ALL FORMULAS BY MODULE

Duplicates are intentional to show frequency of formulas across practice questions vs mocks.

Asset Allocation

1. CME I

• Taylor rule (optimal short-term real rate form used in source):
  $$R_{\text{optimal, real}} = R_{\text{neutral}} + 0.5 \times GDP_{g,\text{forecast}} - GDP_{g,\text{trend}} + 0.5 \times (I_{\text{forecast}} - I_{\text{target}})$$

• Taylor rule (target nominal policy rate form):
  $i^{*} = r_{\text{neutral}} + \pi_{e} + 0.5(\hat{Y}{e} - \hat{Y}{\text{trend}}) + 0.5(\pi_{e} - \pi_{\text{target}})$

• National income identity linking net exports:
  $(X - M) = $S - I$ + (T - G)$

[FORMULAS IN MOCKS]

• Taylor rule (target real short-term policy rate):
  $r^* = r_{\text{neutral}} + 0.5(Y_e - Y_{\text{trend}}) + 0.5(\pi_e - \pi_{\text{target}})$
  (Nominal target rate adds expected inflation; $Y_e$ is expected real GDP growth = nominal GDP growth − expected inflation.)

• Long-run real GDP growth:
  $\text{Real GDP growth} = \text{growth from labor productivity} + \text{growth from labor inputs}$

• Debt-to-GDP ratio:
  $\dfrac{\text{Short-term external debt} + \text{Medium- and long-term external debt}}{\text{Nominal GDP}}$

• Foreign-exchange reserves / short-term debt:
  $\dfrac{\text{FX reserves}}{\text{Short-term external debt}}$

2. CME - II

• Grinold–Kroner expected equity return:
  $E[R_e] \approx D/P + $%\Delta E - %\Delta S$ + %\Delta P/E$
  where $%\Delta E$ (nominal earnings growth) = real GDP growth + long-term inflation + corporate earnings growth premium; $%\Delta S$ = change in shares outstanding (negative for buy-backs).

• CAPM beta:
  $\beta_i = \frac{\operatorname{Cov}(R_i,R_M)}{\operatorname{Var}(R_M)}$

• Singer–Terhaar risk premium (fully integrated):
  $RP_i = \sigma_i \times \rho_{i,M} \times (\text{Sharpe ratio of GIM})$

• Singer–Terhaar risk premium (fully segmented):
  $RP_i = \sigma_i \times (\text{Sharpe ratio})$

• Real-estate expected return:
  $E(R_{re}) = \text{CapRate} + \text{NOI growth rate} - %\Delta\text{CapRate}$
  (NOI growth includes real growth + inflation; a decline in cap rate contributes positively).

• Number of distinct covariance elements in an $N$-asset VCV matrix:
  $\frac{N(N-1)}{2}$

[FORMULAS IN MOCKS]

• Grinold–Kroner expected equity return:
  Expected income return = Dividend yield − Change in shares outstanding
  Expected nominal earnings growth return = Real earnings growth + Inflation
  Expected repricing return = Rate of P/E ratio expansion
  $E(R) =$ income return + nominal earnings growth return + repricing return
  (Note: net share repurchases imply negative change in shares outstanding, which increases income return.)

• Commercial real estate expected return (finite horizon):
  $E(R_e) =$ Current capitalization rate + NOI growth rate − %Δ Cap rate
  where %Δ Cap rate = (Expected cap rate − Current cap rate) / Current cap rate
  (NOI growth rate is typically real NOI growth + inflation.)

• Emerging-market vulnerability ratios:
  Debt-to-GDP = (Short-term external debt + Medium- and long-term external debt) / Nominal GDP
  Foreign-exchange reserves / Short-term debt

• Unsmoothed variance of real-estate returns:
  $\operatorname{Var}(r) = \left[ \frac{1+\lambda}{1-\lambda} \right] \times \operatorname{Var}(R)$
  (Standard deviation is the square root.)

• Singer–Terhaar equity risk premium:
  $\text{Segmented RP} = \sigma \times \text{SR}{\text{segmented}}$
  $\text{Fully integrated RP} = \sigma \times \rho{\text{global}} \times \text{SR}_{\text{global}}$
  $\text{Weighted RP} = \text{Segmented RP} \times $1 - \text{degree of integration}$ + \text{Fully integrated RP} \times \text{degree of integration}$

3. Overview of AA

Economic net worth: $$ \text{Economic Net Worth} = \text{Total Economic Assets} - \text{Total Economic Liabilities} $$ where Total Economic Assets = Financial assets + Extended assets (e.g., human capital, PV of pledged contributions); Total Economic Liabilities = Financial liabilities + Extended liabilities (e.g., PV of future obligations, consumption needs, specific goals).

