ACCA FM · Financial Management

The formula
sheet, explained

You get nine formulas in the exam. Most students can quote all nine and still lose the marks, because knowing the formula is not the same as knowing which number goes where.

Every heading below has the formula, the trap, and a calculator you can push numbers through until it sticks.

Start with the sheet Get the official sheet
Provided in your exam

The nine headings

  1. Economic order quantity
  2. Miller–Orr Model
  3. The Capital Asset Pricing Model
  4. The asset beta formula
  5. The Growth Model
  6. Gordon’s growth approximation
  7. The weighted average cost of capital
  8. The Fisher formula
  9. Purchasing power parity and interest rate parity

Then the present value and annuity tables. That is the whole sheet.

Jump to a formula

Nine formulas,
nine ways to lose marks

Every card gives you the formula as it appears on the sheet, a worked example with real numbers, the mistake the examiner sees most often, and a calculator wired to the same formula so you can test your own figures.

1Economic order quantity 2Miller–Orr 3CAPM 4Asset beta 5Growth model 6Gordon’s growth 7WACC 8Fisher 9PPP & IRP TMaths tables !Not on the sheet
Formulas 1–2

Working capital

Both of these are inventory and cash management. Both are usually worth two or three marks in a Section B or C question, and both are almost free marks if you have practised them once.

Working capital 1

Economic order quantity

The order size that makes total inventory cost as low as possible — the point where the cost of holding stock exactly balances the cost of placing orders.

As given in the exam EOQ = √2C0DCh

C0 cost of placing one order  ·  D annual demand in units  ·  Ch cost of holding one unit for one year

Try it Live
Economic order quantity—
Orders placed per year—
Annual ordering cost—
Annual holding cost—
Total inventory cost—

Check your inputs — demand, order cost and holding cost must all be above zero.

See the worked example

A company uses 40,000 units a year. Each order costs £25 to place. Holding one unit for a year costs 50p.

  1. Put the numbers in: (2 × 25 × 40,000) ÷ 0.50 = 4,000,000
  2. Square root it: √4,000,000 = 2,000 units
  3. Orders per year: 40,000 ÷ 2,000 = 20 orders, costing 20 × £25 = £500
  4. Average inventory is half the order size: 2,000 ÷ 2 = 1,000 units, costing 1,000 × £0.50 = £500

Ordering cost and holding cost come out equal. That is not a coincidence — it is what the EOQ is defined as, and it is a free check on your answer in the exam.

Where the marks go

Holding cost is per unit per year. If the question gives it as a percentage of purchase price, you have to work it out first: 10% of a £5 unit is £0.50, not 10.

If there is a bulk discount, EOQ is only the starting point. You then have to compare total cost at the EOQ against total cost at each discount order level — the answer is often not the EOQ.

Working capital 2

Miller–Orr model

Sets the cash balance you return to, and the upper limit that triggers a move into investments, when cash flows in and out unpredictably. You set the lower limit yourself — the model gives you the other two.

As given in the exam
Return point = Lower limit + (13 × spread)
Spread = 3 3√34 × transaction cost × variance of cash flowsinterest rate

The upper limit is not given a line of its own — it is simply lower limit + spread.

Try it Live
Return point—
Spread—
Upper limit—

Transaction cost, variance and interest rate must all be above zero.

See the worked example

A company keeps a minimum cash balance of £10,000. Each transfer between cash and investments costs £50. The standard deviation of daily cash flows is £1,000, and the daily interest rate is 0.02%.

  1. Variance is the standard deviation squared: £1,000² = 1,000,000
  2. Top of the fraction: 0.75 × 50 × 1,000,000 = 37,500,000
  3. Divide by the daily rate as a decimal: 37,500,000 ÷ 0.0002 = 187,500,000,000
  4. Take the cube root: 5,723.6 — then multiply by 3 for the spread: £17,171
  5. Return point = 10,000 + (17,171 ÷ 3) = £15,724
  6. Upper limit = 10,000 + 17,171 = £27,171
Where the marks go

The formula wants variance, and questions usually give you standard deviation. Square it first. Getting this wrong does not just change the answer — it changes it by a factor of about 100.

The interest rate has to be on the same time basis as the cash flow variance. Daily variance needs a daily rate, so an annual rate has to be converted before it goes in.

And it goes in as a decimal. 0.02% is 0.0002, not 0.02.

Formulas 3–6

The cost of equity

Four of the nine formulas exist to get you to one number: what shareholders require. There are two routes — the market-risk route (CAPM, via beta) and the dividend route (the growth model). Questions frequently ask you to use both and comment on why they differ.

Cost of equity 3

The capital asset pricing model

The required return on a share, built up from the risk-free rate plus a premium for the systematic risk that shareholders cannot diversify away.

