Sunflower Intelligence
Finance & Accounting

F00 · Introductory

Foundations of Finance

9 lessons

F00 · Module 1, lesson 1 · Lesson 1 of 9

What finance is: money, time and risk

Finance studies how people, businesses and governments raise, allocate and manage money over time, under uncertainty. Learn why a dollar today is worth more than a dollar later, how to move an amount forward or back in time, how to put a number on an uncertain outcome, and where finance does its work.

About 75 minutes

Before you start

You will be able to

  • Define finance and explain why time and risk sit at the centre of almost every financial decision.
  • Calculate the future value of an amount today and the present value of an amount received later, over whole years.
  • Calculate the expected value of a simple uncertain outcome and explain a risk premium as compensation for bearing risk.
  • Distinguish personal, corporate, investment and public finance by who makes the decisions and what they decide.
  • Describe the three financial decisions every firm makes and the role finance plays in an economy.

Review first

  • None. This is the first lesson of the course and assumes no background in finance, accounting or economics.
  • Multiplying, dividing and working with percentages (1.5% is 0.015 as a decimal). A calculator with a power key (xʸ or ^) helps.

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

It is January 2026. Ana Lim is head barista trainer at a regional café chain, earning $52,000 a year before tax, and she has $26,000 in a Coastline Bank savings account that pays 1.5% a year. She wants to start her own business, Lantern Coffee Co., which will cater coffee for offices and events and run barista-training workshops, from 1 March. Before she resigns she has a string of questions to answer. Should she give up a safe salary for an uncertain business? How much of her savings should she put in, and how much should she keep for emergencies? Should the business borrow from the bank, and could it afford the interest? None of these questions is about coffee. They are about money, time and risk, and they are what finance studies.

Finance is the study of how people, businesses and governments raise, allocate and manage money over time, under uncertainty. To raise money is to obtain it: from savings, by borrowing, or by selling someone a share of the ownership. To allocate it is to decide where it goes: which investments to make and which to turn down. To manage it is to look after it: keeping enough cash to pay the bills, and controlling the risks. The last two phrases of the definition, over time and under uncertainty, are what set finance apart. Nearly every financial question comes back to one or both of them.

Why time matters

Would you rather have $100 today or $100 a year from now? Almost everyone chooses today, and finance gives a precise reason: money you have now can be put to work. Deposited in Ana’s savings account at 1.5% a year, $100 today becomes $101.50 a year from now. A promise of $100 in a year is therefore worth less than $100 in your hand. This idea is called the time value of money: a dollar today is worth more than a dollar later, because today’s dollar can earn a return in the meantime.

The return that money earns for the passage of time is set by an interest rate: the price of using money for a period, stated as a percentage of the amount per year. A saver who lets a bank use her money receives interest; a borrower who uses someone else’s money pays it. Ana earns 1.5% a year from Coastline Bank on her savings. If Lantern borrows $10,000 from the same bank, as Ana plans, it will pay 6% a year. Notice that the bank pays savers less than it charges borrowers; why, and who gains from the arrangement, comes up at the end of this lesson.

Suppose Ana left $20,000 of her savings in the account instead of putting it into Lantern. After one year it would be worth 20,000×1.015=20,30020{,}000 \times 1.015 = 20{,}300. In the second year the bank pays 1.5% on the whole $20,300, including last year’s $300 of interest, so the balance becomes 20,300×1.015=20,604.5020{,}300 \times 1.015 = 20{,}604.50. Earning interest on earlier interest is called compounding. After three years the balance is $20,913.57. The amount an investment will grow to by a later date is its future value; the amount it is worth today is its present value.

Moving money through time at 1.5% a year. Compounding carries today’s $20,000 forward to $20,913.57 in three years; discounting carries $20,913.57 in three years back to $20,000 today.

The same arithmetic runs backward. If someone promises you $1,015 a year from now, and you could otherwise earn 1.5% at the bank, the promise is worth 1,015÷1.015=1,0001{,}015 \div 1.015 = 1{,}000 dollars today: $1,000 deposited now would grow to exactly that amount. Turning a future amount into its value today is called discounting, and the rate used to do it is the discount rate. Compounding and discounting are the same calculation travelling in opposite directions, as the figure shows.

