ACCA AFM · Advanced Financial Management

The formula
sheet, explained

Eight of these eleven you already met in FM. The sheet has not changed them, but the questions have — and three are entirely new.

This page runs quickly through the familiar ones and goes slowly through M&M, MIRR and Black–Scholes, which is where AFM marks are actually won and lost.

Start with the sheet What’s missing
Provided in your exam

The eleven headings

  1. Modigliani and Miller Proposition 2 (with tax)
  2. The Capital Asset Pricing Model
  3. The asset beta formula
  4. The Growth Model
  5. Gordon’s growth approximation
  6. The weighted average cost of capital
  7. The Fisher formula
  8. Purchasing power parity and interest rate parity
  9. Modified Internal Rate of Return
  10. The Black–Scholes option pricing model
  11. The Put Call Parity relationship

Then the present value and annuity tables.

Jump to a formula

Eleven formulas,
three that are new

The eight carried over from FM get a short card each — the formula, a calculator, and what changes about them at AFM. Modigliani and Miller, MIRR, Black–Scholes and put-call parity get the full treatment.

1M&M Proposition 2 2CAPM 3Asset beta 4Growth model 5Gordon’s growth 6WACC 7Fisher 8PPP & IRP 9MIRR 10Black–Scholes 11Put call parity !Not on the sheet
Formulas 1–3

Gearing and risk

Three ways of moving a cost of capital between one capital structure and another. AFM expects you to know which of the three a given question wants, and to say why.

New at AFM 1

Modigliani and Miller
Proposition 2 (with tax)

Takes the cost of equity a company would have with no debt at all, and adds a premium for the financial risk that borrowing creates. It does in one step what ungearing and regearing a beta does in two.

As given in the exam ke = kei + (1 − T)(kei − kd)VdVe

ke cost of equity in the geared company  ·  kei cost of equity if it were ungeared  ·  kd cost of debt  ·  T corporation tax rate

What changed in September 2024

From the September 2024 sitting the sheet also prints a rearranged version, which solves for the ungeared cost of equity instead. Before that, you had to do the algebra yourself under exam pressure — and a lot of candidates did not.

This is a real second formula, unlike the asset beta formula, where the two terms are just the equity and debt halves of one weighted average. Toggle the calculator below to see the sheet’s two versions working in opposite directions.

Try it Live
Geared cost of equity, ke—
Premium for financial risk—
Gearing, Vd ÷ Ve—

Equity must be above zero.

See the worked example

An ungeared company in the same industry has a cost of equity of 12%. Your company is financed 60% equity and 40% debt by market value, borrows at 6%, and pays tax at 30%.

  1. Gearing: Vd ÷ Ve = 40 ÷ 60 = 0.6667
  2. The spread between ungeared equity and debt: 12 − 6 = 6%
  3. Scale it for tax relief and gearing: 0.7 × 6 × 0.6667 = 2.8%
  4. Add it back: 12 + 2.8 = 14.8%

Flip the calculator to the other direction with 14.8% in the box and you get 12% back. That is the rearranged version doing its job.

Where the marks go

Vd ÷ Ve is debt over equity, not debt over total finance. A company described as 40% geared could mean either, so read the definition the question gives rather than assuming.

This and the asset beta route should give roughly the same answer on the same facts. Where a question gives you a proxy company’s beta, use beta. Where it gives you an ungeared cost of equity directly, use M&M. Choosing the longer route still earns the marks, but it costs time you do not have in AFM.

Carried over from FM 2

The capital asset pricing model

Unchanged from FM. At AFM it is rarely the answer on its own — it is the step that turns a regeared beta into the discount rate for an APV or a project-specific appraisal.

As given in the exam E(ri) = Rf + βi(E(rm) − Rf)
Try it Live
Cost of equity—
Risk premium applied—
Where the marks go

Still the same trap as FM: a market risk premium is already net of the risk-free rate, a market return is not. Toggle the buttons above to see how far apart the two readings land.