Comparative performance ratios (from allocation evaluation): $$ \text{Ratio 1} = \frac{(k - RF)}{\sigma} $$ $$ \text{Ratio 2} = \frac{(k - k_T)}{\sigma} $$ $$ \text{Ratio 3} = \frac{(k - RF)}{\beta} $$ where $k$ = expected annual return, $RF$ = risk-free return, $k_T$ = target return, $\sigma$ = annual volatility, $\beta$ = CAPM beta.

Sharpe ratio (used for ranking allocations): $$ \text{Sharpe ratio} = \frac{\text{Expected return} - \text{Risk-free rate}}{\text{Expected volatility}} $$

[FORMULAS IN MOCKS]

• Economic net worth = Total economic assets − Total economic liabilities
  (Economic assets include financial assets + human capital/PV of earnings + residence + other extended assets; liabilities include mortgage + PV of future consumption + other extended liabilities.)

4. Principles of AA

Safety-first / probability ratio (for ranking portfolios against a required threshold return $R_L$): $$ \frac{E(R_P) - R_L}{\sigma_P} $$

Utility / objective function in mean-variance optimization (returns and variance in percent form): $$ U_m = E(R_{s,m}) - 0.005 \lambda \sigma^2(R_{s,m}) $$ $or equivalently $E(R) - 0.005\lambda\sigma^2$$

Marginal contribution to total risk $MCTR$: $$ \text{MCTR}i = \beta{i,p} \times \sigma_p $$

Absolute contribution to total risk $ACTR$: $$ \text{ACTR}_i = w_i \times \text{MCTR}_i $$

For an optimal portfolio the ratio of excess return to MCTR equals the portfolio Sharpe ratio: $$ \frac{E(R_p) - R_f}{\text{MCTR}_i} = \frac{E(R_p) - R_f}{\sigma_p} $$

Funding ratio: $$ \text{Funding ratio} = \frac{\text{Market value of assets}}{\text{Present value of liabilities}} $$

Reverse-optimization / CAPM equilibrium return: $$ E(R_i) = R_f + \beta_i \times \text{Market risk premium} $$

Inflation-adjusted annual cash-flow (growing-annuity form used for living-expenditure goals): $$ \text{Cash flow} = \text{Capital} \times \frac{k - g}{1 - \left(\frac{1+g}{1+k}\right)^n} \times \frac{1}{1+k} $$ (where $k$ = required success-rate return, $g$ = inflation)

Present-value funding of a future goal: $$ \text{PV} = \frac{\text{FV}}{(1 + k)^n} $$ (where $k$ is the minimum expected return consistent with the required success probability)

[FORMULAS IN MOCKS]

• Mean-variance utility: $U_m = E(R_m) - 0.005 \lambda \sigma_m^2$

• Corner-portfolio weights for target return:
  Target return = $w \times R_3 + $1-w$ \times R_4$
  Solve for $w$.

• Leverage to achieve target return (borrowing at risk-free rate):
  Target return = $w \times R_{\text{tangency}} + $1-w$ \times R_f$
  $w =$ (Target − $R_f$)/( $R_{\text{tangency}}$ − $R_f$)
  (When $w > 1$, the cash weight is negative = leverage.)

• Condition for mean-variance improvement from adding a new asset class:
  Sharpe ratio of new asset > Sharpe ratio of current portfolio × correlation between them.

• Foundation/endowment nominal return objective (spending + inflation + costs):
  $(1 + \text{spending rate})(1 + \text{expected inflation})(1 + \text{investment costs}) - 1$

• Sharpe ratio of a corner portfolio:
  $\text{Sharpe} = \dfrac{E(r) - r_f}{\sigma}$

• Weight of the tangency (corner) portfolio when combined with the risk-free asset to achieve a target return:
  $r_{\text{target}} = E(r_P) \times w + r_f \times (1-w)$
  $\Rightarrow w = \dfrac{r_{\text{target}} - r_f}{E(r_P) - r_f}$

• Resulting portfolio standard deviation $risk-free asset has $\sigma=0$$:
  $\sigma_{\text{port}} = w \times \sigma_P$

• Ratio of excess return to marginal contribution to total risk (optimality condition for risk budgeting):
  $\dfrac{E(r_i) - r_f}{\text{MCTR}_i} = \text{Sharpe of tangency portfolio}$

• Ratio of excess return to MCTR (solving for $r_f$):
  $\dfrac{E(r) - r_f}{\text{MCTR}} = \text{given ratio}$
  $\Rightarrow r_f = E(r) - (\text{ratio} \times \text{MCTR})$

