Interest, compounding and discountingSimple, compound and continuous compounding; present value and the discount factor.
Annuities, perpetuities and loan amortizationPresent value of level payments, growing perpetuities, loan payments and the amortization schedule.
Net present value and internal rate of returnProject cash flows, NPV, the NPV profile, IRR, the profitability index and where the IRR misleads.
The weighted average cost of capitalCost of equity, after-tax cost of debt, the interest tax shield and market-value weights.
Capital structure and the Modigliani–Miller theoremsMM Propositions I and II without taxes, the tax shield, levered cost of equity and WACC with taxes.
Discounted cash flow valuationFree cash flow, enterprise value, the Gordon growth terminal value and equity value.
Relative valuation with multiplesP/E, EV/EBITDA and other multiples, choosing peers, and how multiples relate to DCF value.
Real options: the value of flexibilityOptions to delay, expand and abandon a project, valued on a binomial tree, and the expanded NPV.

Check yourself

Questions from the lessons of this series. Open a lesson's group whenever you want to test yourself.

Annuities, perpetuities and loan amortization2 questions

Re-read the lesson →

Check yourself

A stream pays 500 one year from now and then grows by 3% a year forever. At a discount rate of 8%, what is it worth today?

Your answer

Check yourself

A loan of 100,000 carries 5% interest a year. Which annual payment can never repay it, however long the term?

4,5006,00010,00012,950

Net present value and internal rate of return2 questions

Re-read the lesson →

Check yourself

A project costs 100 today, pays nothing after one year and pays 121 after two years. What is its IRR?

Your answer%

Check yourself

Two mutually exclusive projects: A has the higher IRR, B has the higher NPV at the firm's discount rate. Which one should the firm take?

A, because its percentage return is higherB, because it adds more value at the actual discount rateEither, since both have a positive NPVNeither, until the IRRs agree

The weighted average cost of capital2 questions

Re-read the lesson →

Check yourself

The risk-free rate is 2%, the market risk premium 5% and the firm's equity beta 1.2. What cost of equity does the CAPM give?

Your answer%

Check yourself

Why must the WACC weights use market values rather than balance-sheet (book) values?

Book values are not audited, so they are unreliable.Investors require their returns on what their claims are worth today, which is the market value.Book weights always give a higher WACC, which rejects good projects.Market values only matter for firms that pay corporate tax.

Capital structure and the Modigliani–Miller theorems2 questions

Re-read the lesson →

Check yourself

Without taxes, a firm has $r_U = 12\%$ and borrows at $r_D = 6\%$ with a debt-to-equity ratio $D/E = 1$ . What return do its shareholders require?

Your answer%

Check yourself

With taxes, $V_L = V_U + t\,D$ keeps rising with debt. Why do real firms still not borrow almost everything?

Because the tax shield stops growing once debt exceeds half of firm value.Because heavy debt brings costs of financial distress that outweigh the extra tax shield.Because Proposition I says that debt destroys value.Because the cost of debt is always higher than the cost of equity.

Discounted cash flow valuation2 questions

Re-read the lesson →

Check yourself

A firm has EBIT of 100, a tax rate of 20%, depreciation of 30, capital expenditure of 50 and an increase in net working capital of 5. What is its free cash flow?

Your answer

Check yourself

A spreadsheet DCF gives an enormous or negative terminal value. The analyst has set the perpetual growth rate $g$ equal to or above the WACC. What does this mean?

The firm is extremely valuable and the result should be kept.The assumption is impossible: the Gordon formula needs

g < W

.The terminal value should be discounted at

g

instead of the WACC.Five forecast years are too few; adding years fixes it.

Relative valuation with multiples2 questions

Re-read the lesson →

Check yourself

Which of these multiples pairs a numerator and a denominator that belong to the same investors?

Share price / EBITDAEnterprise value / net incomeEnterprise value / EBITDAShare price / sales

Check yourself

Peers trade at 8 times EBITDA. The target has EBITDA of 80 and net debt of 200. What is its implied equity value?

Your answer

Real options: the value of flexibility2 questions

Re-read the lesson →

Check yourself

A firm holds a three-year licence to build a plant. Waiting costs nothing in forgone cash flows ( $\delta = 0$ ), and today $V > I$ . When should the firm invest?

Now, because the static NPV is positive.Not before the licence expires; then invest if

V_T > I

.As soon as

V

exceeds

I

\sigma V

.Never: with

\delta = 0

the option is worthless.

Check yourself

On a tree with one step per year, $\sigma = 25\%$ , $r = 5\%$ and $\delta = 1\%$ , what is the risk-neutral probability $q$ of an up move?

Your answer

About this tool

Narrative guides that connect the interactive tools into a story: financial instruments and their payoffs, the no-arbitrage principle, risk-neutral pricing, delta hedging and the road to the Black–Scholes formula. Each guide links to the labs where you can try the ideas yourself.

Frequently asked questions

In what order should I read the guides?

Start with instruments and no-arbitrage, then risk-neutral pricing and delta hedging; together they lead to the Black–Scholes formula.

Who are the guides for?

Students of financial and insurance mathematics and anyone who wants intuition behind derivatives pricing, not just the formulas.

