CFA® Level I formula sheet
Every formula that earns its place on Level I — 96 of them, across all 9 topics — each with its variables spelled out and a note on the trap it tends to set. Free to read, no signup. Toggle to Compact for a fast pre-exam scan, or Expanded for the variables and exam-trap notes.
Compact packs the formulas into a dense grid for a fast scan. Expanded adds variables and exam-trap notes.
Quant 18 formulas
Future value (single cash flow)
FV = PV × (1 + r)ⁿ
| FV | Future value |
| PV | Present value (today) |
| r | Periodic interest rate |
| n | Number of compounding periods |
r and n must use the SAME period. With m compounding periods per year, use r/m and n×m.
Present value (single cash flow)
PV = FV(1 + r)ⁿ
| PV | Present value (today) |
| FV | Future value |
| r | Periodic discount rate |
| n | Number of periods |
Discounting is just compounding in reverse. A higher r or longer n lowers PV.
Effective annual rate
EAR = (1 + rs / m)m − 1
| EAR | Effective annual rate |
| rs | Stated (nominal) annual rate |
| m | Compounding periods per year |
As m → ∞ (continuous compounding) EAR = ers − 1. EAR always exceeds the stated rate when m > 1.
Present value of an ordinary annuity
PV = A × 1 − (1 + r)−nr
| PV | Present value of the annuity |
| A | Periodic payment |
| r | Periodic rate |
| n | Number of payments |
Ordinary annuity pays at period END. For an annuity DUE (payment at start), multiply this PV by (1 + r).
Present value of a perpetuity
PV = Ar
| PV | Present value of the perpetuity |
| A | Constant periodic payment |
| r | Periodic discount rate |
Gives the value ONE period before the first payment. Same form as the Gordon growth model with g = 0.
Holding period return
HPR = P₁ − P₀ + IP₀
| HPR | Holding period return |
| P₁ | Ending price |
| P₀ | Beginning price |
| I | Income (dividends or coupons) received |
Not annualized. Chain multiple HPRs as (1+HPR₁)(1+HPR₂)... − 1 to get a multi-period return.
Geometric mean return
RG = [(1 + R₁)(1 + R₂) ... (1 + Rₙ)]1/n − 1
| RG | Geometric mean (compound) return |
| Rₜ | Return in period t |
| n | Number of periods |
Geometric mean ≤ arithmetic mean (equal only with zero variance). Use geometric for past compound growth, arithmetic for a single-period expected return.
Money-weighted return (IRR)
0 = Σ [CFₜ / (1 + IRR)ᵗ]
| CFₜ | Net cash flow at time t (sign-sensitive) |
| IRR | Money-weighted rate of return |
| t | Time index of the cash flow |
MWR is the portfolio IRR — it weights by amount invested, so it is sensitive to deposit/withdrawal timing. TWR removes that and is preferred for comparing managers.
Real (inflation-adjusted) return
1 + real = 1 + nominal1 + inflation
| real | Real rate of return |
| nominal | Nominal rate of return |
| inflation | Inflation rate over the period |
Exact Fisher relation. The approximation real ≈ nominal − inflation only holds for small rates.
Variance (population)
σ² = Σ (Xᵢ − μ)²N
| σ² | Population variance |
| Xᵢ | Observation i |
| μ | Population mean |
| N | Number of observations in the population |
For a SAMPLE, divide by (n − 1), not n — the Bessel correction that makes s² unbiased.
Coefficient of variation
CV = σ|mean|
| CV | Coefficient of variation |
| σ | Standard deviation |
| mean | Mean of the data |
Risk PER UNIT of return — lower is better. Lets you compare dispersion across series with different units or scales.
Standard error of the sample mean
sx = s√n
| sx | Standard error of the mean |
| s | Sample standard deviation |
| n | Sample size |
Shrinks with √n, so quadrupling the sample halves the standard error. Use the population σ when it is known.