New at AFM: for an overseas project, the risk-free rate should be the one in the currency the cash flows are in. Mixing a domestic risk-free rate with foreign cash flows is a discussion mark as well as a calculation one.

Carried over from FM 3

The asset beta formula

Unchanged from FM. One formula, used in both directions: read left to right to ungear, rearranged to regear. The two terms are the equity and debt halves of a weighted average, not separate gearing and ungearing formulas.

As given in the exam βa = Ve(Ve + Vd(1−T))βe + Vd(1−T)(Ve + Vd(1−T))βd
Try it Live
Asset beta, βa—
Debt net of tax, Vd(1−T)—
Equity weighting—

Equity must be above zero.

Where the marks go

Switching the toggle carries the answer into the input box, which is exactly the exam workflow: ungear on the proxy’s gearing, then change the gearing figures to your own and regear. Using the wrong company’s figures at either stage produces a plausible answer and no marks.

At AFM the debt beta is more often non-zero than at FM. When the question gives you one, the second term stops disappearing and has to be carried through.

Formulas 4–6

Valuation
and WACC

All three are identical to the FM versions. What changes is the use: at AFM the growth model values a whole company in an acquisition question, not a single share.

Carried over from FM 4

The growth model

At AFM this is a business valuation tool. Feed it total dividends rather than dividends per share and it values the whole equity of a target company.

As given in the exam
P0 = D0(1 + g)(re − g)
Rearranged for the cost of equity — you do this yourself re = D0(1 + g)P0 + g
Try it Live
Cost of equity, re—
Next year’s dividend, D1—
Dividend yield—

The model only works while the cost of equity is above the growth rate.

Where the marks go

The price must be ex-div, and D0 is the dividend just paid, so it needs growing on one year. Both traps survive the jump from FM.

New at AFM: the model assumes constant growth forever, which is rarely credible for an acquisition target. Saying so — and comparing the answer against an earnings or asset-based valuation — is usually where the discussion marks sit.

Carried over from FM 5

Gordon’s growth approximation

Estimates the growth rate from inside the business: how much profit is held back, multiplied by the return earned on it.

As given in the exam g = bre

b proportion of earnings retained  ·  re the return earned on funds reinvested — not the cost of equity, despite the symbol

Try it Live
Growth rate, g—
Payout ratio—
Where the marks go

b is the retention rate. Given a payout ratio, take it off 100% first.

The symbol re on the sheet is misleading here. It means the accounting return earned on reinvested funds, and putting the cost of equity in instead produces a growth rate that is far too high — often high enough to break the growth model it feeds.

Carried over from FM 6

Weighted average cost of capital

Unchanged. At AFM the interesting part is knowing when not to use it — if a project changes the company’s business risk or its gearing, the existing WACC is the wrong rate and an adjusted present value is usually the better route.

As given in the exam WACC = VeVe + Vdke + VdVe + Vdkd(1−T)
Try it Live
WACC—
Equity weighting—
Post-tax cost of debt—
Contribution from equity—
Contribution from debt—

Total finance must be above zero.

Where the marks go

Market values, never book values. For redeemable debt the (1−T) shortcut does not work — you need the post-tax IRR of the debt’s own cash flows, and that working is not on the sheet.

Formulas 7–8

Inflation and
exchange rates

Identical to FM, but AFM leans on them harder: these are the formulas behind translating an overseas project’s cash flows back into the parent’s currency.

Carried over from FM 7

The Fisher formula

Links the money rate you see to the real rate underneath it. Multiply, never add.

As given in the exam (1 + i) = (1 + r)(1 + h)
Try it Live
Money rate, i—
Simply adding the two would give—
Where the marks go

Never cross the two methods. Either inflate every cash flow at its own rate and discount at the money rate, or keep everything in today’s prices and discount at the real rate. With different inflation rates for different cash flows — which AFM almost always gives you — only the money method works.

Carried over from FM 8

Purchasing power parity
and interest rate parity

Two forecasts of an exchange rate: one from the inflation differential, one from the interest rate differential. Same shape, different inputs.