Privacy & dataNo cookies

This site does not use cookies and asks for no personal data. It keeps a few small settings in your browser's local storage — your chosen theme and language, the sidebar layout, and your progress and answers in the Guides — so it remembers them on your next visit. This data never leaves your device; clearing your browser storage removes it. Inputs you change can also be captured in the page URL, so you can share or bookmark a specific setup. The site uses Vercel Web Analytics, a privacy-friendly, cookieless service that records only anonymous, aggregated usage (such as page views) without identifying you or tracking you across other sites.

Guide

Lessons meant to be read in order. Each one only uses what came before it.

Corporate finance

Time value of money, NPV and IRR, WACC, capital structure and DCF valuation.

Start

0 of 8 read

Interest, compounding and discountingSimple, compound and continuous compounding; present value and the discount factor.
Annuities, perpetuities and loan amortizationPresent value of level payments, growing perpetuities, loan payments and the amortization schedule.
Net present value and internal rate of returnProject cash flows, NPV, the NPV profile, IRR, the profitability index and where the IRR misleads.
The weighted average cost of capitalCost of equity, after-tax cost of debt, the interest tax shield and market-value weights.
Capital structure and the Modigliani–Miller theoremsMM Propositions I and II without taxes, the tax shield, levered cost of equity and WACC with taxes.
Discounted cash flow valuationFree cash flow, enterprise value, the Gordon growth terminal value and equity value.
Relative valuation with multiplesP/E, EV/EBITDA and other multiples, choosing peers, and how multiples relate to DCF value.
Real options: the value of flexibilityOptions to delay, expand and abandon a project, valued on a binomial tree, and the expanded NPV.

Check yourself

Questions from the lessons of this series. Open a lesson's group whenever you want to test yourself.

Annuities, perpetuities and loan amortization2 questions

Re-read the lesson →

Check yourself

A stream pays 500 one year from now and then grows by 3% a year forever. At a discount rate of 8%, what is it worth today?

Your answer

Check yourself

A loan of 100,000 carries 5% interest a year. Which annual payment can never repay it, however long the term?

4,5006,00010,00012,950

Net present value and internal rate of return2 questions

Re-read the lesson →

Check yourself

A project costs 100 today, pays nothing after one year and pays 121 after two years. What is its IRR?

Your answer%

Check yourself

Two mutually exclusive projects: A has the higher IRR, B has the higher NPV at the firm's discount rate. Which one should the firm take?

A, because its percentage return is higherB, because it adds more value at the actual discount rateEither, since both have a positive NPVNeither, until the IRRs agree

The weighted average cost of capital2 questions

Re-read the lesson →

Check yourself

The risk-free rate is 2%, the market risk premium 5% and the firm's equity beta 1.2. What cost of equity does the CAPM give?

Your answer%

Check yourself

Why must the WACC weights use market values rather than balance-sheet (book) values?

Capital structure and the Modigliani–Miller theorems2 questions

Re-read the lesson →

Check yourself

Without taxes, a firm has $r_U = 12\%$ and borrows at $r_D = 6\%$ with a debt-to-equity ratio $D/E = 1$ . What return do its shareholders require?

Your answer%

Check yourself

With taxes, $V_L = V_U + t\,D$ keeps rising with debt. Why do real firms still not borrow almost everything?

Discounted cash flow valuation2 questions

Re-read the lesson →

Check yourself

A firm has EBIT of 100, a tax rate of 20%, depreciation of 30, capital expenditure of 50 and an increase in net working capital of 5. What is its free cash flow?

Your answer

Check yourself

A spreadsheet DCF gives an enormous or negative terminal value. The analyst has set the perpetual growth rate $g$ equal to or above the WACC. What does this mean?

The firm is extremely valuable and the result should be kept.The assumption is impossible: the Gordon formula needs

g < W

.The terminal value should be discounted at

g

instead of the WACC.Five forecast years are too few; adding years fixes it.

Relative valuation with multiples2 questions

Re-read the lesson →

Check yourself

Which of these multiples pairs a numerator and a denominator that belong to the same investors?

Share price / EBITDAEnterprise value / net incomeEnterprise value / EBITDAShare price / sales

Check yourself

Peers trade at 8 times EBITDA. The target has EBITDA of 80 and net debt of 200. What is its implied equity value?

Your answer

Real options: the value of flexibility2 questions

Re-read the lesson →

Check yourself

A firm holds a three-year licence to build a plant. Waiting costs nothing in forgone cash flows ( $\delta = 0$ ), and today $V > I$ . When should the firm invest?

Now, because the static NPV is positive.Not before the licence expires; then invest if

V_T > I

.As soon as

V

exceeds

I

\sigma V

.Never: with

\delta = 0

the option is worthless.

Check yourself

On a tree with one step per year, $\sigma = 25\%$ , $r = 5\%$ and $\delta = 1\%$ , what is the risk-neutral probability $q$ of an up move?

Your answer

About this tool

Frequently asked questions

In what order should I read the guides?

Start with instruments and no-arbitrage, then risk-neutral pricing and delta hedging; together they lead to the Black–Scholes formula.

Who are the guides for?

Students of financial and insurance mathematics and anyone who wants intuition behind derivatives pricing, not just the formulas.