, then discount each period’s figure to today.
Return on a portfolio of two investments
- Return on the whole portfolio%
- Share of the money invested in A, as a decimal (the rest is in B)
- Return on investment A%
- Return on investment B%
Assumes. The shares w and 1 − w are fixed for the period.
The portfolio’s return is a weighted average, but its risk can be lower than either investment’s, because their ups and downs partly cancel.
No-arbitrage price of a safe future payment
- Today’s price of the promise$
- The certain amount to be paid$
- Risk-free rate at which money can be borrowed and lent, as a decimalper year
- Years until paymentyears
Assumes. The payment is certain, and anyone can borrow and lend at the risk-free rate with no transaction costs.
At any other price a riskless profit could be made, so traders would push the price back to P.
Worked examples
Fictional figures
Cliffside Hotel, a fictional hotel, runs a conference business. In October it hosts events invoiced at $14,000, all due in November. During October it collects $5,000 for events hosted in September, pays $10,500 in cash for staff, food and equipment hire, and records $1,000 of depreciation on its conference equipment. What are October’s accounting profit and change in cash, and why do they differ?
- Accounting profit (accrual basis): October’s revenue is the $14,000 of events hosted in October, whether or not they have been paid for, so profit is dollars.
- Cash: $5,000 came in (for September’s events) and $10,500 went out. Depreciation is not a payment. So cash changes by dollars.
- The gap is dollars, and each piece has a cause: the $14,000 of October revenue not yet collected, less the $5,000 collected that was September’s revenue, less the $1,000 of depreciation that used no cash: .
Answer. Profit of $2,500; cash down by $5,500.
What it means. October is profitable, but the conference business needs $5,500 of cash from somewhere to get through it. If November’s collections arrive as planned the gap closes; if customers pay late, or the business keeps growing, it widens. This is why principle 3 says to measure decisions in cash: a business pays its bills with cash, not profit.
Fictional figures
Ana is considering a weekend workshop series once Lantern is running. She estimates (all figures fictional): extra workshop revenue of $1,500 a month, all paid in advance; extra supplies of $250 a month; an assistant for the weekends at $400 a month. Lantern’s kitchen rent of $1,200 a month is paid anyway; Ana’s spreadsheet allocates all of it to the workshops because they use the kitchen. Ana could instead spend the weekends doing private barista consulting for about $300 a month. She has already paid $500 for a workshop brochure design. What is the monthly incremental cash flow, and should she go ahead?
- Revenue, supplies and the assistant all change with the decision: dollars a month.
- The kitchen rent is the same whether or not the workshops run, so it is not incremental. Charging it to the workshops would wrongly make them look like a loss of dollars a month.
- The consulting income Ana gives up is an opportunity cost: it is incremental even though Lantern never pays it. Subtract it: dollars a month.
- The $500 brochure design is sunk: it is spent whether or not the workshops go ahead, so it is left out.
Answer. The incremental cash flow is about $550 a month, so on these estimates the workshops are worth running.
What it means. One error would have reversed the answer: charging the unchanged rent turns the $850 into an apparent loss of $350 a month. Another would have flattered it: forgetting the opportunity cost makes the workshops look $300 a month better than they are. The sunk $500 should not tip the decision either way. Asking “what changes if I do this?” for each line keeps the analysis honest.
Fictional figures
Eastmere University, a fictional university, runs a café in its library. Last year’s accounts show revenue of $180,000; staff, food and other cash costs of $150,000; $15,000 of depreciation on a refit completed three years ago; and $35,000 of central university overheads (finance, cleaning contracts, insurance) allocated to the café. The finance office confirms that closing the café would not reduce the central overheads. A coffee chain offers to lease the space for $24,000 a year. Should the university close its café and lease the space?
| Item | Amount ($) | Incremental if the café closes? |
|---|---|---|
| Revenue | 180,000 | Yes: lost |
| Staff, food and other cash costs | (150,000) | Yes: saved |
| Depreciation on the refit | (15,000) | No: non-cash, and the refit is sunk |
| Allocated central overheads | (35,000) | No: unchanged |
| Lease income offered | Yes: gained only if the café closes | |
| Accounting result as reported | (20,000) |
- Accounting result: dollars. On the accounts the café loses money, which is why closing it is being discussed.