As given in the exam
S1 = S0 × (1 + hc)(1 + hb)
F0 = S0 × (1 + ic)(1 + ib)
Try it Live
Expected future spot, S1—
Movement against the base currency—
Where the marks go

The counter currency is the one there is more than one of, and its rate goes on top. Get it the wrong way round and the currency moves in the wrong direction.

For a multi-year overseas project you need a whole schedule of forecast rates, one per year, with the ratio raised to the power of the year number. Building that schedule as a row in your answer — before you translate anything — is what separates a clean AFM answer from a muddle.

Formula 9

Appraisal

New at AFM, and the answer to a criticism of IRR that FM students are taught to recite without ever seeing the fix.

New at AFM 9

Modified internal rate of return

The ordinary IRR assumes every cash inflow is reinvested at the IRR itself, which for a good project is unrealistically high. MIRR assumes reinvestment at the cost of capital instead, so it gives a more believable return — and unlike IRR, it cannot produce multiple answers.

As given in the exam MIRR = (PVRPVI)1/n (1 + re) − 1

PVR present value of the return phase  ·  PVI present value of the investment phase  ·  n life of the project in years  ·  re cost of capital

Try it Live
MIRR—
PVR ÷ PVI—

Both present values must be positive, and the project life above zero.

See the worked example

A four-year project has an investment phase with a present value of £1,000,000 and a return phase with a present value of £1,500,000. The cost of capital is 10%.

  1. The ratio: 1,500,000 ÷ 1,000,000 = 1.50
  2. Take the nth root: 1.501/4 = 1.1067
  3. Multiply by (1 + cost of capital): 1.1067 × 1.10 = 1.2174
  4. Subtract 1: 21.74%

Since 21.74% comfortably exceeds the 10% cost of capital, the project is worth doing — which is the same conclusion the positive NPV of £500,000 already gave you.

Where the marks go

Enter PVI as a positive number. It is the present value of the outflows, and the formula wants its size, not its sign. A negative figure produces a negative ratio and nonsense.

Both figures are present values — already discounted to today at the cost of capital. Students who put in the raw undiscounted cash flows get an answer that looks fine and is wrong.

Where the investment phase runs over more than one year, discount each outflow and total them before you start. Where returns start after a delay, the same applies to them.

MIRR is a better return measure than IRR, but it is still a percentage. On a choice between mutually exclusive projects, NPV is the decision rule — and saying so is worth a mark.

Formulas 10–11

Option pricing

The two formulas that make AFM look harder than it is. They are long, but they are only substitution — and the second one is three terms.

New at AFM 10

The Black–Scholes
option pricing model

Prices a European call option. In AFM it is used for far more than share options — valuing real options to delay, expand or abandon a project, and valuing the equity of a geared company as a call option over its assets.

As given in the exam
c = PaN(d1) − PeN(d2)e−rt
d1 = ln(Pa/Pe) + (r + 0.5s²)ts√t
d2 = d1 − s√t

Pa current price of the underlying asset  ·  Pe exercise price  ·  r risk-free rate  ·  s volatility (standard deviation of returns)  ·  t time to expiry in years

Try it Live
Value of the call, c—
d1—
d2—
N(d1)—
N(d2)—
Present value of the exercise price—
Value of the matching put, by parity—

Prices, volatility and time must all be above zero.

See the worked example

A share trades at 100p. A call option to buy it at 95p expires in one year. The risk-free rate is 5% and the volatility of the share is 30%.

  1. ln(100 ÷ 95) = 0.0513
  2. (r + 0.5s²)t = (0.05 + 0.5 × 0.09) × 1 = 0.095
  3. d1 = (0.0513 + 0.095) ÷ (0.30 × √1) = 0.4876
  4. d2 = 0.4876 − 0.30 = 0.1876
  5. Round to two decimals for the tables and look up: N(0.49) = 0.5 + 0.1879 = 0.6879; N(0.19) = 0.5 + 0.0753 = 0.5753
  6. PV of the exercise price: 95 × e−0.05 = 90.37
  7. c = (100 × 0.6879) − (90.37 × 0.5753) = 68.79 − 51.99 = 16.8p

The calculator above uses the exact normal function rather than the rounded table, so it shows N(d1) as 0.6871 rather than 0.6879. Both give a call value of about 16.8p — the examiner accepts either, provided your workings are visible.

