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Free CFA Level I Formula Sheet (2026)

Every CFA Level I formula you need on the test, grouped by topic and rendered with full math notation. 107 formulas across 9 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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All CFA Level I Formulas

Quantitative Methods 21 items
Present Value (single sum)
PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}
FV = future value, r = periodic rate, n = number of periods
Future Value of ordinary annuity
FV=PMT×(1+r)n−1rFV = PMT \times \frac{(1+r)^n - 1}{r}
PMT = periodic payment, r = periodic rate, n = periods
Present Value of ordinary annuity
PV=PMT×1−(1+r)−nrPV = PMT \times \frac{1 - (1+r)^{-n}}{r}
PMT = periodic payment, r = periodic rate, n = periods
Present Value of perpetuity
PV=PMTrPV = \frac{PMT}{r}
PMT = periodic payment, r = discount rate
Population variance
σ2=∑i=1N(Xi−μ)2N\sigma^2 = \frac{\sum_{i=1}^{N}(X_i - \mu)^2}{N}
μ\mu = population mean, N = population size
Sample variance
s2=∑i=1n(Xi−Xˉ)2n−1s^2 = \frac{\sum_{i=1}^{n}(X_i - \bar{X})^2}{n-1}
Xˉ\bar{X} = sample mean, n = sample size (uses n−1 for unbiasedness)
Standard error of the sample mean
SE(xˉ)=σ/nSE(\bar{x})=\sigma/\sqrt{n}
SE = standard error, xˉ\bar{x} = sample mean, σ\sigma = population standard deviation, n = sample size. If σ\sigma is unknown, estimate SE with s/ns/\sqrt{n}, where s is the sample standard deviation. Assumes independent observations.
One-sample t-test for a mean
t=xˉ−μ0s/nt=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}
t = test statistic, xˉ\bar{x} = sample mean, μ0\mu_0 = hypothesized population mean, s = sample standard deviation, n = sample size. Use n - 1 degrees of freedom. Exact for an independent normal sample with unknown population variance.
Confidence interval for a population mean (normal approximation)
xˉ±Zα/2σ/n\bar{x}\pm Z_{\alpha/2}\sigma/\sqrt{n}
xˉ\bar{x} = sample mean; σ\sigma = population standard deviation; n = sample size. Zα/2Z_{\alpha/2} = positive normal critical value, upper tail α/2\alpha/2, confidence 1−α1-\alpha. Independent sample, large n; substitute sample standard deviation s if needed.
Confidence interval for a population mean (unknown variance)
xˉ±tα/2,n−1sn\bar{x}\pm t_{\alpha/2,n-1}\frac{s}{\sqrt{n}}
xˉ\bar{x} = sample mean, s = sample standard deviation, n = sample size; tα/2,n−1t_{\alpha/2,n-1} = positive t critical value with n - 1 degrees of freedom and upper-tail probability α/2\alpha/2; 1−α1-\alpha = confidence level. Exact for an independent normal sample.
Standard error of a regression slope coefficient
sb^1=se∑i=1n(Xi−Xˉ)2s_{\hat{b}_1}=\frac{s_e}{\sqrt{\sum_{i=1}^{n}(X_i-\bar{X})^2}}
sb^1s_{\hat{b}_1} = slope standard error; ses_e = standard error of estimate; XiX_i = independent-variable value for observation i; Xˉ\bar{X} = sample mean of X; n = sample size. Simple regression with independent, constant-variance errors.
t-statistic for a regression coefficient
t=b^j−Bjsb^jt=\frac{\hat{b}_j-B_j}{s_{\hat{b}_j}}
t = test statistic; b^j\hat{b}_j = coefficient estimate; BjB_j = null value; sb^js_{\hat{b}_j} = its standard error; j = 0 (intercept), 1 (slope). Simple regression: n observations, n - 2 degrees of freedom; normal, independent, constant-variance errors.