Confidence interval for the mean
CI = X̄ ± z × sx
| X̄ | Sample mean (point estimate) |
| z | Reliability factor (1.65 / 1.96 / 2.58 for 90% / 95% / 99%) |
| sx | Standard error of the mean |
Use t (not z) when the population variance is unknown and the sample is small. Memorize 1.96 for 95%.
Test statistic (t-test of a mean)
t = X̄ − μ₀s / √n
| X̄ | Sample mean |
| μ₀ | Hypothesized population mean |
| s | Sample standard deviation |
| n | Sample size |
df = n − 1. Reject the null if |t| exceeds the critical value. A larger n raises the test statistic, all else equal.
Simple linear regression
Yᵢ = b₀ + b₁ × Xᵢ + εᵢ
| Yᵢ | Dependent variable |
| Xᵢ | Independent variable |
| b₀ | Intercept |
| b₁ | Slope coefficient |
| εᵢ | Residual (error term) |
The slope b₁ = Cov(X,Y) / Var(X). The line is fit by minimizing the sum of squared residuals (OLS).
Coefficient of determination
R² = SSRSST = 1 − SSESST
| R² | Coefficient of determination (0 to 1) |
| SSR | Regression (explained) sum of squares |
| SSE | Sum of squared errors (unexplained) |
| SST | Total sum of squares |
Fraction of variation in Y explained by X. In simple regression R² equals the squared correlation coefficient.
Bayes' formula
P(A | B) = P(B | A) × P(A)P(B)
| P(A | B) | Updated (posterior) probability of A given B |
| P(B | A) | Likelihood of B given A |
| P(A) | Prior probability of A |
| P(B) | Unconditional probability of B |
Updates a prior belief with new information. Watch base rates — ignoring P(A) is the classic Bayesian trap.
Expected value of a portfolio
E(Rₚ) = Σ wᵢ × E(Rᵢ)
| E(Rₚ) | Expected portfolio return |
| wᵢ | Weight of asset i |
| E(Rᵢ) | Expected return of asset i |
Expected return is a simple weighted average — but portfolio variance is NOT, because of covariance terms.
Economics 7 formulas
Price elasticity of demand
Ed = %ΔQ%ΔP
| Ed | Own-price elasticity of demand |
| %ΔQ | Percentage change in quantity demanded |
| %ΔP | Percentage change in price |
Elastic if |Ed| > 1, inelastic if < 1. When demand is elastic, a price cut RAISES total revenue.
Cross-price elasticity of demand
Ec = %ΔQx%ΔPy
| Ec | Cross-price elasticity |
| %ΔQx | Percentage change in quantity of good X |
| %ΔPy | Percentage change in price of good Y |
Positive ⇒ substitutes; negative ⇒ complements. Income elasticity instead uses %Δ income (positive = normal good).
Equation of exchange (quantity theory)
M × V = P × Y
| M | Money supply |
| V | Velocity of money |
| P | Price level |
| Y | Real output |
With V and Y stable, money growth maps one-for-one into inflation — the monetarist neutrality result.
GDP by the expenditure approach
GDP = C + I + G + (X − M)
| C | Consumption |
| I | Gross private investment |
| G | Government spending |
| X − M | Net exports (exports minus imports) |
Equals the income approach by identity. Nominal GDP / real GDP gives the GDP deflator.
Fisher effect
Rₙom = Rᵣeal + E(inflation)
| Rₙom | Nominal interest rate |
| Rᵣeal | Real interest rate |
| E(inflation) | Expected inflation |
The nominal rate also embeds a risk premium in practice. This is the additive approximation of the exact (1+R) form.
Covered interest rate parity
F / S = 1 + rd1 + r𝒻
| F | Forward exchange rate (domestic per foreign) |
| S | Spot exchange rate (domestic per foreign) |
| rd | Domestic interest rate |
| r𝒻 | Foreign interest rate |
An arbitrage condition, so it holds tightly. The higher-rate currency trades at a forward DISCOUNT.
Forward premium / discount
Premium = F − SS
| F | Forward rate |
| S | Spot rate |
Positive value = the base currency trades at a forward premium. Annualize by × (360 / days) when quoted per period.