Why risk matters

The second force is uncertainty. A savings account at a fixed 1.5% is about as predictable as finance gets: Ana knows what she will have in a year. Lantern is not predictable at all. In its first year it might thrive or struggle, and Ana cannot know which in advance. In finance, risk means the possibility that an outcome will differ from what you expect, for better or for worse. The everyday meaning is only the chance of a loss; finance counts surprises in both directions, because a wide spread of possible outcomes is itself what makes planning hard.

To reason about an uncertain outcome you need one number that summarises it. The usual choice is the expected value: weight each possible outcome by its probability and add the results. A coin flip that pays $200 on heads and nothing on tails has an expected value of 0.5×200+0.5×0=1000.5 \times 200 + 0.5 \times 0 = 100 dollars. The expected value is not what you will get. You will get $200 or nothing; $100 is what you would average if you could play the game many times.

Now suppose you are offered the coin flip or a certain $100. Both have the same expected value, yet most people take the certain $100. Preferring a certain amount to a gamble with the same expected value is called risk aversion, and finance assumes that most savers and investors are risk averse. To persuade them to accept a gamble, it must offer more on average: perhaps $240 on heads instead of $200, an expected value of $120. The extra expected reward that people demand for bearing risk, here $20, is a risk premium. Because investors behave this way, risky investments must offer higher expected returns than safe ones to attract money. Higher expected, not guaranteed: risky investments sometimes lose.

Fictional case

Ana tries this on her own decision. She estimates how much Lantern could pay her, for her work and for the $20,000 she puts in, in its first full year (her estimates, and all the figures here, are fictional). In a good year it could pay her $80,000; in a middling year $56,000; in a poor year $12,000. She judges the chances to be 30%, 50% and 20%. The expected value is 0.3×80,000+0.5×56,000+0.2×12,000=54,4000.3 \times 80{,}000 + 0.5 \times 56{,}000 + 0.2 \times 12{,}000 = 54{,}400 dollars. The safe alternative is her $52,000 salary plus the $300 of interest that $20,000 would earn in a year at 1.5%: $52,300. On these estimates, Lantern offers $2,100 more on average, in exchange for a one-in-five chance of earning only $12,000.

Is $2,100 a large enough premium for that risk? Finance cannot answer that for Ana: it depends on how much risk she can bear, on her hope that later years will be better than the first, and on reasons that have nothing to do with money. She uses her current savings account’s 1.5%; a better safe rate would raise what she gives up and shrink Lantern’s $2,100 advantage. What finance does is make the trade-off visible, in numbers she can argue with.

Four branches of finance

The same ideas apply wherever money is raised, allocated and managed, but the people deciding and the decisions they face differ. Finance is usually divided into the branches in the table. The boundaries are conventions rather than rules: OpenStax’s Principles of Finance, for example, names three areas (business finance, investments, and financial markets and institutions), and many economists treat public finance as part of economics.

The branches of finance
BranchWho decidesTypical decisionsExample from the Lantern story
Personal financeIndividuals and householdsBudgeting, saving, borrowing, insurance, saving for retirementHow much of Ana’s $26,000 to keep as an emergency fund ($6,000, in the end)
Corporate financeThe owners and managers of a businessWhich long-term investments to make, how to pay for them, how much cash to keepWhether Lantern buys an espresso machine and grinder for about $12,000, and whether to fund the business with Ana’s $20,000 alone or add a $10,000 bank loan
Investment financeInvestors and the professionals who manage their moneyWhich securities to buy, how to combine them, what they are worthWhether a fund manager buys shares of Harbourline Roasters Inc., a fictional listed coffee roaster
Public financeGovernmentsTaxing, spending and borrowingWhether a city borrows by selling bonds to build a tram line, and how it will repay the lenders

Some words in the table need defining. Securities are tradable claims to money: shares (part-ownership of a company) and bonds (loans that investors can buy and sell). A listed company’s shares trade on a stock exchange; a portfolio is the set of investments an investor holds; and a fund manager is a professional who invests money pooled from many savers, such as a pension fund’s members. Corporate finance is sometimes called business finance, and investment finance is often just called investments. This course touches all four; later courses go deeper into each.