Sharpe ratio
Sp=Rp−RfσpS_p = \frac{R_p - R_f}{\sigma_p}
RpR_p = portfolio return, RfR_f = risk-free rate, σp\sigma_p = portfolio std dev
Excess return per unit of total risk
Holding Period Return (HPR)
HPR=P1−P0+D1P0HPR = \frac{P_1 - P_0 + D_1}{P_0}
P1P_1 = ending price, P0P_0 = beginning price, D1D_1 = cash distributions received
Bayes' Theorem
P(A∣B)=P(B∣A)⋅P(A)P(B)P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}
Updates prior probability P(A) given new information B
P(B)=P(B∣A)P(A)+P(B∣Ac)P(Ac)P(B) = P(B|A)P(A) + P(B|A^c)P(A^c)
Correlation coefficient
ρXY=Cov(X,Y)σXσY\rho_{XY} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}
Ranges from −1 to +1; unitless measure of linear association
Geometric mean return
RˉG=[∏t=1n(1+Rt)]1/n−1\bar{R}_G = \left[\prod_{t=1}^{n}(1+R_t)\right]^{1/n} - 1
RtR_t = period t return, n = number of periods
Money-weighted return (MWR)
∑t=0NCFt(1+r)t=0\sum_{t=0}^{N} \frac{CF_t}{(1+r)^t} = 0
CFtCF_t = cash flow at time t (outflows negative, inflows positive), r = money-weighted return, N = number of periods
Fisher relation (real vs nominal return)
(1+Rnominal)=(1+Rreal)(1+i)(1 + R_{nominal}) = (1 + R_{real})(1 + i); RnominalR_{nominal} = nominal, RrealR_{real} = real, i = inflation. Subtracting inflation is only an approximation; it breaks down when inflation is high.
Time-weighted return (TWR)
(1+TWR)=∏i=1n(1+ri)(1 + TWR) = \prod_{i=1}^{n}(1 + r_i)
rir_i = holding-period return for sub-period i, n = number of sub-periods between external cash flows
Leveraged portfolio return
rLP=rp+VDVE(rp−rd)r_{LP}=r_p+\frac{V_D}{V_E}(r_p-r_d)
rLPr_{LP} = return on the investor's equity, rpr_p = return on the total portfolio, VDV_D = borrowed funds, VEV_E = investor's own equity, rdr_d = borrowing rate. Leverage raises the return only when rp>rdr_p>r_d.
Economics 6 items
GDP expenditure approach
GDP=C+I+G+(X−M)GDP = C + I + G + (X - M)
C = consumption, I = investment, G = government spending
X = exports, M = imports, (X−M) = net exports
Money multiplier
m=1reserve requirementm = \frac{1}{\text{reserve requirement}}
Maximum deposit expansion from a given reserve base
ΔMoney supply=m×ΔReserves\Delta\text{Money supply} = m \times \Delta\text{Reserves}
Fisher effect
(1+rnom)=(1+rreal)(1+π)(1 + r_{nom}) = (1 + r_{real})(1 + \pi)
Approx: rnom≈rreal+πr_{nom} \approx r_{real} + \pi
rnomr_{nom} = nominal rate, rrealr_{real} = real rate, π\pi = expected inflation
Breakeven and shutdown points
Breakeven: P=ATCP = ATC (price covers all costs).
Shutdown (short run): P<AVCP < AVC. If AVC<P<ATCAVC < P < ATC, keep operating short-run because contribution covers some fixed cost.
Price elasticity of demand
Ed=%ΔQd%ΔPE_d = \frac{\%\Delta Q_d}{\%\Delta P}
∣Ed∣>1|E_d| > 1: elastic (revenue rises when P falls). ∣Ed∣<1|E_d| < 1: inelastic. =1= 1: unit-elastic (revenue maximized).
Cross-rate calculation
AC=AB×BC\frac{A}{C}=\frac{A}{B}\times\frac{B}{C}. Multiply or divide rates to cross. Here bid(A/C)=bid(A/B)bid(B/C)\mathrm{bid}(A/C)=\mathrm{bid}(A/B)\mathrm{bid}(B/C) and ask(A/C)=ask(A/B)ask(B/C)\mathrm{ask}(A/C)=\mathrm{ask}(A/B)\mathrm{ask}(B/C). Invert: new bid =1/old ask=1/\text{old ask}, new ask =1/old bid=1/\text{old bid}.