Where the marks go

The d1 and d2 figures are not the answer. They are inputs to a normal distribution lookup, and N(d) is a probability between 0 and 1. Students who put d1 straight into the top line lose the whole calculation.

Volatility goes in as a decimal, and it is squared inside d1. A 30% volatility means s = 0.30 and s² = 0.09, not 0.3² read as 0.9.

Time is measured in years. Three months is t = 0.25, and √t = 0.5 — not 0.25.

The formula prices a European call, exercisable only at expiry. It also assumes constant volatility and no dividends. Naming those limitations is reliably worth marks, and takes far less time than the calculation.

The look-up table gives the area between the mean and d. For a positive d, N(d) is 0.5 plus that figure. For a negative d, it is 0.5 minus it.

New at AFM 11

The put call parity relationship

Once you have priced a call, the matching put comes free. Black–Scholes only prices calls, so this is how AFM questions get to a put value without a second long calculation.

As given in the exam p = c − Pa + Pee−rt

Same call, same underlying asset, same exercise price, same expiry date. The relationship only holds when all four match.

Try it Live
Value of the put, p—
Present value of the exercise price—

The exercise price must be above zero and time cannot be negative.

See the worked example

Using the call value of 16.8p from the example above, with the same 100p share, 95p exercise price, 5% risk-free rate and one year to expiry:

  1. Present value of the exercise price: 95 × e−0.05 = 90.37
  2. p = 16.80 − 100 + 90.37 = 7.2p

A sense check: the share is above the exercise price, so the right to sell at 95p is worth less than the right to buy at it. A put worth less than the call is what you would expect here.

Where the marks go

Discount the exercise price. The final term is Pe multiplied by e−rt, not Pe on its own, and dropping the discounting is the most common slip in a formula this short.

The signs matter and are easy to reverse under pressure: minus the asset price, plus the discounted exercise price. Getting them the wrong way round gives a negative option value, which is impossible — an option is a right, never an obligation, so it can never be worth less than nothing.

The other half

What the sheet
does not give you

Eleven formulas, and AFM still expects a great deal more from memory. Several of the items below carry more marks in a typical sitting than anything the exam gives you.

  • Adjusted present valueBase case NPV at the ungeared cost of equity, plus the present value of the financing side effects. Nothing about it is given to you.
  • Free cash flowBoth to the firm and to equity, and the build-up from operating profit that gets you there.
  • Business valuationEarnings-based, asset-based and free cash flow methods, plus the P/E multiple approach.
  • The binomial modelUp and down factors, the risk-neutral probability, and the lattice itself.
  • DurationMacaulay duration and modified duration, and the price change they imply.
  • Hedging with futuresNumber of contracts, basis, and basis risk.
  • Interest rate swapsThe gains to each party and how they are split.
  • Money market hedgeThe four-step sequence, in both directions.
  • Value at riskWhich needs the normal distribution and a confidence level.
  • Yield to maturityThe redemption yield on a bond, as an IRR.
  • Islamic financeNot numerical, but examinable, and regularly examined.
  • Ratios throughoutGearing, interest cover, EPS, P/E — assumed knowledge from FR and FM.

The takeaway. Eight of the eleven formulas you are given are ones you already used at FM. Reading them fast is a five-minute job. The AFM marks are in the three new ones and in everything on this second list — particularly APV and free cash flow, which between them anchor a large share of Section A.

Eleven formulas
will not pass AFM

Every formula above has appeared in a real AFM question, buried inside a fifty-mark scenario. Work through them in the aCOWtancy AFM Exam Centre. It is part of the free tier — all 15 papers, no payment.

Practise AFM questions