• Present value of a goals-based multi-period cash-flow stream (end-of-year payments):
  $\text{PV} = \sum_{t=1}^{N} \dfrac{\text{PMT}}{(1 + r_{\min})^t}$
  $or financial-calculator inputs: $N$, $I/Y = r_{\min}$, $\text{PMT}$, $\text{FV}=0$, solve $\text{PV}$$

5. Asset Alloc w/RW Constraints

Tactical incremental return contribution of an asset class:

$(TAA weight−Policy weight)×Period return$

After-tax rebalancing range: $$ \text{After-tax rebalancing range} = \frac{\text{Pre-tax rebalancing range}}{1 - t} $$

After-tax return on an asset (when gains are taxed at rate $t$): $$ \text{After-tax return} = \text{Pre-tax return} \times (1 - t) $$

Portfolio Construction

Overview of Equity

• Explicit trading costs = Brokerage commissions + Transaction taxes + Exchange fees + Regulatory fees
  (expressed in percent or basis points converted to percent)

• Herfindahl–Hirschman Index $HHI$ = $\sum w_i^2$
  where $w_i$ are the portfolio weights (including the sum of squared weights of remaining holdings)

• Effective number of stocks = $\dfrac{1}{\text{HHI}}$

Overview of Fixed Income

• Coupon income = $\dfrac{\text{Annual average coupon payment}}{\text{Current average bond price}}$

• Rolldown return = $\dfrac{\text{Expected average bond price at end of horizon} - \text{Current average bond price}}{\text{Current average bond price}}$
  (assumes unchanged yield curve over the strategy horizon)

• Rolling yield = Coupon income + Rolldown return

• Expected change in price based on yield or spread view ≈ $(- \text{Modified Duration} \times \Delta y) + \left[\frac{1}{2} \times \text{Convexity} \times (\Delta y)^{2}\right]$

• Leveraged portfolio return = $\dfrac{(\text{Return on invested funds} \times \text{Value of portfolio assets}) - (\text{Borrowing rate} \times \text{Value of borrowed funds})}{\text{Value of portfolio equity}}$

• Total expected return = Rolling yield $+/-$ $E$(Change in price based on investor’s benchmark yield view) $+/-$ $E$(Change in price due to investor’s view of credit spread) $+/-$ $E$(Currency gains or losses)

• Level of inflation protection for coupons = Percentage of portfolio in floating-coupon bonds + Percentage of portfolio in inflation-linked bonds

[FORMULAS IN MOCKS]

• Rolling yield = Coupon income + Rolldown return
  Coupon income = Annual coupon payment / Current bond price
  Rolldown return = (Expected price in one year assuming unchanged yield curve − Current price) / Current price

• Expected return due to change in benchmark YTM:
  $(- \text{ModDur} \times \Delta \text{Yield}) + [0.5 \times \text{Convexity} \times (\Delta \text{Yield})^2]$

• Expected return due to change in spread:
  $(- \text{ModDur} \times \Delta \text{Spread}) + [0.5 \times \text{Convexity} \times (\Delta \text{Spread})^2]$

• Leveraged return:
  $\dfrac{r_I \times V_E + V_B - (V_B \times r_B)}{V_E}$

• Rolling yield:
  $\text{Rolling yield} = \text{Coupon income} + \text{Rolldown return}$
  $\text{Coupon income} = \frac{\text{Annual coupon payment}}{\text{Current bond price}}$
  $\text{Rolldown return} = \frac{P_1 - P_0}{P_0}$ (assuming unchanged yield curve)

• Expected return due to change in benchmark YTM or spread:
  $(-\text{ModDur} \times \Delta\text{Yield}) + \left[ 0.5 \times \text{Convexity} \times (\Delta\text{Yield})^2 \right]$

• Leveraged return:
  $r_L = r_1 + \frac{V_B}{V_E} (r_1 - r_B)$

Alternatives

• Expected distribution = $[\text{Prior-year NAV} \times (1 + \text{Growth rate})] \times \text{Distribution rate}$

• Expected NAV = $\text{Prior-year NAV} \times $1 + \text{Growth rate}$ + \text{Capital contributions} - \text{Distributions}$

• $\text{NAV}t = [\text{NAV}{t-1} \times (1 + G)] + C_t - D_t$
  where $G$ is the growth rate, $C_t$ is the capital contribution, and $D_t$ is the distributions

Private Wealth

• After-tax distribution income (annual yield) = $B \left{ [1 + r(1 - t_x)]^T - 1 \right}$
  (where $B$ is beginning portfolio value/basis, $r$ is pretax yield, $t_x$ is tax rate on distributions, $T=1$ for annual)