Corporate Issuers 6 items
WACC
WACC=wdrd(1−t)+wprp+wereWACC = w_d r_d (1-t) + w_p r_p + w_e r_e
w = weights (market value), r = required returns
d = debt, p = preferred, e = equity, t = tax rate
Free Cash Flow to the Firm (FCFF)
FCFF=NI+NCC+Int(1−t)−ΔWC−CapExFCFF = NI + NCC + Int(1-t) - \Delta WC - CapEx
NI = net income, NCC = non-cash charges, Int = interest expense
ΔWC\Delta WC = change in working capital, t = tax rate
Net Present Value (NPV)
NPV=∑t=0nCFt(1+r)tNPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}
CFtCF_t = expected after-tax cash flow at time t, r = required rate of return
Accept when NPV > 0: the investment adds value in currency terms
Degree of Financial Leverage (DFL)
DFL=EBITEBIT−IDFL = \frac{EBIT}{EBIT - I}
I = interest expense
% change in EPS per 1% change in EBIT
Return on Invested Capital (ROIC)
ROIC=After-tax operating profitAverage invested capitalROIC = \frac{\text{After-tax operating profit}}{\text{Average invested capital}}
Invested capital=Total long-term liabilities+Total equity\text{Invested capital}=\text{Total long-term liabilities}+\text{Total equity} (Corporate Issuers convention).
ROIC above the cost of capital means the company is creating value.
Free Cash Flow to Equity (FCFE)
FCFE=FCFF−Int(1−t)+ΔDebtFCFE = FCFF - \text{Int}(1-t) + \Delta\text{Debt}
Cash available to equity holders after all obligations and reinvestment
Financial Statement Analysis 8 items
3-factor DuPont decomposition
ROE=NISales×SalesAverage total assets×Average total assetsAverage total equityROE = \frac{NI}{\text{Sales}} \times \frac{\text{Sales}}{\text{Average total assets}} \times \frac{\text{Average total assets}}{\text{Average total equity}}
Net profit margin×Asset turnover×Financial leverage\text{Net profit margin}\times\text{Asset turnover}\times\text{Financial leverage}
Current ratio
Current ratio=Current assetsCurrent liabilities\text{Current ratio} = \frac{\text{Current assets}}{\text{Current liabilities}}
Measures short-term liquidity; higher = more liquid
Inventory turnover
Inventory turnover=COGSAverage inventory\text{Inventory turnover} = \frac{\text{COGS}}{\text{Average inventory}}
Days on hand (DOH): DOH=365Inventory turnoverDOH = \frac{365}{\text{Inventory turnover}}
Receivables turnover and DSO
Receivables turnover=RevenueAvg Accounts Receivable\text{Receivables turnover} = \frac{\text{Revenue}}{\text{Avg Accounts Receivable}}
DSO=365Receivables turnover\text{DSO} = \frac{365}{\text{Receivables turnover}}
Days Sales Outstanding: average collection period
Return on Assets (ROA)
ROA=Net incomeAverage total assetsROA = \frac{\text{Net income}}{\text{Average total assets}}
Alternative: ROA=Net profit margin×Asset turnoverROA = \text{Net profit margin} \times \text{Asset turnover}
2-factor DuPont decomposition
Return on Equity (ROE)
ROE=Net incomeAverage total equityROE = \frac{\text{Net income}}{\text{Average total equity}}
DuPont: ROE=Net margin×Asset turnover×LeverageROE=\text{Net margin}\times\text{Asset turnover}\times\text{Leverage}. Sustainable growth: g=b×NIBeginning book value of equityg=b\times\frac{NI}{\text{Beginning book value of equity}}, where bb is the earnings retention ratio.
Cash flow interest coverage ratio
If interest paid is in CFO: Coverage=(CFO+Interest paid+Taxes paid)/Interest paid\text{Coverage}=(\text{CFO}+\text{Interest paid}+\text{Taxes paid})/\text{Interest paid}. If interest is financing: Coverage=(CFO+Taxes paid)/Interest paid\text{Coverage}=(\text{CFO}+\text{Taxes paid})/\text{Interest paid}. CFO is cash from operations; distinct from accounting EBIT/InterestEBIT/\text{Interest}.