As given in the exam E(ri) = Rf + βi(E(rm) − Rf)

Rf risk-free rate  ·  βi equity beta of the share  ·  E(rm) return on the market as a whole

Try it Live
Cost of equity—
Risk premium applied—
See the worked example

Treasury bills yield 4%. The company’s equity beta is 1.4. The return on the market is expected to be 11%.

  1. The market risk premium is the market return less the risk-free rate: 11 − 4 = 7%
  2. Scale it by beta: 1.4 × 7 = 9.8%
  3. Add the risk-free rate back: 4 + 9.8 = 13.8%
Where the marks go

This is the single most reliable trap on the paper. If the question says market risk premium or equity risk premium, that figure is already the (E(rm) − Rf) part — do not subtract the risk-free rate from it a second time. If it says market return or return on the market, you do subtract.

Switch the toggle above between the two and watch the answer move. In the example, reading 11% as a premium instead of a return gives 19.4% rather than 13.8%.

Cost of equity 4

The asset beta formula

Strips the effect of borrowing out of a beta, so you can move a beta from one company’s capital structure to another’s. Equity beta carries business risk and financial risk; asset beta carries business risk only.

As given in the exam — one formula, not two βa = Ve(Ve + Vd(1−T))βe + Vd(1−T)(Ve + Vd(1−T))βd

Ve market value of equity  ·  Vd market value of debt  ·  T corporation tax rate  ·  βd debt beta, taken as zero unless the question says otherwise

Read this before you use it

The two terms are not “one for gearing and one for ungearing”. They are the equity share and the debt share of a weighted average. Asset beta is simply the weighted average of the equity beta and the debt beta.

Ungearing and regearing are the same formula used in opposite directions. Ungear by reading it left to right; regear by rearranging it to make βe the subject. When the debt beta is zero — which is nearly always — the whole second term vanishes and you are left with a single fraction.

Step 1Take the proxy

Find a listed company in the same line of business. Take its equity beta.

Step 2Ungear it

Using the proxy’s gearing and tax rate. You now have a pure business-risk beta.

Step 3Regear it

Using your gearing and tax rate. You now have an equity beta for your company.

Step 4Into CAPM

Turn that beta into a cost of equity, then into a project discount rate.

Try it Live
Asset beta, βa—
Debt net of tax, Vd(1−T)—
Equity weighting—

Equity must be above zero and the two values cannot both be zero.

See the worked example — ungear, then regear

You are appraising a project in a new industry. A listed company in that industry has an equity beta of 1.5 and is financed 70% equity, 30% debt by market value. Your own company is financed 60% equity, 40% debt. Tax is 20% for both. Debt betas are zero.

  1. Ungear on the proxy’s gearing. Debt net of tax = 30 × 0.8 = 24. Total = 70 + 24 = 94.
  2. βa = (70 ÷ 94) × 1.5 = 1.117
  3. Regear on your gearing. Debt net of tax = 40 × 0.8 = 32. Total = 60 + 32 = 92.
  4. 1.117 = (60 ÷ 92) × βe, so βe = 1.117 × (92 ÷ 60) = 1.713
  5. Now into CAPM. At a 4% risk-free rate and a 7% market risk premium: 4 + (1.713 × 7) = 16.0%

Your company carries more debt than the proxy, so it ends up with a higher equity beta and a higher cost of equity. That is the financial risk showing through, and saying so in a sentence is often worth a mark on its own.

Where the marks go

Use the proxy’s equity and debt when you ungear, and your own when you regear. Mixing the two up is the most common way this question is lost, and it produces an answer that looks perfectly plausible.

Gearing given as a debt-to-equity ratio of 30:70 means Ve is 70 and Vd is 30. Given as “debt is 30% of total finance”, the same. Given as “gearing of 30% measured as debt to equity”, Vd is 30 against Ve of 100. Read the definition, do not assume it.

Use market values, never book values.

Cost of equity 5

The growth model

Values a share as the present value of its future dividends, growing forever at a constant rate. Rearranged, it gives you the cost of equity from the share price — which is how FM usually asks for it.

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

D0 the dividend just paid  ·  g constant annual growth rate  ·  P0 ex-div market value of one share

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.

See the worked example

A company has just paid a dividend of 20p per share. Dividends have grown steadily at 5% a year and are expected to continue. The shares trade at £2.50 ex-div.

  1. Grow the dividend on one year: 20p × 1.05 = 21p
  2. Divide by the share price: 21p ÷ 250p = 0.084, or 8.4%
  3. Add the growth rate back: 8.4 + 5 = 13.4%
Where the marks go

The price has to be ex-div. If the question gives a cum-div price, subtract the dividend that is about to be paid before you use it — a cum-div price of £2.70 with a 20p dividend imminent is an ex-div price of £2.50.