• Real purchasing power of fixed nominal income = $\dfrac{\text{Nominal income}}{(1 + \pi)^T}$

• Future value interest factor adjusted for tax and inflation (capital gains with deferred tax treated as $t=0$) = $\left[ \dfrac{1 + r}{1 + \pi} \right]^T$

• Average tax rate = $\dfrac{\text{Total tax liability}}{\text{Total taxable income}}$

• After-tax nominal return = $r(1 - t)$

• After-tax real return = $\dfrac{1 + r(1 - t)}{1 + \pi} - 1$
  (or approximately after-tax nominal return minus inflation)

• Inflation-adjusted future value (pretax) = $\text{Notional} \times \left[ \dfrac{1 + R_P}{1 + i} \right]^T$

• Inflation-adjusted after-tax future value = $\text{Notional} \times \left[ \dfrac{1 + R_P(1 - \text{Tax rate})}{1 + i} \right]^T$

• Investable net worth = Liquid investments + Cash – Short-term liabilities (excludes primary residence, vehicles, business assets/liabilities, checking for day-to-day use)

[FORMULAS IN MOCKS]

• After-tax future value of non-dividend-paying equities (capital-gains only): $FV_{cg} = PV \bigl[(1 + r)^n $1 - t_{cg}$ + t_{cg}\bigr]$ .

• Accrual-equivalent after-tax return $R_{AE}$ solves $PV (1 + R_{AE})^n = FV$. Accrual-equivalent tax rate $T_{AE}$ solves $r(1 - T_{AE}) = R_{AE}$.

• Real after-tax required return = (Cash need / Asset base).
  Nominal after-tax required return ≈ Real after-tax return + Inflation (arithmetic) or $(1 + real)(1 + inflation) - 1$ (geometric).

• Shortfall risk ≈ Nominal pre-tax expected return − 2 × Nominal expected standard deviation. $Q68, Q96$

• Real after-tax return of a portfolio = Nominal pre-tax return × (1 − tax rate) − Inflation. $Q68, Q96$

• Minimum portfolio size that meets next-year net cash need at real after-tax return $r$: $\text{min Portfolio} = \text{net cash need} / r$.
  Maximum donation = Current portfolio − min Portfolio.

• Accrual-equivalent after-tax annual return: $\left( \frac{\text{Ending Value}}{\text{Beginning Value}} \right)^{1/n} - 1$

• Real after-tax portfolio return: Nominal pre-tax return $\times $1 - \text{tax rate}$ - \text{inflation}$

• Shortfall risk: Nominal pre-tax expected return $- 2 \times$ standard deviation

• Minimum portfolio size that meets next-year net cash need: $\text{Net cash need} / \text{real after-tax expected return}$

• Maximum immediate donation: Current portfolio $-$ minimum portfolio

• Worst-case (downside) return = Expected nominal total return − 2 × standard deviation

• After-tax standard deviation = $\sigma \times (1 - TR)$ where $TR$ is the tax rate

• Taxable-account terminal value = $BV \times [1 + r(1 - t)]^n$

• Tax-deferred terminal value (after tax on gain) = $BV \times (1 + r)^n \times 1 - t + BV \times t$ (or equivalent net-of-tax gain + basis)

• Maximum spending rate that preserves real portfolio value:

• Expected nominal return that preserves real value of portfolio = (1 + spending rate) × (1 + inflation) × (1 + management fee) – 1 Multiplicative: $\text{spending} = \frac{1 + r}{(1 + i)(1 + f)} - 1$
  Additive approximation: $\text{spending} \approx r - i - f$
  where $r$ = expected nominal return, $i$ = inflation, $f$ = management fee

• After-tax expected return on a taxable asset = Pre-tax expected return $\times (1 - t)$

• After-tax risk of a taxable asset = Pre-tax standard deviation $\times (1 - t)$ $Q163, Q164$

• After-tax future value (accrual taxes):
  $FV_t = \left[ 1 + R(1 - t_x) \right]^T$

• Tax drag (as % of investment gains):
  $\text{Tax drag} = \frac{FV_{\text{no tax}} - FV_{\text{tax}}}{FV_{\text{no tax}} - \text{Initial value}}$

Institutional

• Portfolio expected return = $\sum (w_i \times r_i)$

• Modified duration of bank equity capital after restructuring:
  $D_E = \left(\dfrac{A}{E}\right) D_A - \left(\dfrac{L}{E}\right) D_L \times \left(\dfrac{\Delta y_L}{\Delta y_A}\right)$
  where $A$ = assets, $E$ = equity, $L$ = liabilities, $D_A$ = modified duration of assets, $D_L$ = modified duration of liabilities, and $\Delta y_L / \Delta y_A$ is the relative yield change of liabilities to assets.