Cash return on assets
Cash ROA=CFOAverage total assets\text{Cash ROA} = \frac{CFO}{\text{Average total assets}}
CFO = cash flow from operations; denominator uses average of beginning and ending total assets
Equity Investments 20 items
Gordon Growth Model (DDM)
V0=D1r−g=D0(1+g)r−gV_0 = \frac{D_1}{r - g} = \frac{D_0(1+g)}{r - g}
D1D_1 = next dividend, r = required return, g = constant growth rate
Requires r > g
Justified P/E (leading)
P0E1=1−br−g\frac{P_0}{E_1} = \frac{1 - b}{r - g}
b = retention ratio (1−b = payout ratio), r = required return, g = ROE × b
Enterprise Value (EV)
EV=Market cap+Debt+Preferred+Minority interest−(Cash+Short-term investments)EV = \text{Market cap} + \text{Debt} + \text{Preferred} + \text{Minority interest} - (\text{Cash} + \text{Short-term investments})
EV/EBITDAEV/EBITDA = enterprise value multiple
Valuation metric neutral to capital structure
Price-to-Book ratio
P/B=Market price per shareBook value per share\text{P/B} = \frac{\text{Market price per share}}{\text{Book value per share}}
Justified P/B: ROE−gr−g\frac{ROE-g}{r-g}
P/B>1\text{P/B}>1 means the market value of equity exceeds its book value.
P/E ratio (trailing & leading)
Trailing: P0EPS0\frac{P_0}{EPS_{0}}
uses last 12 months EPS.
Leading: P0EPS1\frac{P_0}{EPS_{1}}
uses next 12 months / forecast EPS. Forward-looking variant.
Equity value per share from enterprise value
P0=EV−Debt−Preferred+CashSharesP_0=\dfrac{EV-Debt-Preferred+Cash}{Shares}
EVEV = enterprise value, DebtDebt = interest-bearing debt, PreferredPreferred = market value of preferred shares, CashCash = cash and short-term investments, including cash equivalents, SharesShares = diluted shares outstanding
Terminal value via Gordon growth applied to FCFF
TVn=FCFFn+1WACC−gTV_n=\dfrac{FCFF_{n+1}}{WACC-g}
FCFFn+1FCFF_{n+1} = next-period free cash flow to firm, WACCWACC = weighted average cost of capital, gg = sustainable long-run FCFF growth rate, with g<WACCg<WACC
Residual income
RIt=NIt−(r×BVt−1)RI_t = NI_t - (r \times BV_{t-1})
NI = net income, r = cost of equity, BV = book value of equity at start of period
Arbitrage pricing theory (APT) expected return
E(Ri)=Rf+∑k=1Kβi,k λkE(R_i) = R_f + \sum_{k=1}^{K} \beta_{i,k}\,\lambda_k
RfR_f = risk-free rate, βi,kβ_{i,k} = sensitivity of asset i to factor k, λkλ_k = risk premium per unit exposure to factor k
Two-stage dividend discount model
V0=∑t=1nDt(1+r)t+Dn+1/(r−gs)(1+r)nV_0 = \sum_{t=1}^{n} \frac{D_t}{(1+r)^t} + \frac{D_{n+1}/(r - g_s)}{(1+r)^n}
DtD_t = dividend at time t, r = cost of equity, gsg_s = stable growth, n = explicit horizon
Carhart four-factor model
E(Ri)−Rf=βi,M[E(Rm)−Rf]+βi,SSMB+βi,VHML+βi,WWMLE(R_i) - R_f = \beta_{i,M}[E(R_m)-R_f] + \beta_{i,S}\text{SMB} + \beta_{i,V}\text{HML} + \beta_{i,W}\text{WML}
SMB = size, HML = value, WML = momentum premiums; β = loadings
Single-stage FCF perpetuity enterprise value
EV=FCFF0(1+g)WACC−gEV=\frac{FCFF_0(1+g)}{WACC-g}, for g<WACCg<WACC
EVEV = enterprise value; FCFF0FCFF_0 = current free cash flow to the firm; gg = constant growth rate; WACCWACC = weighted average cost of capital
Total return on an equity security
Rtotal=P1−P0+D1P0=Rprice+D1P0R_{total} = \frac{P_1 - P_0 + D_1}{P_0} = R_{price} + \frac{D_1}{P_0}
P0P_0 = beginning price, P1P_1 = ending price, D1D_1 = dividends received
Price return on an equity security
Rprice=P1−P0P0R_{price} = \frac{P_1 - P_0}{P_0}
P0P_0 = beginning price, P1P_1 = ending price
Average daily volume (ADV)
ADV=∑i=1nVin\text{ADV} = \frac{\sum_{i=1}^{n} V_i}{n}
ViV_i = shares traded on day i, n = number of trading days in the window
Free float shares
Float=Shares Outstanding−Restricted Shares\text{Float}=\text{Shares Outstanding}-\text{Restricted Shares}
Restricted = insider lock-ups, strategic stakes, and government holdings
Justified trailing P/E from Gordon growth
P0E0=(1−b)(1+g)r−g\frac{P_0}{E_0} = \frac{(1-b)(1+g)}{r-g}
b = retention ratio, (1-b) = payout ratio, r = required return on equity, g = sustainable growth rate
Implied price via method of comparables
Ptarget=Mpeer×FtargetP_{target} = M_{peer} \times F_{target}
MpeerM_{peer} = peer-group median multiple, FtargetF_{target} = target's per-share fundamental (EPS, BVPS, etc.)
