ICWIM14 min readUpdated June 2026

ICWIM Calculation Formulas Cheat Sheet

ICWIM Chapter 5 (Economics & Investment Analysis) is 21% of the exam — the single heaviest chapter — and most of those marks come from straightforward calculations. Get the formulas to reflex speed and you've banked 15+ marks before reading any conceptual answer carefully. This cheat sheet covers every formula tested, with worked examples and the traps to avoid.

How to use this. Don't try to memorise everything in one sitting. Pass 1: read through and mark the formulas you DON'T recognise. Pass 2: chapter-focused practice, returning here for the formula reference. Pass 3 (exam week): pure reflex drill — for each formula, write it from memory, check, repeat.

Time value of money

Present Value (PV)

PV = FV ÷ (1 + r)ⁿ

The value today of a future cash flow. r = discount rate per period; n = number of periods.

Worked example. What is the PV of £10,000 receivable in 5 years, at a discount rate of 4% pa?
PV = 10,000 ÷ (1.04)⁵ = 10,000 ÷ 1.2167 = £8,219.27

Future Value (FV)

FV = PV × (1 + r)ⁿ

The future value of an amount invested today.

Worked example. £5,000 invested today at 6% pa for 10 years:
FV = 5,000 × (1.06)¹⁰ = 5,000 × 1.7908 = £8,954.24

Annuity FV (regular contributions)

FV = P × [((1 + r)ⁿ − 1) ÷ r]

For a fixed periodic contribution P at rate r for n periods.

Trap: assume END-of-period contributions unless the question specifies "annuity due" (beginning of period), in which case multiply the result by (1+r).

Holding period & multi-period returns

Holding Period Return (HPR)

HPR = (Ending value − Beginning value + Income) ÷ Beginning value

Total return over the holding period, including income.

Annualised return

R_ann = (1 + HPR)^1/n − 1

Converts an n-year HPR to a per-annum geometric rate.

Time-Weighted Rate of Return (TWRR)

TWRR = [(1 + r₁) × (1 + r₂) × ... × (1 + r_n)]^1/n − 1

The geometric mean of sub-period returns. Used to compare PORTFOLIO MANAGERS — neutralises the effect of cash flows.

Money-Weighted Rate of Return (MWRR)

The IRR of all cash flows (contributions, withdrawals, ending value). Used to assess the INVESTOR's experience including the impact of timing decisions.

Classic trap: "Which return measure neutralises the effect of cash flows?" → TWRR. "Which reflects the investor's actual experience?" → MWRR. They go opposite directions on the question of "whose performance are we measuring?".

Risk measures

Standard deviation

σ = √[ Σ(r_i − r̄)² ÷ (n − 1) ]

Measures dispersion of returns around the mean. The standard "risk" measure for an asset.

Variance

σ² = Σ(r_i − r̄)² ÷ (n − 1)

The square of standard deviation. Used in portfolio mathematics because variances of uncorrelated assets are additive (std devs are not).

Covariance

Cov(A, B) = Σ(r_A,i − r̄_A)(r_B,i − r̄_B) ÷ (n − 1)

Correlation

ρ_AB = Cov(A, B) ÷ (σ_A × σ_B)

Standardised covariance, bounded −1 to +1. The single most important diversification driver.

Portfolio risk & return

Two-asset portfolio return

R_P = w_A × R_A + w_B × R_B

Weighted average of asset returns.

Two-asset portfolio variance

σ²_P = w_A²σ_A² + w_B²σ_B² + 2 w_A w_B ρ_AB σ_A σ_B

The cross-term is what diversification operates on — when ρ < 1, portfolio risk < weighted average of individual risks.

Worked example. 50/50 portfolio of A (σ = 20%) and B (σ = 15%), correlation 0.3.
σ²_P = 0.5²(0.20)² + 0.5²(0.15)² + 2(0.5)(0.5)(0.3)(0.20)(0.15)
= 0.01 + 0.0056 + 0.0045 = 0.0201
σ_P = √0.0201 = 14.18%
(Note: significantly less than the 17.5% weighted-average of std devs, thanks to correlation < 1.)

CAPM & the security market line

CAPM

E(R_i) = R_f + β_i × (R_m − R_f)

Expected return on asset = risk-free rate + beta × market risk premium. The cornerstone of modern portfolio pricing.

Worked example. Risk-free rate 3%, market return 9%, asset beta 1.4.
E(R) = 3% + 1.4 × (9% − 3%) = 3% + 8.4% = 11.4%

Beta (linear regression form)

β_i = Cov(R_i, R_m) ÷ σ_m²

Beta measures systematic (market) risk. β = 1 → asset moves 1:1 with market. β > 1 → more volatile than market. β < 1 → less volatile.

Jensen's Alpha

α = R_actual − [R_f + β × (R_m − R_f)]

Excess return over what CAPM predicts. Positive α = manager outperformed risk-adjusted expectation.

