Returns of Financial Assets and Instruments: formula sheet

Key formulas

Holding period return

HPR = (P1 − P0 + D1) ÷ P0 = (P1 + D1) ÷ P0 − 1

P0 is the starting price, P1 the ending price, D1 income received in the period.

Multi-period HPR

HPR = (1 + R1) × (1 + R2) × … × (1 + Rn) − 1

Link returns by multiplying growth factors. Never add them.

Arithmetic mean return

Mean = (R1 + R2 + … + Rn) ÷ n

Use for the expected return of a single period.

Geometric mean return

RG = [(1 + R1)(1 + R2)…(1 + Rn)]^(1/n) − 1

Use for compound growth over past periods. Each 1 + R must be positive.

Harmonic mean

XH = n ÷ Σ(1 ÷ Xi)

Use for average price when a fixed money amount buys each time. All values must be positive.

Ordering of means

Harmonic ≤ Geometric ≤ Arithmetic

Holds for positive values. All three are equal only if all observations are equal.

Money-weighted return (IRR)

0 = CF0 + CF1 ÷ (1 + r) + CF2 ÷ (1 + r)² + … + CFn ÷ (1 + r)ⁿ

Deposits into the portfolio are negative, withdrawals, income received and the ending value are positive. The r that solves it is the MWR for the whole period, per period of the cash flow spacing.

Sub-period holding period return

HPR = (Ending value + Cash flow received) ÷ Beginning value − 1

Use the value just before any new deposit or withdrawal at the end of the sub-period, and the value just after the previous one at the start.

Time-weighted return (total)

TWR = [(1 + HPR1) × (1 + HPR2) × … × (1 + HPRn)] − 1

Geometric linking. Do not add or average the HPRs.

Annualized time-weighted return

Annualized TWR = (1 + TWR)^(1 ÷ years) − 1

Use when the total period is longer than one year. For example, two years gives a square root.

Which measure to use

Manager evaluation → TWR; investor's own return → MWR

If the manager controls the timing of cash flows, MWR can also be appropriate.

Net return (approximate)

Net return ≈ Gross return − Fees and expenses

Works when fees are stated as a percentage of assets. Check whether fees are deducted from value or from return.

After-tax return

After-tax return = Pre-tax return × (1 − tax rate)

Use only when one tax rate applies to the whole return. Otherwise tax each component separately.

Real return (exact)

Real return = (1 + nominal return) ÷ (1 + inflation rate) − 1

Use this when asked for the exact real return.

Real return (approximation)

Real return ≈ Nominal return − Inflation rate

Only a close estimate when rates are small. It overstates the exact real return whenever inflation is positive, and the gap grows as rates rise.

Leveraged return

rL = [rP × (VP) − (VB × rB)] ÷ VE, equivalently rL = rP + (VB ÷ VE) × (rP − rB)

VP = total position, VE = own equity, VB = borrowed amount, rP = asset return, rB = borrowing rate. VP = VE + VB.

Quick revision

• HPR = (ending value − beginning value + income) ÷ beginning value.
• Arithmetic mean = sum of returns ÷ number of periods; it is never lower than the geometric mean.
• Geometric mean = [(1 + r1)(1 + r2)...(1 + rn)]^(1/n) − 1, and it measures compound growth.
• The two means are equal only when all returns are identical.
• Money-weighted return is the IRR of the cash flows and is affected by the timing and size of contributions.
• Time-weighted return chains sub-period returns and removes the effect of cash flow timing.
• GIPS require time-weighted returns for performance presentation.
• Time-weighted return is the better measure of a manager's skill when the manager does not control the flows.
• Annualize a return over a period shorter than a year by compounding: (1 + r)^(periods per year) − 1, where r is the periodic return for that shorter period.
• Effective annual rate = (1 + periodic rate)^m − 1, where m is compounding periods per year. If you are given a stated annual rate r, the periodic rate is r ÷ m, so EAR = (1 + r/m)^m − 1.
• Gross return is before fees; net return is after fees and expenses.
• Real return ≈ nominal return − inflation; the exact form is (1 + nominal) ÷ (1 + inflation) − 1.
• Leveraged return = [gain on total position − borrowing cost] ÷ your own equity.

Common mistakes

• Averaging returns arithmetically to describe compound growth. Fix: If the question asks for past growth of wealth over time, use the geometric mean.
• Forgetting to add 1 before multiplying returns. Fix: Always convert to growth factors, 1 + R, then subtract 1 at the end.
• Averaging the sub-period returns to get the time-weighted return. Fix: Always link with (1 + HPR) factors. Use the arithmetic mean only when the question asks for it.
• Using MWR to rank managers. Fix: MWR is distorted by client cash flows the manager does not control. Use TWR for manager comparison unless the manager controls flow timing.
• Dividing leveraged gain by total position instead of equity. Fix: Always divide the net gain after interest by your own equity.
• Using nominal − inflation when the question asks for the exact real return. Fix: Use (1 + nominal) ÷ (1 + inflation) − 1 whenever the stem does not say approximate.

Exam tips

• Questions are three-option MCQs. The wrong options are often the arithmetic mean when geometric is asked, so work out both and match the wording.
• Use the ordering harmonic ≤ geometric ≤ arithmetic to eliminate options without full calculation.
• Watch for the word 'income' or 'dividend' in HPR questions. It must go in the numerator.
• Practise the yx key on your approved calculator until the root step takes under 20 seconds.
• Conceptual questions may ask which mean suits a purpose. Match: expected return to arithmetic, past compound growth to geometric, average cost to harmonic.
• If a question asks about the manager's skill or comparing managers, the answer is almost always time-weighted return.
• Predict direction before calculating: big deposit before a good period makes MWR above TWR, before a bad period makes it below.
• Expect the arithmetic average of sub-period returns as a wrong option. Eliminate it and then check the other two.