In depth

Future value over several years

Each year at an interest rate rr, a balance is multiplied by (1+r)(1 + r). After nn years it has been multiplied nn times, so an amount PV\mathit{PV} today grows to

FV=PV (1+r)n\mathit{FV} = \mathit{PV}\,(1 + r)^n

The rate must be written as a decimal (1.5% is 0.015), and rr and nn must use the same period: an annual rate with a number of years. The formula assumes the rate stays the same throughout, that interest is added once a year and left in the account, and that nothing else is paid in or taken out. For Ana’s $20,000 over three years: (1.015)3=1.0456784…(1.015)^3 = 1.0456784\ldots and 20,000×1.0456784=20,913.5720{,}000 \times 1.0456784 = 20{,}913.57, as in the figure. The factor (1+r)n(1 + r)^n is called the growth factor.

What $10,000 grows to at 1.5% and at 6% a year, compounded once a year
YearsAt 1.5% (Ana’s savings rate)At 6% (Lantern’s loan rate)
0$10,000.00$10,000.00
1$10,150.00$10,600.00
2$10,302.25$11,236.00
3$10,456.78$11,910.16
4$10,613.64$12,624.77
5$10,772.84$13,382.26

Each figure is the one above it multiplied by 1.015 or 1.06, rounded to the cent only for display.

Two patterns stand out. A higher rate grows money much faster: over five years 6% adds $3,382.26, while 1.5% adds only $772.84. And at 6% the growth itself grows: five years of $600 interest on the original $10,000 would come to only $3,000, so the remaining $382.26 is interest earned on earlier interest. The effect is small over a few years and very large over decades. The lesson on interest and inflation, in module 2, compares compound interest with simple interest, which is paid only on the original amount, and shows what happens when interest is added more often than once a year.

Present value and discounting

Rearranging the future value formula answers the opposite question: how much is an amount FV\mathit{FV}, received nn years from now, worth today?

PV=FV(1+r)n\mathit{PV} = \dfrac{\mathit{FV}}{(1 + r)^n}

The present value is the amount you would need to invest today, at rate rr, to end up with FV\mathit{FV} after nn years. The table shows the present value of $10,000 to be received in five years at four different discount rates. The further away the money is, or the higher the discount rate, the less it is worth today.

Present value of $10,000 received in five years
Discount rate0%2%6%10%
Present value$10,000.00$9,057.31$7,472.58$6,209.21

Which discount rate should you use? The right rate is the return you could earn elsewhere on an investment of similar risk, because that is what you give up by waiting for this money. For a safe promise, that is roughly the rate on a safe deposit. For a risky one it is higher, as the next section shows.

Comparing amounts at different dates

Suppose you can have $1,000 now or $1,100 in two years, and you could earn 4% a year in the meantime. You can compare the two at either date, and both routes give the same answer. Carried forward, $1,000 today becomes 1,000×1.042=1,081.601{,}000 \times 1.04^2 = 1{,}081.60 in two years, less than $1,100. Carried back, $1,100 in two years is worth 1,100÷1.042=1,017.011{,}100 \div 1.04^2 = 1{,}017.01 today, more than $1,000. Either way the later $1,100 is worth more, by about $17 in today’s money. At a higher rate the answer can flip, which is why the rate matters as much as the amounts.

Expected value and the risk premium

When an uncertain outcome can take kk values x1,x2,…,xkx_1, x_2, \ldots, x_k with probabilities p1,p2,…,pkp_1, p_2, \ldots, p_k that add up to 1, its expected value is p1x1+p2x2+⋯+pkxkp_1 x_1 + p_2 x_2 + \cdots + p_k x_k. The probabilities may come from data, such as how often similar businesses succeed, or, as with Ana’s estimates, from careful judgement. Either way the expected value is only as good as the estimates behind it.

The same idea applies to returns. The risk-free rate is the return on an investment whose payoff is certain, such as an insured short-term deposit or a short-term loan to a government that is almost sure to repay. A risky investment’s risk premium is its expected return minus the risk-free rate. If a safe deposit pays 3% and a fund of company shares is expected to return 7% a year (both fictional figures), the fund’s risk premium is 4 percentage points. Investors accept the fund’s ups and downs only because, on average, they expect to be paid for them.