Performance ratios

Sharpe Ratio

Sharpe = (R_P − R_f) ÷ σ_P

Excess return per unit of TOTAL risk. Higher = better. Used for assessing standalone investments.

Treynor Ratio

Treynor = (R_P − R_f) ÷ β_P

Excess return per unit of SYSTEMATIC risk only. Used when the portfolio is one part of a diversified mix (idiosyncratic risk already diversified away).

Information Ratio

IR = (R_P − R_benchmark) ÷ Tracking Error

How much excess return the manager delivers per unit of active risk taken. The skill ratio for active managers vs benchmark.

Memory anchor for the three ratios:

Sharpe = excess return ÷ TOTAL risk (σ)
Treynor = excess return ÷ SYSTEMATIC risk (β)
Information Ratio = excess return vs BENCHMARK ÷ tracking error

Sharpe and Treynor use the risk-free rate; IR uses the benchmark.

Bond pricing & yields

Bond price (clean)

P = Σ [ C ÷ (1 + y)^t ] + [ F ÷ (1 + y)ⁿ ]

Bond price = PV of coupons + PV of redemption. C = periodic coupon, y = yield per period, F = face value, n = total periods.

Current Yield

CY = Annual coupon ÷ Current market price

Simple income yield. Ignores capital gains/losses to redemption.

Yield to Maturity (YTM)

The internal rate of return of holding the bond to maturity. The single discount rate that makes PV of cash flows equal current price. No closed-form formula — solved iteratively (or by financial calculator).

Accrued Interest

AI = Coupon × (days since last coupon ÷ days in period)

The portion of the coupon earned but not yet paid as of the settlement date. Added to clean price to give dirty (full) price.

Duration (Macaulay)

D = Σ [ t × (CF_t ÷ (1+y)^t) ] ÷ Price

Weighted-average time to receive cash flows, weighted by PV. A bond's "average maturity".

Modified Duration

ModDur = Macaulay Duration ÷ (1 + y)

The percentage change in bond price for a 1% change in yield. Negative relationship: yields up = price down.

Equity valuation

Gordon Growth Model (Dividend Discount)

P₀ = D₁ ÷ (r − g)

Price = next dividend ÷ (required return − growth rate). Assumes constant growth in perpetuity. g must be less than r for the formula to make sense.

Worked example. Next year's dividend £2.00, required return 8%, growth 3%.
P = 2.00 ÷ (0.08 − 0.03) = 2.00 ÷ 0.05 = £40.00

Price/Earnings ratio

P/E = Price per share ÷ EPS

Multiple of earnings the market is paying for the stock.

Dividend Yield

DY = Annual dividend per share ÷ Share price

Ratios you may also see

Ratio	Formula	Interpretation
ROE	Net income ÷ Equity	Profitability per £ of shareholder capital
ROA	Net income ÷ Total assets	Profitability per £ of asset base
Gearing	Debt ÷ (Debt + Equity)	Financial leverage
Interest cover	EBIT ÷ Interest expense	Ability to service debt
Current ratio	Current assets ÷ Current liabilities	Short-term liquidity
Quick ratio	(Current assets − Inventory) ÷ Current liabilities	Stricter liquidity test

Statistics quick-reference

Concept	Definition
Arithmetic mean	Sum ÷ n. Bias upward for return series — use geometric for multi-period returns.
Geometric mean	[Π (1 + r_i)]^1/n − 1. The correct way to annualise multi-period returns.
Median	Middle value of sorted data. Robust to outliers.
Mode	Most frequent value. Rare in finance; appears in skewness questions.
Skewness	Asymmetry of distribution. Mean > median = positive skew; mean < median = negative skew.
Kurtosis	"Fat tails". Excess kurtosis (vs normal) common in real returns.
Normal distribution	68% within 1σ, 95% within 2σ, 99.7% within 3σ.

Most-tested traps

Confusion	The fix
Sharpe vs Treynor	Sharpe uses σ; Treynor uses β. Sharpe for standalone, Treynor for portion-of-portfolio.
TWRR vs MWRR	TWRR judges the MANAGER; MWRR captures the INVESTOR's experience.
Macaulay vs Modified duration	Macaulay is in years; Modified is dimensionless % sensitivity to yield. ModDur = Macaulay ÷ (1+y).
Arithmetic vs Geometric mean	Geometric for multi-period returns. Arithmetic biases upward when returns vary.
YTM vs Current Yield	CY ignores capital gain/loss to redemption. YTM includes it.
Clean vs Dirty price	Dirty = Clean + Accrued interest. Quoted price is usually clean; settlement is dirty.
r > g in Gordon	The formula collapses if r ≤ g. If a question implies g > r, it's testing whether you spot the issue.

Drill these to reflex speed

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