FRA 13 formulas
Accounting equation
Assets = Liabilities + Owners' equity
| Assets | Resources controlled by the firm |
| Liabilities | Present obligations |
| Owners' equity | Residual claim of owners |
The balance sheet identity that must always hold. Equity = contributed capital + retained earnings − treasury stock.
Current ratio
Current ratio = Current assetsCurrent liabilities
| Current assets | Assets expected to convert to cash within a year |
| Current liabilities | Obligations due within a year |
A liquidity measure. The quick ratio is stricter — it strips inventory from the numerator.
Quick (acid-test) ratio
Quick ratio = Cash + Marketable securities + ReceivablesCurrent liabilities
| Cash | Cash and equivalents |
| Marketable securities | Short-term investments |
| Receivables | Accounts receivable |
| Current liabilities | Obligations due within a year |
Excludes inventory and prepaids — the least liquid current assets. Use when inventory turns slowly.
Cash ratio
Cash ratio = Cash + Marketable securitiesCurrent liabilities
| Cash | Cash and equivalents |
| Marketable securities | Short-term investments |
| Current liabilities | Obligations due within a year |
The most conservative liquidity ratio — only the assets that are already cash or near-cash.
Inventory turnover
Inventory turnover = COGSAverage inventory
| COGS | Cost of goods sold |
| Average inventory | (Beginning + ending inventory) / 2 |
Days of inventory on hand = 365 / inventory turnover. Use COGS (not sales) so cost bases match.
Receivables turnover
Receivables turnover = RevenueAverage receivables
| Revenue | Net credit sales (or total revenue) |
| Average receivables | (Beginning + ending receivables) / 2 |
Days sales outstanding = 365 / receivables turnover. The cash conversion cycle = DOH + DSO − days payable.
Gross profit margin
Gross margin = Gross profitRevenue
| Gross profit | Revenue minus cost of goods sold |
| Revenue | Net sales |
Net profit margin instead uses net income. Margin compression with stable revenue points to rising input costs.
Net profit margin
Net margin = Net incomeRevenue
| Net income | Bottom-line profit after all expenses and taxes |
| Revenue | Net sales |
The first term of the DuPont decomposition of ROE.
Return on equity
ROE = Net incomeAverage shareholders' equity
| Net income | Income available to common shareholders |
| Average shareholders' equity | (Beginning + ending equity) / 2 |
Decompose with DuPont to see whether ROE comes from margin, efficiency, or leverage.
DuPont decomposition (3-part ROE)
ROE = (Net income / Revenue) × (Revenue / Average assets) × (Average assets / Average equity)
| Net income / Revenue | Net profit margin (profitability) |
| Revenue / Average assets | Total asset turnover (efficiency) |
| Average assets / Average equity | Financial leverage (equity multiplier) |
Same firms can have equal ROE for very different reasons. The 5-part version splits margin into tax and interest burdens × EBIT margin.
Basic earnings per share
Basic EPS = Net income − Preferred dividendsWeighted average shares outstanding
| Net income | Net income for the period |
| Preferred dividends | Dividends declared on preferred stock |
| Weighted average shares outstanding | Time-weighted common shares for the period |
Subtract preferred dividends only for non-convertible preferred. Diluted EPS adds in convertibles and options (treasury-stock method).
Financial leverage ratio
Financial leverage = Average total assetsAverage total equity
| Average total assets | (Beginning + ending assets) / 2 |
| Average total equity | (Beginning + ending equity) / 2 |
Also called the equity multiplier — the leverage term in DuPont. Higher means more debt funding.
Interest coverage ratio
Interest coverage = EBITInterest expense
| EBIT | Earnings before interest and taxes |
| Interest expense | Periodic interest cost on debt |
A solvency measure — how many times operating earnings cover interest. Lower coverage flags default risk.