Time and risk come together in the discount rate. The rate used to discount a risky amount is the risk-free rate plus an allowance for its risk. When the amount is a promise that might not be kept, as here, the allowance must cover both the expected shortfall and a premium for bearing the uncertainty, as the bank example at the end of this lesson shows; when the amount is already an expected value, only the premium is needed. Either way, riskier money is discounted more heavily. A promise of $10,000 in one year from a government that is almost sure to pay, discounted at 2%, is worth 10,000÷1.02=9,803.9210{,}000 \div 1.02 = 9{,}803.92 dollars today. The same promise from a new café, discounted at 12% (both rates fictional), is worth 10,000÷1.12=8,928.5710{,}000 \div 1.12 = 8{,}928.57. The amounts and the dates are identical; only the risk differs.

What finance does inside a firm

In a business, finance answers three recurring questions. The investment decision asks which long-term assets and projects to put money into; the analysis behind it is called capital budgeting. For Lantern, it is whether an espresso machine and grinder costing about $12,000 will earn back more than they cost, in present-value terms. The financing decision asks how to pay for those investments: with the owners’ money (equity) or with borrowed money (debt), and in what mix, which is called the firm’s capital structure. For Lantern, it is $20,000 from Ana and $10,000 from Coastline Bank. And working capital management asks how to run the day-to-day money: keeping enough cash to pay suppliers and staff while waiting for customers to pay. The lesson on capital and liquidity, in module 2, defines debt, equity and working capital properly.

For a business run for profit, finance textbooks usually state the goal of these decisions as maximising the value of the firm to its owners. That goal is debated, and the lesson on governance and agency, in module 3, sets it beside the view that a firm should serve all its stakeholders. In a small firm like Lantern one person makes all three decisions; in a large company they are spread across a finance team led by a chief financial officer.

What finance does in an economy

Across a whole economy, some people and organisations have more money than they want to spend now (savers) and others have uses for money that they do not yet have (borrowers and businesses with projects). The financial system, the banks, markets and other institutions that connect the two, moves savings to the people who can put them to productive use. Ana’s story shows it in miniature: the $26,000 she deposits at Coastline Bank is part of the pool from which the bank lends to businesses like Lantern.

The financial system does more than move money. Interest rates and market prices signal how valuable money is now compared with later, and how risky each use of it is, so that money tends to flow to the uses that promise the best return for their risk. It lets people share risks they could not bear alone, as insurance does. And it lets savers get their money back when they need it. When it breaks down, as in a banking crisis, businesses that depend on credit cannot borrow, and the damage spreads well beyond finance. Module 3, Markets and the people in them, looks at the markets and participants in detail.

Formulas

Future value of a single amount (annual compounding)

FV=PV (1+r)n\mathit{FV} = \mathit{PV}\,(1 + r)^n
FV\mathit{FV}
Future value: what the amount will be worth after n years$
PV\mathit{PV}
Present value: the amount invested today$
rr
Interest rate per year, as a decimal (1.5% is 0.015)per year
nn
Number of yearsyears

Assumes. The rate stays the same, interest is added once a year and left to earn interest, and nothing else is paid in or taken out.

The lesson on interest and inflation covers interest added more often than once a year. F04 Mathematics of Finance covers series of payments.

Present value of a single amount (annual discounting)

PV=FV(1+r)n\mathit{PV} = \dfrac{\mathit{FV}}{(1 + r)^n}
PV\mathit{PV}
Present value: what the future amount is worth today$
FV\mathit{FV}
The amount to be received in the future$
rr
Discount rate per year, as a decimal: the return available elsewhere at similar riskper year
nn
Number of years until the amount is receivedyears

Assumes. One amount, received at the end of year n; the discount rate is the same in every year.

A higher discount rate or a longer wait gives a lower present value.

Expected value of an uncertain outcome

E[X]=p1x1+p2x2+⋯+pkxkE[X] = p_1 x_1 + p_2 x_2 + \cdots + p_k x_k
E[X]E[X]
Expected value of the uncertain outcome Xsame unit as the outcomes
xix_i
The i-th possible outcomee.g. $
pip_i
The probability of the i-th outcome; the probabilities add up to 1
kk
The number of possible outcomes

Assumes. The list of outcomes is complete and the probabilities add up to 1 (100%).

An average over many repetitions, not a prediction of any single result.

Risk premium

Risk premium=E[R]−rf\text{Risk premium} = E[R] - r_f
E[R]E[R]
Expected return on the risky investment% per year
rfr_f
Risk-free rate: the return on an investment whose payoff is certain% per year

Assumes. Both returns are measured over the same period.