Cumulative voting total votes available
V=S×NV = S \times N
V = total votes a shareholder may cast, S = shares owned, N = number of director seats up for election
Voting power share in a dual-class structure
VP=sAvA+sBvBSAvA+SBvBVP=\frac{s_Av_A+s_Bv_B}{S_Av_A+S_Bv_B}
sAs_A and sBs_B = shareholder's holdings; SAS_A and SBS_B = total shares outstanding in each class; vAv_A and vBv_B = votes per share.
Fixed Income 18 items
Bond price
P=∑t=1nC(1+r)t+FV(1+r)nP = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{FV}{(1+r)^n}
C = coupon payment, r = periodic YTM, n = periods, FV = face value
Current yield
Current yield=Annual couponPrice\text{Current yield} = \frac{\text{Annual coupon}}{\text{Price}}
Simplest yield measure; ignores capital gains/losses and time value
Macaulay duration
DMac=∑t=1nt⋅CFt(1+r)tPD_{Mac} = \frac{\sum_{t=1}^{n} t \cdot \frac{CF_t}{(1+r)^t}}{P}
Present-value-weighted average time to receive cash flows. With tt indexing coupon periods and rr the periodic YTM, the formula gives coupon periods; divide by coupon payments per year to obtain years.
Modified duration
DMod=DMac1+rD_{Mod} = \frac{D_{Mac}}{1 + r}
%ΔP≈−DMod×Δy\%\Delta P \approx -D_{Mod} \times \Delta y
r = periodic YTM, Δy\Delta y = change in yield
Forward rate from spot rates
(1+z2)2=(1+z1)(1+1f1)(1 + z_2)^2 = (1 + z_1)(1 + {_1f_1})
General: (1+zn)n=(1+zn−1)n−1(1+n−1f1)(1+z_n)^n = (1+z_{n-1})^{n-1}(1 + {_{n-1}f_1})
z = spot rate, f = implied forward rate
Price value of a basis point (PVBP)
PVBP=Py−0.01%−Py+0.01%2PVBP = \frac{P_{y-0.01\%}-P_{y+0.01\%}}{2}, where PP denotes full price.
Approximation: PVBP≈Dmod×P×0.0001PVBP \approx D_{\text{mod}}\times P\times0.0001, using annualized modified duration and full price.
Dollar price change for a 1 bp yield move.
Floating-rate note price using discount margin
PV=∑t=1N(MRR+QM)FV/m(1+(MRR+DM)/m)t+FV(1+(MRR+DM)/m)NPV = \sum_{t=1}^{N}\frac{(MRR+QM)FV/m}{(1+(MRR+DM)/m)^t} + \frac{FV}{(1+(MRR+DM)/m)^N}
MRR = reference rate, QM = quoted margin, DM = discount margin, m = periods/yr, FV = face
Bond equivalent yield for money market instruments
BEY=FV−PVPV⋅365daysBEY = \frac{FV - PV}{PV} \cdot \frac{365}{days}
An add-on rate on a 365-day year: the interest earned is divided by PV, the price paid. A discount rate (T-bills, commercial paper) divides by FV instead.
FV = amount paid at maturity, PV = price, days = days to maturity
Debt-to-EBITDA leverage ratio
Debt/EBITDA=Total DebtEBITDA\text{Debt/EBITDA} = \dfrac{\text{Total Debt}}{\text{EBITDA}}
Total Debt = all interest-bearing debt; EBITDA = earnings before interest, taxes, depreciation, amortization. Lower is stronger.