D0 is the dividend just paid, so it needs growing by one year. If the question hands you next year’s dividend directly, that is already D1 — do not grow it again.

Where growth is not given, you are expected to estimate it, either from the historic dividend record or from Gordon’s growth approximation below.

Cost of equity 6

Gordon’s growth approximation

Estimates the growth rate for the model above from inside the business rather than from its dividend history. Growth comes from reinvesting profit, so it depends on how much is held back and how well it is put to work.

As given in the exam g = bre

b proportion of earnings retained  ·  re the return earned on funds reinvested

Try it Live
Growth rate, g—
Payout ratio—
See the worked example

A company pays out 40% of its earnings as dividends and reinvests the rest, earning 12% on reinvested funds.

  1. Retention rate: 100% − 40% = 60%, so b = 0.6
  2. g = 0.6 × 0.12 = 0.072, or 7.2%

That 7.2% is then the g you feed into the growth model.

Where the marks go

The sheet prints the second term as re, which invites students to plug in the cost of equity. It is not the cost of equity — it is the accounting rate of return the company earns on the money it reinvests. Questions usually describe it as return on reinvestment or return on shareholders’ funds.

b is the retention rate, not the payout ratio. If the question gives a payout ratio, take it off 100% first.

Formula 7

Putting it together

Everything above feeds this. WACC is where the cost of equity meets the cost of debt, and it is the discount rate most FM investment appraisal questions are built around.

Cost of capital 7

Weighted average cost of capital

The average return the company must earn to keep all its providers of finance content, weighted by how much of the company each of them funds.

As given in the exam WACC = VeVe + Vdke + VdVe + Vdkd(1−T)

ke cost of equity  ·  kd cost of debt before tax  ·  T corporation tax rate

Try it Live
WACC—
Equity weighting—
Post-tax cost of debt—
Contribution from equity—
Contribution from debt—

Total finance must be above zero.

See the worked example

A company has 4 million shares trading at £3.00, and irredeemable debt with a market value of £4m. Its cost of equity is 13.8% and the pre-tax cost of debt is 7%. Tax is 20%.

  1. Market value of equity: 4m × £3.00 = £12m. Total finance = £12m + £4m = £16m.
  2. Weightings: equity 12⁄16 = 0.75, debt 4⁄16 = 0.25
  3. Post-tax cost of debt: 7% × (1 − 0.20) = 5.6%
  4. WACC = (0.75 × 13.8) + (0.25 × 5.6) = 10.35 + 1.40 = 11.75%
Where the marks go

Weightings are market values, not the balance sheet. Equity is the number of shares times the share price — not share capital plus reserves. Debt trading at £95 per £100 nominal has a market value of 95% of nominal.

The (1−T) in the formula only handles debt where the cost is a simple interest yield. For redeemable debt you have to find the post-tax cost of debt separately as an IRR of the after-tax cash flows, then bring that figure in as kd(1−T) already calculated. That IRR working is not on the formula sheet.

Preference shares are a third source. They get their own weighting and their cost gets no tax relief.

Formulas 8–9

Inflation and
exchange rates

Three formulas that all do the same job in different clothes: they take a rate today and adjust it for the difference between two economies. Once you see that, they stop being three things to memorise.

Investment appraisal 8

The Fisher formula

Links the money rate you actually see to the real rate underneath it, once general inflation is taken out. It is the bridge between the two ways of doing an NPV.

As given in the exam (1 + i) = (1 + r)(1 + h)

i money (nominal) rate  ·  r real rate  ·  h general rate of inflation

Try it Live
Money rate, i—
Simply adding the two would give—
See the worked example

A company’s real cost of capital is 6%. General inflation is running at 4%. What money rate should be used to discount inflated cash flows?

  1. (1 + i) = 1.06 × 1.04 = 1.1024
  2. i = 1.1024 − 1 = 0.1024, or 10.24%

Not 10%. The extra 0.24% is inflation applied to the real return itself, and rounding it away loses the mark.

Where the marks go

You cannot add and subtract these rates. Multiply.

Keep the two methods clean and never cross them. Either inflate every cash flow at its own specific inflation rate and discount at the money rate, or leave all cash flows in today’s prices and discount at the real rate. The real-rate method only works if every cash flow inflates at the same general rate — the moment the question gives different inflation rates for sales and costs, you must use the money method.

Tax and capital allowances are always calculated on the inflated figures, whichever method you use.

Foreign exchange 9

Purchasing power parity
and interest rate parity

Two ways of forecasting an exchange rate. Purchasing power parity uses the inflation differential; interest rate parity uses the interest rate differential. They are the same shape, and only the inputs change.