[FORMULAS IN MOCKS]

• Modified duration of equity capital:
  $D_E^* = \left( \frac{A}{E} \right) D_A^* - \left( \frac{A}{E} - 1 \right) D_L^* \left( \frac{\Delta i}{\Delta y} \right)$
  where $\frac{A}{E}$ is the assets-to-equity ratio, $D_A^$ modified duration of assets, $D_L^$ modified duration of liabilities, and $\frac{\Delta i}{\Delta y}$ is the change in liability yield per unit change in asset yield.

Trading

• Inside (market) bid–ask spread = Highest bid price − Lowest ask price

• Effective bid–ask spread (buy order) = $2 \times \bigl(\text{Trade price} - \dfrac{\text{Ask price} + \text{Bid price}}{2}\bigr)$
  or equivalently $2 \times (\text{Trade price} - \text{Midpoint of the market at order entry})$

• Average effective spread = $\dfrac{\sum \text{Effective spreads of individual trades}}{N}$

• Volume-weighted average execution price for a multi-level fill against the book = $\dfrac{\sum (\text{Size}_i \times \text{Price}_i)}{\text{Total shares}}$

• Effective-spread transaction cost per share (midquote benchmark) = $\dfrac{\text{Trade price} - \text{Prevailing bid}}{2}$ (for buys)
  Total transaction cost estimate = Effective-spread cost per share $\times$ Total shares traded

[FORMULAS IN MOCKS]

• Effective spread (sell order): $2 \times (\text{Midpoint of market at order entry} - \text{Trade price})$

• VWAP transaction cost estimate: $(\text{Day VWAP} - \text{Trade VWAP}) \times \text{Shares sold}$

• Effective spread (buy order):
  $2 \times \left[ \text{Trade price} - \frac{\text{Bid} + \text{Ask}}{2} \right]$

Performance

Portfolio Performance Evaluation

Brinson-Hood-Beebower $BHB$ selection effect:
$S_i = W_i \times (R_i - B_i)$

BHB allocation effect:
$A_i = $w_i - W_i$ \times B_i$

BHB interaction effect:
$I_i = $w_i - W_i$ \times (R_i - B_i)$

Brinson-Fachler allocation effect:
$A_i = $w_i - W_i$ \times (B_i - B)$

where $w_i$ = portfolio weight in sector $i$, $W_i$ = benchmark weight in sector $i$, $R_i$ = portfolio return in sector $i$, $B_i$ = benchmark return in sector $i$, $B$ = overall benchmark return.

Combined selection + interaction:
$W_i(R_i - B_i) + (w_i - W_i)(R_i - B_i)$

Sortino ratio:
$SR_D = \frac{r_p - r_T}{\sigma_D}$
$$\sigma_D$ = target semi-deviation below target return $r_T$$

Sharpe ratio:
$SR = \frac{r_p - r_f}{\sigma}$

Misfit active return (definition): difference between broad market benchmark return and manager’s normal portfolio return.

[FORMULAS IN MOCKS]

• Brinson–Hood–Beebower selection effect: $W_b \times (R_p - R_b)$

• Brinson–Fachler allocation effect: $(W_p - W_b) \times (R_b - B)$

• Interaction effect: $(W_p - W_b) \times (R_p - R_b)$

• Sortino ratio: $\dfrac{\text{Average fund return} - \text{Minimum acceptable return}}{\text{Target semi-standard deviation}}$

• Implementation shortfall $IS$: Execution cost + Opportunity cost + Fees
  Execution cost = $\sum s_j p_j - $\sum s_j$ p_d$
  Opportunity cost = $(S - \sum s_j)(p_n - p_d)$
  Fees = shares executed × commission per share

• Factor contribution to active return:
  $\text{Contribution} = $\text{Portfolio sensitivity} - \text{Benchmark sensitivity}$ \times \text{Factor return}$

• Active return:
  $A = P - B$

• Style return:
  $S = B - M$

• Excess return:
  $E = P - M$

Investment Manager Selection

Investment management fee (base + sharing, subject to cap):

$\text{Fee} = \text{Base Fee} + [\text{Sharing Fee} \times (\text{Gross Active Return} - \text{Base Fee})]$

Capped at Maximum Annual Fee if exceeded.

Net active return:

$\text{Net Active Return} = \text{Gross Active Return} - \text{Billed Fee}$

where Billed Fee = Base + (active return above base × sharing %).

High-water mark adjustment: Prior losses must be recouped before sharing fee applies on subsequent gains (e.g., adjust gross return downward by prior loss amount before calculating sharing portion).