EBITDA-to-interest coverage ratio
EBITDA/Interest=EBITDAInterest Expense\text{EBITDA/Interest} = \dfrac{\text{EBITDA}}{\text{Interest Expense}}
EBITDA = earnings before interest, taxes, depreciation, amortization; Interest Expense = period interest. Higher is stronger.
Effective convexity
EffCon=P−+P+−2P0P0×(Δy)2\text{EffCon}=\frac{P_{-}+P_{+}-2P_{0}}{P_{0}\times(\Delta y)^2}
P−P_{-} and P+P_{+} = full prices after parallel downward and upward benchmark yield-curve shifts of Δy\Delta y, respectively, allowing cash flows to change; P0P_0 = initial full price; Δy\Delta y = benchmark yield-curve shift (decimal)
Effective duration
EffDur=P−−P+2×P0×Δy\text{EffDur}=\frac{P_{-}-P_{+}}{2\times P_{0}\times\Delta y}
P−P_{-} and P+P_{+} = full prices after parallel downward and upward benchmark yield-curve shifts of Δy\Delta y, respectively, allowing cash flows to change; P0P_0 = initial full price; Δy\Delta y = benchmark yield-curve shift (decimal)
Approximate convexity
ApproxCon=P−+P+−2P0P0×(Δy)2\text{ApproxCon} = \frac{P_{-} + P_{+} - 2P_{0}}{P_{0} \times (\Delta y)^{2}}
P₋ = price after yield falls by Δy, P₊ = price after yield rises, P₀ = starting full price
Bond percentage price change with convexity adjustment
%ΔP≈−ModDur×Δy+12×Con×(Δy)2\%\Delta P \approx -\text{ModDur} \times \Delta y + \tfrac{1}{2} \times \text{Con} \times (\Delta y)^{2}
ModDur = modified duration, Con = annual convexity, Δy = yield change (decimal)
Repo repurchase price
Repurchase price=Purchase price0×[1+Repo rate×Days360]\text{Repurchase price} = \text{Purchase price}_0 \times \left[1 + \text{Repo rate} \times \frac{\text{Days}}{360}\right]
Purchase price0\text{Purchase price}_0 = cash lent at the start (collateral value less the haircut); Repo rate = annual rate; Days = repo term in days on a 360-day year
Full price of a bond between coupon dates
PVFull=PV×(1+r)t/TPV^{Full} = PV \times (1+r)^{t/T}
PV = value of the remaining cash flows at the last coupon date, r = market discount rate per period, t = days since the last coupon, T = days in the coupon period. Flat price =PVFull−AI= PV^{Full} - AI, with AI=tT×PMTAI = \frac{t}{T} \times PMT
Periodicity conversion of an annual yield
(1+APRmm)m=(1+APRnn)n\left(1+\frac{APR_m}{m}\right)^{m}=\left(1+\frac{APR_n}{n}\right)^{n}
APRmAPR_m = annual rate stated for m compounding periods per year; APRnAPR_n = equivalent annual rate for n periods per year. More frequent compounding pairs with a lower stated annual rate
Par rate from spot rates
100=PMT(1+z1)1+PMT(1+z2)2+⋯+PMT+100(1+zN)N100 = \frac{PMT}{(1+z_1)^1} + \frac{PMT}{(1+z_2)^2} + \cdots + \frac{PMT+100}{(1+z_N)^N}
Solve for PMT; the par rate per period is PMT/100. ztz_t = spot rate for period t, N = periods to maturity, 100 = par value
Derivatives 13 items
Put-call parity
For European calls and puts on the same underlying with no income or benefit, and the same XX and TT: C+X(1+r)T=P+S0C+\frac{X}{(1+r)^T}=P+S_0.
CC = call price, PP = put price, S0S_0 = spot price, XX = exercise price
rr = risk-free rate, TT = time to expiration
Forward contract price
With no additional ownership costs or benefits: F0=S0(1+r)TF_0=S_0(1+r)^T.
With continuous dividends: F0=S0e(r−q)TF_0=S_0e^{(r-q)T}.
S0S_0 = spot, rr = risk-free rate, TT = time, qq = dividend yield
Forward price with discrete income or cost
F0(T)=(S0−PV(I)+PV(C))(1+r)TF_0(T) = (S_0 - PV(I) + PV(C))(1 + r)^T
I = discrete income (dividends, coupons) over T; C = carrying cost (storage). PV at risk-free rate. Income reduces the forward; cost raises it.