- Incremental cash from keeping the café: only revenue and the cash costs change, so dollars a year. Depreciation uses no cash and the refit is sunk; the overheads are paid either way.
- Opportunity cost of keeping it: the $24,000 a year of lease income given up.
- Comparison: keeping the café brings dollars a year more cash than leasing the space.
Answer. Keep the café: it adds $30,000 a year of cash, $6,000 more than the $24,000 lease, despite showing a $20,000 accounting loss.
What it means. The accounting loss answers a different question (how much of the university’s total cost the café “bears”) from the decision (what changes if it closes). Closing it would cut the university’s cash by $30,000 a year while bringing in $24,000. The margin is thin, so the university should test the estimates, and it should weigh things the cash flows omit, such as students’ convenience.
Explore
Common mistakes
“We have spent too much to stop now.”
Money already spent is a sunk cost: it is gone whatever you decide. The only question is whether the future cash flows from continuing are worth more than the future cash flows from stopping. Continuing a failing project to justify past spending, the sunk cost fallacy, turns a past loss into a bigger one.
Leaving out costs that involve no payment
An opportunity cost is real even though no money changes hands. Ana’s $52,000 salary does not appear in Lantern’s accounts, but it is the largest cost of her decision to run Lantern. Ask what the best alternative use of the time, money or space would have earned.
Charging allocated costs to a decision that does not change them
A share of rent, head-office costs or overheads allocated to a product or department is not incremental if the total would be paid anyway. Eastmere’s café “loses” $20,000 only because it carries $35,000 of overheads that closing it would not save.
“If it is profitable, it has the cash.”
Profit is measured on the accrual basis; cash depends on when customers and suppliers actually pay and when assets are bought. A business can be profitable and run out of cash, as Lantern’s example month shows. Plan cash separately from profit.
“I own twenty shares, so I am diversified.”
Diversification depends on how closely the investments move together, not on how many there are. Twenty shares in one industry share the same risks. And even a widely diversified portfolio still carries systematic risk: when the whole market falls, it falls too.
“Buying low and selling high is arbitrage.”
Arbitrage, strictly, is riskless: buying and selling the same cash flows at the same time for different prices. Buying something because you expect its price to rise is speculation, and it carries risk. When two seemingly identical things have different prices, look first for a difference between them.
Guided practice
Work through these with hints when you need them. Every hint and every worked solution is there to be used.
Loading four questions…
On your own
No hints this time. Answer, check, and read the feedback; the solution appears once you have answered or if you ask for it. Your first answers count toward mastery.
Loading five questions…
In practice
Fictional case
Fenwick Foods, a fictional maker of snack bars, built a new production line two years ago for $4 million. The line is underused, and its accounts show a loss of $300,000 a year, after $400,000 of depreciation and $250,000 of head-office costs allocated to it. A rival manufacturer has offered $1.5 million for the line. Fenwick’s finance manager is asked whether to sell.
She starts by setting aside the two numbers that dominate the board’s discussion. The $4 million spent two years ago is sunk: it cannot be recovered by keeping the line or by selling it, so it is irrelevant to the choice, however painful it is to sell for less. And the $300,000 accounting loss is the wrong measure, because it includes depreciation, which uses no cash, and allocated head-office costs, which her colleagues confirm would not fall if the line were sold.
She rebuilds the figures in cash. Adding back the depreciation and the unchanged head-office costs, the line contributes dollars of cash a year. Production expects it to run for five more years. (The accounts spread its cost over the ten-year life estimated when it was built, hence $400,000 a year; that old estimate does not matter now, because the decision depends only on the cash expected from here on.) Discounting each year’s $350,000 at 10%, a rate that includes a premium for the line’s exposure to the economy as a whole, gives present values of $318,181.82, $289,256.20, $262,960.18, $239,054.71 and $217,322.46: a total of $1,326,775.37.