As given in the exam
S1 = S0 × (1 + hc)(1 + hb)
F0 = S0 × (1 + ic)(1 + ib)

S0 spot rate today  ·  S1 expected future spot  ·  F0 forward rate  ·  c the counter (variable) currency  ·  b the base currency

Try it Live
Expected future spot, S1—
Movement against the base currency—
See the worked example

The spot rate is $1.5000 = £1. US inflation is expected to run at 5% and UK inflation at 2%. What is the expected spot rate in one year?

  1. The rate is quoted as dollars per pound, so the dollar is the counter currency and the pound is the base.
  2. S1 = 1.5000 × (1.05 ÷ 1.02) = $1.5441

It takes more dollars to buy a pound, so the dollar has weakened. That is the direction you would expect — the currency with higher inflation loses value. If your answer moves the other way, you have the two rates the wrong way round.

Interest rate parity works identically. With US interest at 4% and UK interest at 6%, the one-year forward rate is 1.5000 × (1.04 ÷ 1.06) = $1.4717.

Where the marks go

The counter currency is the one there is more than one of — in $1.5000/£1 that is the dollar. Its rate goes on top. Get this the wrong way round and the currency moves in the wrong direction, which is visible to the marker.

Interest rate parity gives a forward rate, which you can actually contract at. Purchasing power parity gives an expected future spot rate, which is a forecast and may not happen. Questions ask you to distinguish these, and it is a written mark rather than a calculation one.

For more than one year, the ratio is raised to the power of the number of years, not multiplied by it.

After the formulas

The maths tables

Two tables follow the formula sheet. Both are just a formula worked out for you, and knowing which one you are looking at saves more marks than knowing the numbers.

Table 1

Present value

The value today of one single amount received in n years. Use it for a one-off receipt — a scrap value, a sale proceed, a single tax payment.

DF = (1 + r)−n
Table 2

Annuity

The value today of the same amount every year for n years, starting at the end of year 1. Use it whenever a cash flow repeats unchanged.

AF = 1 − (1 + r)−nr
Build your own

Discount factor lookup

Set a rate and a number of years. Both factors are calculated from the formulas above, so you can check any figure the tables give you — and get rates the tables do not carry.

Try it Live
Present value factor — year n only—
Annuity factor — years 1 to n—
Perpetuity factor — forever not in the tables—

Rate must be above zero for the perpetuity factor, and n at least 1.

YearPV factorCumulative = annuity factor
Where the marks go

An annuity factor already adds up all the individual years. Do not multiply an annuity factor by the number of years as well — that double counts.

If an annuity starts later than year 1, take the annuity factor for the full run and subtract the factor for the years you are skipping. Years 3 to 7 at 10% is the 7-year factor less the 2-year factor.

The tables round to three decimal places and stop at 15 years and 20%. The examiner accepts answers from either the tables or a calculator, so small differences are fine — but show your workings so the marker can follow you.

The other half

What the sheet
does not give you

Nine formulas are provided. The FM syllabus needs considerably more than nine. Everything below has to come out of your own head on the day, and between them these account for more marks than the formulae sheet does.

  • PerpetuityA cash flow forever: 1 ÷ r. Growing forever: 1 ÷ (r − g). Used constantly and never given to you.
  • IRR by interpolationIRR ≈ a + [NPVa ÷ (NPVa − NPVb)] × (b − a), where a and b are the two rates you tried
  • Cost of redeemable debtThe post-tax IRR of the debt’s own cash flows. No formula, just an NPV at two rates and interpolation.
  • Cost of preference sharesfixed dividend ÷ market price, with no tax relief.
  • Cost of bank loansThe interest rate itself, adjusted for tax.
  • Working capital ratiosInventory days, receivable days, payable days, and the cash operating cycle that strings them together.
  • Early settlement discountsThe annualised cost of offering or taking one, and whether it beats the overdraft rate.
  • Investor ratiosEPS, P/E ratio, dividend yield, dividend cover, earnings yield.
  • Gearing and interest coverBoth the debt-to-equity and debt-to-total-capital versions — and you need to say which you have used.
  • Money market hedgeA sequence of steps rather than a formula, and one of the most heavily tested areas of Section C.
  • Tax-allowable depreciationReducing balance or straight line, with a balancing allowance or charge in the final year.
  • Asset replacementEquivalent annual cost: NPV of one cycle ÷ annuity factor.

The takeaway. Students spend revision time memorising the nine formulas that are handed to them and almost none on the dozen that are not. Flip that round. Learn the sheet well enough to read it quickly, then put the hours into everything on this list.

Knowing it and
using it are different

Every one of these formulas has appeared in a real ACCA question, with the numbers buried in a scenario. Work through those questions in the aCOWtancy FM Exam Centre. It is part of the free tier — all 15 papers, no payment.

Practise FM questions