Measured in percentage points. The risk premium is what investors expect to earn on average for bearing risk; it is not guaranteed in any year.

Worked examples

A first example: Leaving the money in the bank

Fictional figures

Suppose Ana decided not to open Lantern and left $20,000 in her Coastline Bank savings account at 1.5% a year, paid once a year, for three years. How much would she have at the end, and how much of that would be interest?

  1. Identify the inputs for the future value formula: PV=20,000\mathit{PV} = 20{,}000, r=0.015r = 0.015 (1.5% as a decimal), n=3n = 3.
  2. Find the growth factor: (1.015)3=1.015×1.015×1.015=1.0456784…(1.015)^3 = 1.015 \times 1.015 \times 1.015 = 1.0456784\ldots
  3. Apply FV=PV(1+r)n\mathit{FV} = \mathit{PV}(1 + r)^n: 20,000×1.0456784=20,913.5720{,}000 \times 1.0456784 = 20{,}913.57 dollars, rounding to the cent only at the end.
  4. Interest earned is the future value minus the amount deposited: 20,913.57−20,000=913.5720{,}913.57 - 20{,}000 = 913.57. Check by going year by year: 20,300, then 20,604.50, then 20,913.57.

Answer. About $20,913.57 after three years, of which $913.57 is interest.

What it means. This is what Ana gives up, in interest alone, by putting $20,000 into Lantern for three years. It is a small amount next to her salary, but it sets a floor: money invested in Lantern should earn more than the bank would pay, or the bank is the better home for it.

A harder example: Cash now, or a larger promise later?

Fictional figures

A fictional events company, Marlow Events, owes Lantern for catering. It offers either $5,000 today or $5,600 in two years. Lantern could earn 3% a year on cash it holds. (a) If Marlow is certain to pay, which offer is worth more? (b) Lantern then learns that small events companies often fail, and judges that there is a 10% chance Marlow will not be around to pay anything in two years. Which offer is worth more now?

  1. (a) Move the later amount to today with the present value formula: 5,600÷1.032=5,600÷1.0609=5,278.545{,}600 \div 1.03^2 = 5{,}600 \div 1.0609 = 5{,}278.54 dollars.
  2. (a) Compare at the same date: $5,278.54 today is more than $5,000 today, so if payment is certain the later offer is worth more, by about $279 in today’s money.
  3. (b) Replace the promised amount with its expected value: 0.9×5,600+0.1×0=5,0400.9 \times 5{,}600 + 0.1 \times 0 = 5{,}040 dollars, received in two years.
  4. (b) Discount the expected amount at 3%: 5,040÷1.0609=4,750.685{,}040 \div 1.0609 = 4{,}750.68 dollars, less than the $5,000 on offer today.
  5. (b) And this still values the risky promise as if it were safe. Because Lantern is risk averse, it should discount the expected $5,040 at more than 3%, which makes the later offer look worse still.

Answer. (a) Take the $5,600 in two years: it is worth $5,278.54 today. (b) Take the $5,000 now: the risky promise is worth at most $4,750.68 today.

What it means. Time alone favoured waiting; time and risk together favour taking the cash. That is the pattern of most financial decisions: first put every amount on the same date, then ask how sure each amount is.

Applied: a business decision: Should Brightwell sell its receivable?

Fictional figures

Brightwell Energy, a fictional battery-storage installer, has finished a project for a client who has agreed to pay $250,000 in three years. A finance company offers to buy that right to payment today for $200,000. For a payment from a reliable client, Brightwell judges that 7% a year is the return available elsewhere at similar risk, so it discounts at 7%. Should it sell? How does the answer change if the client looks shaky and Brightwell uses 10% instead?

  1. Present value at 7%: (1.07)3=1.225043(1.07)^3 = 1.225043, so 250,000÷1.225043=204,074.47250{,}000 \div 1.225043 = 204{,}074.47 dollars.
  2. Compare at today’s date: the offer of $200,000 is about $4,074 less than the payment is worth to Brightwell, so on these numbers it should keep the receivable and wait.
  3. Present value at 10%: (1.10)3=1.331(1.10)^3 = 1.331, so 250,000÷1.331=187,828.70250{,}000 \div 1.331 = 187{,}828.70 dollars.
  4. Now the offer is about $12,171 more than the present value, so selling is the better choice: Brightwell swaps an uncertain payment in three years for certain cash today.