Capture ratio:

$\text{Capture Ratio} = \frac{\text{Upside Capture}}{\text{Downside Capture}}$

Modified Dietz sub-period return (with external cash flows):

$r = \frac{EMV - BMV - CF}{BMV + \sum (CF_i \times w_i)}$

where wi w_i wi​ is the weighting factor (proportion of period cash flow was invested). Sub-period returns are geometrically linked.

[FORMULAS IN MOCKS]

• Capture ratio: $\dfrac{\text{Upside capture}}{\text{Downside capture}}$

• Performance-based fee (sharing structure):
  Base fee + (Sharing percentage × (Gross active return − Base fee))
  Subject to maximum annual fee cap

• Capture ratio:
  $\text{Capture ratio} = \dfrac{\text{Upside capture}}{\text{Downside capture}}$

Overview of the Global Investment Performance Standards

Modified Dietz sub-period return with external cash flow:
$$ $r = \frac{EMV - BMV - CF}{BMV + \sum (CF_i \times w_i)}$ $$
where $w_i$ is the weighting factor (proportion of the period the cash flow was in the portfolio). Sub-period returns are geometrically linked to obtain the full-period return.

Composite return (asset-weighted):
$$ $r_{comp} = \sum (w_p \times r_p)$ $$
where $w_p$ is the weight of each portfolio based on beginning assets (or beginning assets plus weighted cash flows).

Derivatives

Option Strategies

• Collar position delta:
  $$\text{Collar delta}=\Delta_{\text{stock}}+\Delta_{\text{put}}-\Delta_{\text{call}}$$

• Synthetic long stock (put-call parity):
  $$S_0=c_0-p_0+\frac{X}{(1+r)^T}$$

• Synthetic short stock:
  $$-S_0=p_0-c_0-\frac{X}{(1+r)^T}$$

• Protective-put maximum loss:
  $$\text{Max loss}=S_0-X+p_0$$

• Protective-put breakeven:
  $$\text{Breakeven}=S_0+p_0$$

• Bull-call-spread maximum profit:
  $$\text{Max profit}=(X_H-X_L)-(c_L-c_H)$$

• Bull-call-spread breakeven:
  $$\text{Breakeven}=X_L+(c_L-c_H)$$

• Bear-put-spread net cost / maximum loss:
  $$\text{Net cost}=p_H-p_L$$
  Maximum loss occurs when underlying $\ge X_H$.

• Straddle / strangle cost and breakevens:
  $$\text{Cost}=c+p$$
  Long-straddle breakevens $=X\pm\text{cost}$
  Long-strangle upper breakeven $=X_{\text{call}}+\text{cost}$

• Covered-call effective sale price:
  $$\text{Effective sale price}=X+c_0$$
  Breakeven $=S_0-c_0$

• Covered-call position delta:
  $$\text{Position delta}=(1-\Delta_{\text{call}})\times\text{shares}$$

• Variance of option gamma: highest for ATM options near expiration.

[FORMULAS IN MOCKS]

• Collar delta: $\text{Collar delta} = \Delta_{\text{underlying}} + \Delta_{\text{put}} - \Delta_{\text{call}}$

• Straddle cost (per share): ATM put premium + ATM call premium

• Strangle cost (per share): OTM put premium + OTM call premium
  Percentage increase to break even: $\frac{\text{Strangle cost}}{\text{Current share price}}$

• Short seagull spread cost (per share): ATM put premium − OTM put premium − OTM call premium

• Long straddle maximum loss = total premium paid
  Downside breakeven = strike − total premium
  Upside breakeven = strike + total premium

• Put-spread net cost: long-put premium − short-put premium

• Bull-call-spread maximum gain = difference in strikes − net premium paid
  Breakeven = lower strike + net premium paid

• Covered-call gain (stock + short OTM call, stock finishes below strike):
  $\text{Gain per share} = $S_T - S_0$ + \text{Call premium received}$

• Combined position delta (long stock + short option):
  $\Delta_{\text{position}} = 1 + \Delta_{\text{option}}$
  $for a short put: $1 - (\text{negative put delta}) = 1 + |\Delta_{\text{put}}|$$

• Bull put spread (short higher-strike put + long lower-strike put) breakeven:
  $\text{Breakeven} = \text{Higher strike} - \text{Net premium received}$

• Short seagull spread net cost (per share):
  $\text{Net cost} = \text{ATM put premium} - \text{OTM put premium} - \text{OTM call premium}$

• Long straddle maximum loss:
  $\text{Max loss} = \text{Call premium} + \text{Put premium}$
  Upside breakeven $= \text{Strike} + \text{Total premium}$
  Downside breakeven $= \text{Strike} - \text{Total premium}$

Swaps, Forwards, and Futures Strategies

• Basis-point-value hedge ratio (bond futures):
  $$BPVHR=\left{\frac{BPV_T-BPV_P}{BPV_{CTD}}\right}\times CF$$
  where $$BPV_T=MDUR_T\times0.0001\times MV_P$$ and $$BPV_P=MDUR_P\times0.0001\times MV_P$$ (or given directly).