Corporate Issuers 8 formulas
Weighted average cost of capital
WACC = (E/V) × re + (D/V) × rd × (1 − t)
| E/V | Weight of equity (market value) |
| D/V | Weight of debt (market value) |
| re | Cost of equity |
| rd | Pre-tax cost of debt |
| t | Marginal tax rate |
Use MARKET-value weights and TARGET capital structure, not book values. The (1 − t) applies only to debt (the tax shield).
After-tax cost of debt
rdₐfter-tax = rd × (1 − t)
| rd | Pre-tax (yield) cost of debt |
| t | Marginal tax rate |
Use the yield to maturity of debt, not the coupon rate. Interest is tax-deductible; dividends are not.
Cost of equity (CAPM)
re = R𝒻 + β × (E(Rₘ) − R𝒻)
| re | Required return on equity |
| R𝒻 | Risk-free rate |
| β | Equity beta (systematic risk) |
| E(Rₘ) − R𝒻 | Equity risk premium |
Only systematic (market) risk is priced; unsystematic risk is diversifiable. This is the SML applied to one stock.
Degree of operating leverage
DOL = %ΔEBIT%ΔSales
| DOL | Degree of operating leverage |
| %ΔEBIT | Percentage change in operating income |
| %ΔSales | Percentage change in unit sales |
Driven by fixed operating costs — higher fixed costs ⇒ higher DOL ⇒ more volatile EBIT. Equals Q(P−V) / [Q(P−V) − F].
Degree of financial leverage
DFL = %ΔEPS%ΔEBIT
| DFL | Degree of financial leverage |
| %ΔEPS | Percentage change in earnings per share |
| %ΔEBIT | Percentage change in operating income |
Driven by fixed financing costs (interest). DTL = DOL × DFL captures total sensitivity of EPS to a sales change.
Free cash flow to the firm
FCFF = CFO + Int × (1 − t) − FCInv
| CFO | Cash flow from operations |
| Int | Interest expense (cash) |
| t | Marginal tax rate |
| FCInv | Fixed-capital (capex) investment |
Add interest back (after tax) because FCFF is pre-financing — it belongs to ALL capital providers. Discount FCFF at the WACC.
Free cash flow to equity
FCFE = FCFF − Int × (1 − t) + Net borrowing
| FCFF | Free cash flow to the firm |
| Int | Interest expense (cash) |
| t | Marginal tax rate |
| Net borrowing | New debt issued minus debt repaid |
FCFE is what is available to EQUITY after debt holders are paid, so discount it at the cost of equity, not WACC.
Net present value
NPV = Σ [CFₜ / (1 + r)ᵗ] − Outlay
| CFₜ | After-tax cash flow at time t |
| r | Project discount rate (often WACC) |
| Outlay | Initial investment at time 0 |
Accept if NPV > 0. When NPV and IRR conflict on mutually exclusive projects, follow NPV.
Equity 17 formulas
Gordon (constant) growth dividend discount model
V₀ = D₁r − g
| V₀ | Intrinsic value today |
| D₁ | Next year's expected dividend |
| r | Required return on equity |
| g | Constant dividend growth rate |
Requires r > g and stable growth. Note D₁ = D₀ × (1 + g) — the model uses NEXT year's dividend.
Sustainable growth rate
g = b × ROE
| g | Sustainable growth rate |
| b | Earnings retention ratio (1 − payout ratio) |
| ROE | Return on equity |
The growth a firm can fund without new external equity. Feeds g into the Gordon model.
Required return (one-period DDM rearranged)
r = (D₁ / P₀) + g
| r | Required return on equity |
| D₁ / P₀ | Forward dividend yield |
| g | Constant growth rate (= capital gains yield) |
Total return = dividend yield + price-appreciation (growth) yield. Just the Gordon model solved for r.
Justified leading P/E
P₀E₁ = 1 − br − g
| P₀ / E₁ | Forward (leading) price-to-earnings ratio |
| 1 − b | Dividend payout ratio |
| r | Required return on equity |
| g | Constant growth rate |
Derived from the Gordon model — higher growth or lower required return justifies a higher multiple. Leading uses E₁; trailing uses E₀.