Option payoff at expiration
Long call: max⁡(ST−X,0)\max(S_T - X, 0); Long put: max⁡(X−ST,0)\max(X - S_T, 0)
STS_T = price at expiry, X = strike. Short positions are the negative of long. Subtract premium paid for profit.
Intrinsic value and time value
Call intrinsic: max⁡(S−X,0)\max(S - X, 0); Put intrinsic: max⁡(X−S,0)\max(X - S, 0)
Time value = Option price − intrinsic. ATM/OTM intrinsic = 0; deep ITM time value → 0 near expiry.
Lower bound on European options (no dividends)
Call: c≥max⁡(S0−X(1+r)−T,0)c \geq \max(S_0 - X(1+r)^{-T}, 0)
Put: p≥max⁡(X(1+r)−T−S0,0)p \geq \max(X(1+r)^{-T} - S_0, 0)
Enforces no-arbitrage. Below these, the option is mispriced relative to the synthetic.
Value of a long forward contract at time t
Vt=Ft−F0(1+r)T−tV_t = \frac{F_t - F_0}{(1 + r)^{T - t}}
FtF_t = current forward price, F0F_0 = original forward price, r = risk-free rate, T - t = time remaining to expiration
Swap fixed rate (price) at initiation
s∗=1−D(tn)∑i=1nD(ti)s^{*}=\frac{1-D(t_n)}{\sum_{i=1}^{n}D(t_i)}
D(ti)D(t_i) = discount factor at settlement ii; nn = number of equal settlement periods; s∗s^{*} = fixed rate per settlement period. For mm equal periods per year, the annual quoted rate is m×s∗m\times s^{*}.
Swap value to the fixed-receiver after initiation
Vswap, fixed receiver=PV(fixed leg)−PV(floating leg)V_{\text{swap, fixed receiver}}=PV(\text{fixed leg})-PV(\text{floating leg})
PVs use current discount factors and remaining interest payments; neither leg includes repayment of notional.
Value of a long forward during its life (with income and costs)
Vt(T)=St−PVt(I)+PVt(C)−F0(T)(1+r)−(T−t)V_t(T)=S_t-PV_t(I)+PV_t(C)-F_0(T)(1+r)^{-(T-t)}
StS_t = spot price at t; PVt(I)PV_t(I), PVt(C)PV_t(C) = present values at t of remaining income and carrying costs; F0(T)F_0(T) = forward price set at initiation; r = risk-free rate; T−tT-t = time remaining. The short's value is the negative.
FRA cash settlement to the fixed-rate payer
SettlementA=(MRRB−A−IFRA,B−A)×Notional×Period1+MRRB−A×Period\text{Settlement}_A=\frac{(MRR_{B-A}-IFR_{A,B-A})\times\text{Notional}\times\text{Period}}{1+MRR_{B-A}\times\text{Period}}
MRRB−AMRR_{B-A} = reference rate set at A for the B - A period, IFR = FRA fixed rate, Period = accrual fraction. Paid at A, when the period starts; positive means the fixed payer receives.
Put-call forward parity (forward contract in place of the underlying)
F0(T)(1+r)−T+p0=c0+X(1+r)−TF_0(T)(1+r)^{-T}+p_0=c_0+X(1+r)^{-T}
F0(T)F_0(T) = forward price for expiration T; p0p_0, c0c_0 = European put and call prices with the same exercise price X and expiration T; r = risk-free rate. A long forward plus a bond paying F0(T)F_0(T) replaces the asset.
Risk-neutral probability (one-period binomial model)
π=1+r−RdRu−Rd\pi=\frac{1+r-R^d}{R^u-R^d}
π\pi = risk-neutral probability of an up move (1−π1-\pi for a down move), r = risk-free rate for the period, Ru=S1u/S0R^u=S_1^u/S_0 = up gross return, Rd=S1d/S0R^d=S_1^d/S_0 = down gross return.
Alternative Investments 7 items
NAV per share
NAV=Total assets−Total liabilitiesShares outstandingNAV = \frac{\text{Total assets} - \text{Total liabilities}}{\text{Shares outstanding}}
Used for mutual funds, ETFs, private equity fund valuation
Incentive fee with a hard vs. soft hurdle
Hard hurdle: fee =p×max⁡(0,r−h)= p \times \max(0, r - h) per unit of capital, charged only on the return above the hurdle.