• Equity-index futures contracts for beta adjustment or equitization:
  $$N_f=\left(\frac{\beta_T-\beta_S}{\beta_f}\right)\times\frac{S}{F}$$
  where $F=$ futures price $\times$ multiplier. Positive $N_f$ = buy; negative = sell.

• Interest-rate-swap notional to change portfolio modified duration:
  $$(MV_P\times MDUR_P)+(N_S\times MDUR_S)=MV_P\times MDUR_T$$
  Solve for $N_S$.

• Variance-swap settlement:
  Variance notional $=\dfrac{\text{Vega notional}}{2\times\text{Volatility strike}}$
  Settlement $=\text{Variance notional}\times(\text{Realized variance}-\text{Variance strike})$
  where Variance $=(\text{Volatility})^2$.

• Fed-funds-futures implied rate and rate-change probability:
  Implied FFE rate $=100-\text{futures price}$
  Probability of a 25 bp move $=\dfrac{\text{Implied FFE}-\text{Current midpoint}}{\text{New midpoint after move}-\text{Current midpoint}}$.

• Equity-swap net payment (cash-settled):
  Net cash $=\text{Notional}\times(\text{Floating-rate return}-\text{Equity total return})$.

• CTD selection (invoice price minus dirty price cost):
  Amount received $=(\text{Futures settlement}/100)\times CF\times\text{Contract size}$
  Bond cost $=(\text{Dirty price}/100)\times\text{Contract size}$.
  Lowest (most negative) difference = cheapest-to-deliver.

[FORMULAS IN MOCKS]

• Interest-rate swap notional to change modified duration:
  $NS = \left( \frac{\text{MDUR}_T - \text{MDUR}_P}{\text{MDUR}_S} \right) \times MVP$
  where $\text{MDUR}_S$ is typically negative for a pay-fixed swap

• Variance notional: $\text{Variance notional} = \frac{\text{Vega notional}}{2 \times \text{Strike}}$
  Value of variance swap at time $t$:
  $\text{VarSwap}_t = \text{Variance notional} \times PV_t(T) \times \Bigl{ \frac{t}{T} [\text{RealizedVol}(0,t)]^2 + \frac{T-t}{T} [\text{ImpliedVol}(t,T)]^2 - \text{Strike}^2 \Bigr}$
  $PV_t(T) = \frac{1}{1 + r \times (T-t)}$

• Equity-index futures contracts to change beta:
  $N_f = \left( \frac{\beta_T - \beta_S}{\beta_f} \right) \times \left( \frac{S}{F} \right)$
  where $F =$ futures price $\times$ multiplier

• Bond futures hedge ratio (BPV method):
  $\text{BPV}_{HR} = \left( \frac{\text{BPV}_T - \text{BPV}P}{\text{BPV}{\text{CTD}}} \right) \times \text{Conversion factor}$
  $\text{BPV}_T = \text{MDUR}_T \times 0.0001 \times MVP$

• Fed-funds futures implied average rate:
  $\text{Implied rate} = 100 - \text{Futures price}$

• Probability of a rate hike:
  $\dfrac{\text{Implied average rate} - \text{Current midpoint}}{\text{Hike midpoint} - \text{Current midpoint}}$

• Basis-point-value hedge ratio (duration adjustment):
  $\text{BPVHR} = \dfrac{\text{BPV}{\text{target}} - \text{BPV}{\text{current}}}{\text{BPV}_{\text{CTD}}} \times \text{CF}$
  $\text{BPV} = \text{Modified duration} \times 0.0001 \times \text{Portfolio value}$

Currency

• Domestic-currency return on a foreign asset/portfolio:
  $$R_{DC}=(1+R_{FC})(1+R_{FX})-1$$
  where $R_{FX}$ is the percentage change of the foreign currency against the domestic currency (domestic currency is the price currency in the P/B quote).