Enterprise value
EV = Market cap + Total debt + Preferred − Cash
| Market cap | Market value of common equity |
| Total debt | Market value of interest-bearing debt |
| Preferred | Market value of preferred equity |
| Cash | Cash and short-term investments |
The takeover cost of the whole firm. EV/EBITDA is capital-structure neutral, so it compares firms with different leverage.
Residual income
RI = Eₜ − r × Bt−1
| RI | Residual income for the period |
| Eₜ | Net income (earnings) in period t |
| r | Required return on equity |
| Bt−1 | Beginning book value of equity |
Economic profit AFTER charging for the cost of equity capital. A firm can post positive net income yet negative residual income.
Residual income valuation
V₀ = B₀ + Σ [RIₜ / (1 + r)ᵗ]
| V₀ | Intrinsic value today |
| B₀ | Current book value of equity |
| RIₜ | Residual income in period t |
| r | Required return on equity |
Recognizes value earlier than DDM/FCFE, so it relies less on a large terminal value — useful for non-dividend payers.
Preferred stock value (perpetual)
V₀ = Dₚrₚ
| V₀ | Value of the preferred share |
| Dₚ | Fixed annual preferred dividend |
| rₚ | Required return on the preferred |
Non-callable, non-convertible perpetual preferred is just a perpetuity. A maturity or call feature changes the model.
Justified trailing P/E
P₀E₀ = (1 − b)(1 + g)r − g
| b | Retention ratio (1 − dividend payout) |
| g | Constant dividend growth rate |
| r | Required return on equity |
Built on TRAILING earnings, so the numerator carries the extra (1 + g). The leading version (on E₁) drops it — the exam tests which earnings base is used.
Justified price-to-book (P/B)
P₀B₀ = ROE − gr − g
| ROE | Return on equity |
| g | Sustainable growth rate |
| r | Required return on equity |
A firm is justified above book value only when ROE exceeds r. If ROE = r, justified P/B = 1.
Justified price-to-sales (P/S)
P₀S₀ = (E₀/S₀)(1 − b)(1 + g)r − g
| E₀/S₀ | Net profit margin |
| b | Retention ratio |
| g | Growth rate |
| r | Required return on equity |
P/S is driven by net profit margin. Two firms with equal sales but different margins are not comparable on P/S alone.
Dividend yield (trailing and leading)
Trailing = D₀P₀ ; Leading = D₁P₀
| D₀ | Most recent annual dividend (already paid) |
| D₁ | Next expected annual dividend |
| P₀ | Current price |
Trailing uses the dividend just paid; leading uses the next expected dividend. The exam swaps D₀ and D₁ as a distractor.
PEG ratio
PEG = P/Eg
| P/E | Price-to-earnings multiple |
| g | Expected earnings growth, in percent (e.g. 10, not 0.10) |
g is a whole-number percent, not a decimal. Lower PEG implies cheaper growth, but it assumes a linear P/E-to-growth relationship.
Book value of equity per share
BVPS = Total equity − Preferred equityShares outstanding
| Total equity | Common + preferred shareholders’ equity |
| Preferred equity | Book value of preferred claims |
| Shares outstanding | Common shares outstanding |
Preferred equity is subtracted because BVPS is a COMMON-shareholder figure.
Two-stage (multistage) DDM
V₀ = Σ [Dₜ / (1 + r)ᵗ] + Dn+1 / (r − g)(1 + r)ⁿ
| Dₜ | Dividend in year t of the high-growth stage |
| Dn+1 | First dividend of the constant-growth stage |
| r | Required return on equity |
| g | Constant (terminal) growth rate |
| n | Length of the high-growth stage |
The terminal value [Dn+1 / (r − g)] is dated at time n, so it is discounted back n periods — not n+1. This off-by-one is the classic trap.
Terminal value (Gordon growth)
TVₙ = D_ n+1r − g
| Dn+1 | Dividend one period after the terminal date |
| r | Required return on equity |
| g | Constant perpetual growth rate |
TVₙ is dated at time n (end of the explicit forecast), then discounted to today. Requires r > g.