Soft hurdle: once r>hr > h, fee =p×r= p \times r, charged on the entire gain. Below the hurdle, neither structure pays.
Hedge fund fee structure (2-and-20)
Mgmt fee =m×AUM=m\times AUM (e.g. 2%). Incentive fee =p×max⁡(0,Fee-eligible profit)=p\times\max(0,\text{Fee-eligible profit}) (e.g. 20%), where eligible profit reflects any agreed hurdle, high-water mark, and management-fee treatment.
Net investor return equals gross return less both fees.
High-water mark (HWM) incentive fee
With no hurdle and incentive fees net of management fees: Incentive fee=pmax⁡(0,NAVgross−Management fee−HWM)\text{Incentive fee}=p\max(0,\text{NAV}_{gross}-\text{Management fee}-\text{HWM}). HWM: highest prior calculation-date NAV net of all fees, starting at initial NAV. After losses, incentive fees resume above HWM, avoiding repeat fees on gains.
Loan-to-Value (LTV)
LTV=Loan amountProperty valueLTV = \frac{\text{Loan amount}}{\text{Property value}}
Higher LTV = more leverage and credit risk. Typical max ≈ 80% commercial; 95%+ residential with mortgage insurance.
Management fee base: hedge funds vs. private capital
Hedge funds: fee =m×AUM= m \times \text{AUM} (net asset value, marked to market).
Private capital funds: fee =m×Committed capital= m \times \text{Committed capital} during the investment period (often shifting to invested capital afterward), so fees accrue even before capital is deployed.
Multiple of invested capital (MOIC)
MOIC=Realized value+Unrealized valueTotal invested capitalMOIC=\frac{\text{Realized value}+\text{Unrealized value}}{\text{Total invested capital}}
Realized value = value of exited investments; Unrealized value = residual value of assets still held; Total invested capital = paid-in capital less management fees and fund expenses. Ignores cash flow timing, unlike IRR.
Portfolio Management 8 items
Capital Market Line (CML)
E(Rp)=Rf+E(Rm)−Rfσm⋅σpE(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m} \cdot \sigma_p
Sharpe ratio of market is slope; uses total risk σp\sigma_p (not beta)
CAPM / Security Market Line (SML)
E(Ri)=Rf+βi[E(Rm)−Rf]E(R_i) = R_f + \beta_i [E(R_m) - R_f]
βi=Cov(Ri,Rm)σm2\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2}
Uses systematic risk only; SML plots expected return vs beta
Two-asset portfolio variance
σp2=w12σ12+w22σ22+2w1w2σ1σ2ρ12\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_1 \sigma_2 \rho_{12}
w = weights, σ\sigma = std devs, ρ12\rho_{12} = correlation coefficient
Information ratio
IRi=αiσeiIR_i=\frac{\alpha_i}{\sigma_{e_i}}, where αi=E(Ri)−[Rf+βi(E(Rm)−Rf)]\alpha_i=E(R_i)-[R_f+\beta_i(E(R_m)-R_f)] and σei\sigma_{e_i} is the standard deviation of the asset’s nonsystematic return.
Treynor ratio
Tp=Rp−RfβpT_p = \frac{R_p - R_f}{\beta_p}
Excess return per unit of systematic risk (beta)
Compare with Sharpe (uses total risk σp\sigma_p)
Jensen's alpha
αp=Rp−[Rf+βp(Rm−Rf)]\alpha_p = R_p - [R_f + \beta_p(R_m - R_f)]
Actual return minus CAPM-expected return
α>0\alpha > 0 means manager added value beyond compensation for risk
M-squared (M²) performance measure
M2=Rf+(Rp−Rf)⋅σmσpM^2=R_f+(R_p-R_f)\cdot\frac{\sigma_m}{\sigma_p}
RpR_p = portfolio return, RfR_f = risk-free rate, RmR_m = market return, σp\sigma_p = portfolio standard deviation, σm\sigma_m = market standard deviation
Beta from correlation and standard deviations
βi=ρi,m⋅σiσm=Cov(Ri,Rm)σm2\beta_i = \rho_{i,m} \cdot \frac{\sigma_i}{\sigma_m} = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2}
ρ = correlation with market, σi = asset σ, σm = market σ, Cov = covariance
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