Answer. At 7% the receivable is worth $204,074.47 today, more than the $200,000 offer: keep it. At 10% it is worth $187,828.70, less than the offer: sell it.

What it means. The decision turns on the discount rate, and the discount rate turns on Brightwell’s judgement of the client’s risk. Present value does not remove judgement; it shows exactly which judgement the decision depends on.

Explore

Start with Ana’s $20,000 at 1.5% for three years. Raise the rate to 6% and then lengthen the time to 30 years: watch how compounding pulls the future value away from the simple-interest figure. The present value row shows the same calculation in reverse: what that amount is worth today if it arrives after the number of years you chose.

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Common mistakes

  • “A dollar is a dollar, whenever it arrives.”

    Money available sooner can earn a return in the meantime, so a dollar today is worth more than a dollar next year. Adding or comparing amounts that arrive at different dates, such as “$5,000 now or $5,600 in two years, so the second is $600 more”, ignores this. Move every amount to the same date first.

  • Multiplying the yearly interest by the number of years

    At 1.5% a year, $20,000 does not become 20,000×(1+0.015×3)=20,90020{,}000 \times (1 + 0.015 \times 3) = 20{,}900 after three years. With interest added once a year and left in the account, each year’s interest earns interest too, so the balance is 20,000×1.0153=20,913.5720{,}000 \times 1.015^3 = 20{,}913.57. Use the exponent.

  • Typing the percentage as a whole number

    In the formulas, rr is a decimal: 1.5% is 0.015 and 6% is 0.06. Typing 1.5 makes the calculation assume 150% a year, so $20,000 appears to grow to more than $300,000 in three years. If an answer looks absurd, check the rate first.

  • “The expected value is what will happen.”

    An expected value is a probability-weighted average. A coin flip paying $200 or nothing has an expected value of $100, but no single flip pays $100. Use the expected value to compare choices, and look separately at the range of outcomes, which is where the risk is.

  • “Higher expected return means a better investment.”

    A higher expected return usually comes with more risk; that is the risk premium at work. Comparing a risky investment’s expected return with a safe account’s rate, without asking how much more risk you would be taking, compares unlike things. The right question is whether the extra return is enough for the extra risk.

Guided practice

Work through these with hints when you need them. Every hint and every worked solution is there to be used.

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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.

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In practice

Fictional case

Coastline Bank’s small-business lending officer is deciding what rate to charge Lantern on its two-year, $10,000 loan. All the figures that follow are fictional, like the bank itself.

She starts from time. Instead of lending to Lantern, the bank could lend the same $10,000 to the government for two years by buying a government bond, which in this example pays 4.0% a year and is treated as safe. Any loan to Lantern must pay at least that much, or the bank would be better off with the bond: 4.0% is the bank’s risk-free rate for this decision.

Then she turns to risk. Lantern is a new business with no trading history, and new businesses sometimes fail. From the bank’s records of similar loans she estimates a 2% chance each year that a borrower like Lantern stops paying, and that if it does, the bank recovers about half of what it is owed, for example by having the espresso equipment sold. The expected loss is therefore about 0.02×0.5=0.010.02 \times 0.5 = 0.01, or 1% of the loan a year. Charging 4% plus 1% would only break even on average; it would not pay the bank for the uncertainty itself, nor for the cost of assessing and administering a small loan. She adds 1 percentage point for those, giving 4.0+1.0+1.0=6.04.0 + 1.0 + 1.0 = 6.0, the 6% in Lantern’s loan agreement.

Every part of that rate is one of this lesson’s ideas: a risk-free rate for the time value of money, an expected loss computed as an expected value, and a premium for bearing risk. The officer would apply the same reasoning to a hotel, an airline or a university; only the estimates change. A riskier borrower would pay a higher rate, and a borrower whose risk the bank could not judge at all might not get a loan at any rate.

The same transaction also shows the financial system at work. Ana, as a saver, receives 1.5% on her deposit; Lantern, as a borrower, pays 6%. The gap is not pure profit for the bank: it covers the bank’s costs, the loans that go bad and the risk it bears, so that savers can earn a safe return without having to judge each borrower themselves. Ana, in effect, is both a saver and, through her company, a borrower in the same bank.

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.