The rival’s offer is an opportunity cost of keeping the line: $1.5 million today, certain. It is worth about dollars more than the present value of keeping the line, so on these estimates Fenwick should sell. She also checks the offer against the law of one price: similar used lines have recently changed hands for comparable amounts, which suggests the offer is a fair market price rather than a bargain or a trap.
Her report notes what would change the answer: if the line’s cash flows were 15% higher, or it could run for longer, keeping it would win. It also notes what does not matter. The board worried that the line’s product was risky because it depends on one supermarket chain. That dependence belongs in the cash-flow forecast, as the chance of losing the contract, but it does not justify a higher discount rate, because Fenwick’s shareholders can diversify that specific risk away by holding many other investments, a point F05 Corporate Finance takes further. In one decision she used all six principles: time value, risk, cash, incremental flows (including an opportunity cost and a sunk cost), diversification and the law of one price.
Knowledge check
Six questions, one attempt each, no hints. Score 80% or more to complete the lesson; you can take it again at any time.
Loading six questions…
Summary
- Time value of money: move amounts to the same date before comparing them.
- Risk and return: a higher expected return is the reward for bearing more risk; a promise of high return with low risk deserves suspicion.
- Cash, not profit: decisions are measured in cash and its timing; a profitable business can still run short of cash.
- Incremental cash flows: count only what a decision changes, including opportunity costs; ignore sunk costs and costs that stay the same.
- Diversification: combining investments that do not move together reduces unsystematic risk, but not the systematic risk that affects the whole economy.
- No arbitrage: identical cash flows should have one price; present value is the price at which no riskless profit is possible.
- Together the principles say: a decision adds value if the present value of its expected incremental cash flows, discounted for time and systematic risk, exceeds its cost.
Key formulas
- Incremental cash flow of a decision
- Return on a portfolio of two investments
- No-arbitrage price of a safe future payment
Terms introduced
- Incremental cash flow
- The difference between the cash flows with a decision and without it; the only cash flows that should influence the decision.
- Sunk cost
- Money already spent that cannot be recovered whatever is decided now. It is not an incremental cash flow and should be ignored.
- Sunk cost fallacy
- Letting money already spent influence a decision, typically by continuing a failing project so that past spending is not “wasted”.
- Diversification
- Spreading money across investments whose results do not move in lockstep, so that losses on some are partly offset by gains on others.
- Unsystematic risk
- Risk specific to one company or industry, such as a product recall or a lost customer, which can be largely removed by diversification. Also called specific or diversifiable risk.
- Systematic risk
- Risk that affects nearly all investments at once, such as a recession or a change in interest rates, and cannot be removed by diversification. Also called market or non-diversifiable risk.
- Law of one price
- The principle that identical goods or assets, producing identical cash flows, should sell for the same price, because otherwise traders could profit by buying the cheaper and selling the dearer.
- Arbitrage
- Buying and selling the same, or equivalent, cash flows at the same time for different prices, making a profit with no risk and none of one’s own money.
Sources
- Greenlaw, S. A., Shapiro, D. & MacDonald, D. (2022). Principles of Economics (3rd ed.), section 2.1, “How Individuals Make Choices Based on Their Budget Constraint” (opportunity cost and sunk costs). OpenStax. openstax.org/books/principles-economics-3e/pages/2-1-how-ind…
- Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 15.2, “Risk and Return to Multiple Assets”. OpenStax. openstax.org/books/principles-finance/pages/15-2-risk-and-re…
- Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 7.1, “Now versus Later Concepts”. OpenStax. openstax.org/books/principles-finance/pages/7-1-now-versus-l…
- US Securities and Exchange Commission, Office of Investor Education and Advocacy. “Diversify Your Investments”. Investor.gov. investor.gov/introduction-investing/basics/save-invest/diver…
- Persson, K. G. (2008). “Law of One Price”. In R. Whaples (ed.), EH.Net Encyclopedia, 10 February 2008. eh.net/encyclopedia/the-law-of-one-price