Single-stage FCFE valuation
V₀ = FCFE₁r − g
| FCFE₁ | Next-year free cash flow to equity |
| r | Required return on equity |
| g | Constant growth rate of FCFE |
FCFE replaces dividends when payout does not reflect capacity to pay. Same Gordon structure, different cash-flow base.
Fixed Income 10 formulas
Bond price (discounted cash flows)
P = Σ [C / (1 + y)ᵗ] + FV(1 + y)N
| P | Bond price |
| C | Periodic coupon payment |
| y | Yield to maturity per period |
| FV | Face (par) value |
| N | Number of periods to maturity |
Price moves inversely to yield. Premium when coupon > yield, discount when coupon < yield, par when equal.
Current yield
Current yield = Annual couponBond price
| Annual coupon | Total annual coupon payments |
| Bond price | Current market (full or flat) price |
Ignores capital gain/loss to maturity and reinvestment, so it sits between coupon rate and YTM for a discount bond.
Macaulay duration
MacDur = Σ [t × (CFₜ / (1 + y)ᵗ)]Price
| MacDur | Macaulay duration (in periods) |
| t | Time to each cash flow |
| CFₜ | Cash flow at time t |
| y | Yield per period |
| Price | Full bond price |
The cash-flow-weighted average time to receipt — also the investment horizon where price and reinvestment risk offset.
Modified duration
ModDur = MacDur1 + y
| ModDur | Modified duration |
| MacDur | Macaulay duration |
| y | Yield to maturity per period |
The % price change for a 1% yield change. Use EFFECTIVE duration for bonds with embedded options (cash flows aren't fixed).
Approximate modified duration
ApproxModDur = PV_− − PV_+2 × PV₀ × Δy
| PV_− | Price if yield falls by Δy |
| PV_+ | Price if yield rises by Δy |
| PV₀ | Initial price |
| Δy | Yield change (in decimal) |
Replace prices with option-adjusted values and you get effective duration — the right measure for callable/putable bonds.
Price change from duration and convexity
%ΔPrice = −ModDur × Δy + 0.5 × Convexity × (Δy)²
| ModDur | Modified (or effective) duration |
| Convexity | Convexity measure |
| Δy | Change in yield (decimal) |
Duration is the linear estimate; convexity is the curvature correction and is always additive for option-free bonds (it helps you).
Money duration (dollar duration)
Money duration = ModDur × Full price
| ModDur | Modified duration |
| Full price | Full (dirty) price of the position |
The price change in CURRENCY units, not percent, for a yield change. PVBP = money duration × 0.0001.
Price value of a basis point
PVBP = PV_− − PV_+2
| PV_− | Price if yield falls 1 bp |
| PV_+ | Price if yield rises 1 bp |
The money change in a bond's price for a 1-basis-point yield move — a hedging workhorse.
Yield spread (G-spread)
Yield = Benchmark rate + Spread
| Yield | Bond's yield to maturity |
| Benchmark rate | Government bond yield of matched maturity |
| Spread | Credit/liquidity spread over the benchmark |
G-spread is over a govt yield; I-spread is over a swap rate; the Z-spread is added to every spot rate to reprice the bond.
Forward rate (no-arbitrage)
(1 + z₂)² = (1 + z₁) × (1 + f1,1)
| z₁ | 1-year spot rate |
| z₂ | 2-year spot rate |
| f1,1 | 1-year rate, 1 year forward |
Forward rates are implied by spot rates. An upward-sloping spot curve makes forward rates exceed spot rates.
Derivatives 7 formulas
Forward / futures price (no income)
F₀ = S₀ × (1 + r)T
| F₀ | Forward price set today |
| S₀ | Spot price today |
| r | Risk-free rate |
| T | Time to settlement (years) |
Pure cost-of-carry. Subtract the future value of any income (dividends/coupons) and add storage/convenience adjustments.