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Summary

  • Finance is the study of how people, businesses and governments raise, allocate and manage money over time, under uncertainty.
  • Because money can earn a return, a dollar today is worth more than a dollar later: the time value of money.
  • Future value: FV = PV(1 + r)ⁿ. Present value: PV = FV ÷ (1 + r)ⁿ. Compounding moves money forward; discounting moves it back. Compare amounts only at the same date.
  • Risk is the possibility that outcomes differ from what you expect. The expected value weights each outcome by its probability; it is an average, not a prediction.
  • Most investors are risk averse, so risky investments must offer a risk premium: a higher expected return than the risk-free rate. Riskier amounts are discounted at higher rates.
  • Personal, corporate, investment and public finance apply the same ideas to households, businesses, investors and governments.
  • Inside a firm, finance makes the investment, financing and working capital decisions; in an economy, the financial system moves savings to productive uses, prices time and risk, and shares risk.

Key formulas

Terms introduced

Finance
The study of how people, businesses and governments raise, allocate and manage money over time, under uncertainty.
Time value of money
The principle that a dollar available today is worth more than a dollar available later, because today’s dollar can earn a return in the meantime.
Interest rate
The price of using money for a period, stated as a percentage of the amount per year: received by savers and lenders, paid by borrowers.
Compounding
Earning interest on interest already earned, because each period’s interest is added to the balance and earns interest in later periods.
Future value
What an amount invested today will be worth at a later date, at a given rate of return: FV = PV(1 + r)ⁿ.
Present value
What an amount to be received in the future is worth today, at a given discount rate: PV = FV ÷ (1 + r)ⁿ.
Discounting
Converting an amount to be received in the future into its value today by dividing by (1 + r) for each period of waiting.
Discount rate
The rate used to discount future amounts: the return available elsewhere on an investment of similar risk. Riskier amounts are discounted at higher rates.
Risk
In finance, the possibility that an outcome will differ from what is expected, for better or for worse.
Expected value
The probability-weighted average of the possible outcomes of an uncertain event: each outcome multiplied by its probability, then added.
Risk aversion
Preferring a certain amount to a gamble with the same expected value. Most savers and investors are assumed to be risk averse.
Risk premium
The extra expected return that investors demand for bearing risk: the expected return on a risky investment minus the risk-free rate.
Risk-free rate
The return on an investment whose payoff is certain, such as an insured short-term deposit or a short-term loan to a government that is almost sure to repay.
Personal finance
The branch of finance concerned with the decisions of individuals and households: budgeting, saving, borrowing, insurance and retirement.
Corporate finance
The branch of finance concerned with the decisions of businesses: which investments to make, how to pay for them and how to manage day-to-day cash. Also called business finance.
Investment finance
The branch of finance concerned with the decisions of investors: which securities to buy, how to combine them and what they are worth. Also called investments.
Public finance
The branch of finance (often treated as part of economics) concerned with how governments tax, spend and borrow.
Investment decision
A firm’s choice of which long-term assets and projects to put money into. The analysis behind it is capital budgeting.
Capital budgeting
The process a firm uses to decide which long-term investments are worth making, by comparing what they cost with the value of what they are expected to return.
Financing decision
A firm’s choice of how to pay for its investments: with owners’ money (equity), borrowed money (debt) or a mix, called its capital structure.
Financial system
The banks, markets and other institutions that move money from savers to borrowers and businesses, price time and risk, and help people share risk.

Sources

  1. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 1.1, “What Is Finance?”. OpenStax. openstax.org/books/principles-finance/pages/1-1-what-is-fina…
  2. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 1.8, “Concepts of Time and Value”. OpenStax. openstax.org/books/principles-finance/pages/1-8-concepts-of-…
  3. 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…
  4. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 1.2, “The Role of Finance in an Organization”. OpenStax. openstax.org/books/principles-finance/pages/1-2-the-role-of-…
  5. US Securities and Exchange Commission, Office of Investor Education and Advocacy. “What is Risk?”. Investor.gov. investor.gov/introduction-investing/investing-basics/what-ri…
  6. Knight, F. H. (1921). Risk, Uncertainty, and Profit. Boston: Houghton Mifflin. Part III, chapter VIII, “Structures and Methods for Meeting Uncertainty”. Library of Economics and Liberty. econlib.org/library/Knight/knRUP7.html