Value of a long forward during its life
Vₜ = Sₜ − [F₀ / (1 + r)T−t]
| Vₜ | Value of the long forward at time t |
| Sₜ | Spot price at time t |
| F₀ | Originally contracted forward price |
| r | Risk-free rate |
| T − t | Time remaining to settlement |
Value is zero at initiation. At expiry VT = ST − F₀, the contract's settlement payoff.
Put-call parity
c + PV(K) = p + S₀
| c | Price of a European call |
| p | Price of a European put |
| PV(K) | Present value of the strike K, = K / (1+r)T |
| S₀ | Current price of the underlying |
A fiduciary call (call + risk-free bond) equals a protective put (put + stock). Same strike and expiry; European options only.
Put-call-forward parity
c + PV(K) = p + PV(F₀)
| c | Price of a European call |
| p | Price of a European put |
| PV(K) | Present value of the strike |
| PV(F₀) | Present value of the forward price = S₀ for a no-income asset |
The parity restated with a forward replacing the spot — handy when the underlying pays a carry/income stream.
Call option intrinsic value
Call intrinsic = max(0, Sₜ − K)
| Sₜ | Current price of the underlying |
| K | Exercise (strike) price |
Total option value = intrinsic value + time value. A put's intrinsic value is max(0, K − Sₜ).
Option value bounds (European call)
max(0, S₀ − PV(K)) ≤ c ≤ S₀
| c | European call price |
| S₀ | Current underlying price |
| PV(K) | Present value of the strike |
A call is never worth more than the underlying, nor less than its discounted intrinsic value. Longer time and higher volatility raise option value.
Binomial option hedge ratio (delta)
h = c_+ − c_−S_+ − S_−
| h | Hedge ratio (option delta) |
| c_+ | Option value in the up state |
| c_− | Option value in the down state |
| S_+ | Underlying price in the up state |
| S_− | Underlying price in the down state |
The number of underlying units that replicates the option. The binomial price uses risk-neutral probabilities, not real ones.
Alternatives 6 formulas
Net asset value per share (fund)
NAV = Assets − LiabilitiesShares outstanding
| Assets | Market value of fund holdings |
| Liabilities | Fund liabilities |
| Shares outstanding | Units/shares of the fund |
Open-end funds transact at NAV; closed-end funds and ETFs can trade at a premium or discount to NAV.
Real estate capitalization rate
Cap rate = NOIProperty value
| NOI | Net operating income (annual) |
| Property value | Market value or purchase price |
Cap rate ≈ discount rate − growth, so a lower cap rate implies a higher price (and lower expected return).
Net operating income
NOI = Effective gross income − Operating expenses
| Effective gross income | Potential rent less vacancy/collection loss, plus other income |
| Operating expenses | Property operating costs (excludes debt service and depreciation) |
NOI is BEFORE financing and taxes — it isolates the property's operating performance for cap-rate valuation.
Hedge fund management fee
Management fee = Mgmt fee rate × Assets under management
| Mgmt fee rate | Annual management fee percentage |
| Assets under management | Fund AUM (often beginning- or average-period value) |
The '2' in '2-and-20'. Charged on AUM regardless of performance; specify whether it is on beginning or ending value.
Hedge fund incentive (performance) fee
Incentive fee = Incentive rate × max(0, Profit − Hurdle)
| Incentive rate | Performance fee percentage (the '20') |
| Profit | Gain above the prior high-water mark |
| Hurdle | Minimum return before the fee applies |
A high-water mark blocks paying twice for the same gains after a drawdown. Watch whether the hurdle is hard or soft.
Committed-capital private equity multiples
TVPI = Cumulative distributions + Residual NAVPaid-in capital
| Cumulative distributions | Cash returned to LPs (drives DPI) |
| Residual NAV | Value of investments still held |
| Paid-in capital | Capital actually drawn from LPs |
TVPI = DPI (realized) + RVPI (unrealized). A multiple, not a rate — it ignores the TIMING that IRR captures.
Portfolio Management 10 formulas
Two-asset portfolio variance
σₚ² = w₁² σ₁² + w₂² σ₂² + 2 w₁ w₂ ρ1,2 σ₁ σ₂
| w₁, w₂ | Portfolio weights of assets 1 and 2 |
| σ₁, σ₂ | Standard deviations of assets 1 and 2 |
| ρ1,2 | Correlation between the two assets |
Diversification benefit grows as correlation falls. Cov(1,2) = ρ σ₁ σ₂, so the last term can also be written 2 w₁ w₂ Cov.
Capital asset pricing model (SML)
E(Rᵢ) = R𝒻 + βᵢ × (E(Rₘ) − R𝒻)
| E(Rᵢ) | Expected return of asset i |
| R𝒻 | Risk-free rate |
| βᵢ | Beta of asset i (systematic risk) |
| E(Rₘ) − R𝒻 | Market risk premium |
The Security Market Line prices BETA. Assets above the SML are undervalued (offer excess return for their risk).
Beta
βᵢ = Cov(Rᵢ, Rₘ)σₘ²
| βᵢ | Beta of asset i |
| Cov(Rᵢ, Rₘ) | Covariance of asset and market returns |
| σₘ² | Variance of market returns |
Equivalently β = ρi,m × σᵢ / σₘ. The market's beta is 1.0; cash has beta 0.
Capital market line
E(Rₚ) = R𝒻 + [(E(Rₘ) − R𝒻) / σₘ] × σₚ
| E(Rₚ) | Expected portfolio return |
| R𝒻 | Risk-free rate |
| E(Rₘ) − R𝒻 | Market risk premium |
| σₘ | Standard deviation of the market |
| σₚ | Standard deviation of the portfolio |
The CML uses TOTAL risk (σ) and applies only to efficient portfolios; the SML uses beta and prices any asset.
Sharpe ratio
Sharpe = Rₚ − R𝒻σₚ
| Rₚ | Portfolio return |
| R𝒻 | Risk-free rate |
| σₚ | Standard deviation of portfolio returns (total risk) |
Excess return per unit of TOTAL risk — best for evaluating a standalone portfolio. Higher is better.
Treynor ratio
Treynor = Rₚ − R𝒻βₚ
| Rₚ | Portfolio return |
| R𝒻 | Risk-free rate |
| βₚ | Portfolio beta (systematic risk) |
Excess return per unit of SYSTEMATIC risk — appropriate for a well-diversified portfolio or a sub-portfolio of a larger one.
Jensen's alpha
αₚ = Rₚ − [R𝒻 + βₚ × (Rₘ − R𝒻)]
| αₚ | Jensen's alpha (excess return vs CAPM) |
| Rₚ | Portfolio return |
| βₚ | Portfolio beta |
| Rₘ − R𝒻 | Market risk premium |
Return above what CAPM predicts for the portfolio's beta. Positive alpha = outperformance on a risk-adjusted basis.
M-squared (M²)
M² = (Rₚ − R𝒻) × (σₘ / σₚ) − (Rₘ − R𝒻)
| Rₚ | Portfolio return |
| R𝒻 | Risk-free rate |
| σₘ | Standard deviation of the market |
| σₚ | Standard deviation of the portfolio |
| Rₘ | Market return |
Sharpe expressed in return (percentage) units — the excess return over the market after scaling the portfolio to the market's risk.
Roy's safety-first ratio
SFRatio = E(Rₚ) − RLσₚ
| E(Rₚ) | Expected portfolio return |
| RL | Minimum acceptable (threshold) return |
| σₚ | Standard deviation of the portfolio |
Maximize SFRatio to minimize the probability of falling below RL. With RL = R𝒻 it reduces to the Sharpe ratio.
Expected return on the minimum-variance set (multi-asset)
E(Rₚ) = Σ wᵢ × E(Rᵢ)
| E(Rₚ) | Expected portfolio return |
| wᵢ | Weight of asset i (weights sum to 1) |
| E(Rᵢ) | Expected return of asset i |
Return is linear in weights, but risk is not — which is why the efficient frontier bows to the left as correlations fall.
Knowing the formula is step one. Knowing when the exam wants it is what passes — and that comes from working problems, not memorizing a sheet.
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