• Multi-asset domestic-currency return: #sunday $$R_{DC}=w_1(1+R_{FC,1})(1+R_{FX,1})+w_2(1+R_{FC,2})(1+R_{FX,2})-1$$

• Domestic-currency return variance approximation:
  $$\sigma^2(R_{DC})\approx\sigma^2(R_{FC})+\sigma^2(R_{FX})+2\sigma(R_{FC})\sigma(R_{FX})\rho(R_{FC},R_{FX})$$

• Minimum-variance hedge ratio:
  $$h=\rho(R_{DC};R_{FX})\times\frac{\sigma(R_{DC})}{\sigma(R_{FX})}$$

• Carry-trade profit (interest differential):
  $$\text{Profit}=\text{high-yield currency rate}-\text{low-yield currency rate}$$

• Bull-call-spread net premium, maximum gain and breakeven:
  $$\text{Net premium}= \text{premium paid on long call}-\text{premium received on short call}$$
  $$\text{Maximum gain}=(K_{\text{high}}-K_{\text{low}})-\text{net premium}$$
  $$\text{Breakeven}=K_{\text{low}}+\text{net premium}$$

• One-year exchange-rate return from forward:
  $$\text{Exchange-rate return}=\frac{\text{Forward rate}-\text{Spot rate}}{\text{Spot rate}}$$

• All-in forward rate (points quoted per 10 000):
  $$\text{All-in forward}=\text{spot}\pm\frac{\text{forward points}}{10,000}$$

• Discounted cash-settlement amount on early close-out of forward:
  $$\text{Present value}=\frac{\text{notional}\times(\text{close-out forward}-\text{original forward})}{1+r\times\frac{t}{360}}$$

[FORMULAS IN MOCKS]

• Minimum-variance hedge ratio:
  $h = \rho(R_{\text{DC}}; R_{\text{DC/FC}}) \times \left( \frac{\sigma(R_{\text{DC}})}{\sigma(R_{\text{DC/FC}})} \right)$
  Hedge position = $h \times$ foreign-currency asset value

• Domestic-currency return: $(1 + R_{\text{FC}})(1 + R_{\text{FX}}) - 1$

• Forward premium/discount: $\frac{F}{S} - 1$

• Unhedged domestic-currency return:
  $R_{\text{DC}} = (1 + R_{\text{FC}})(1 + R_{\text{FX}}) - 1$
  where $R_{\text{FX}} = (S_T - S_0)/S_0$

• Minimum-variance hedge ratio (currency context, if appearing): $h = \rho \times (\sigma_{\text{DC}} / \sigma_{\text{FX}})$

Portfolio Management $PMP$

Index-Based Strategies

• Herfindahl–Hirschman Index $HHI$ = sum of the squared constituent weightings
  Effective number of stocks $\approx 1/\text{HHI}$

• Sector (or factor) contribution to return = Sector (or factor) weight $\times$ Sector (or factor) return

• Active factor-weighting contribution to excess return = $\sum(\text{Portfolio weight}_i - \text{Benchmark weight}_i) \times \text{Return}_i$ (for factors whose weights differ)

• Number of equity index futures contracts needed = $\dfrac{\text{Target exposure amount}}{\text{Futures contract unit value}}$
  Contract unit value = Futures price $\times$ Multiplier

Active Equity Strategies

• Forward P/E = Price / 1-Year Forward EPS

• PEG ratio = Forward P/E / Expected EPS growth rate (in percent)

• Information coefficient (quantitative methods) relates firm returns to model factors using past data

• Factor-mimicking portfolio $FMP$ is a theoretical long/short, dollar-neutral portfolio with unit exposure to a chosen factor and zero exposure to other factors

**[FORMULAS IN MOCKS]

• • Sharpe ratio: $SR = \frac{r_p - r_f}{\sigma_p}$ where $r_p$ = portfolio return, $r_f$ = risk-free rate, $\sigma_p$ = portfolio volatility.

• Cyclically adjusted P/E $CAPE$ = real equity price index / 10-year average real earnings.

• Maximum position size = Strategy AUM × Position-size limit

• Average daily volume = Average market capitalization × Average percentage traded each day

• Daily purchase limit = Average daily volume × Daily volume limit

• Days to purchase full position = Maximum position size / Daily purchase limit

• Portfolio active return = $\sum h_{A_i} r_{A_i}$

• Portfolio active risk (uncorrelated) = $\sqrt{\sum h_{A_i}^2 \sigma_{A_i}^2}$

• Information ratio = Active return / Active risk

• PEG ratio:
  $\text{PEG} = \dfrac{P/E}{\text{Expected EPS growth rate}}$

Active Equity Portfolio Construction

• Expected active return (fundamental law of active management):
  $$E(R_A)=IC\times\sqrt{BR}\times\sigma_{RA}\times TC$$
  where $IC$ = information coefficient, $BR$ = breadth (number of independent decisions), $\sigma_{RA}$ = active risk, $TC$ = transfer coefficient.

• Active share:
  $$AS=\frac12\sum_{i=1}^N|w_{P,i}-w_{B,i}|$$

• Active return contribution:
  $$R_A=\sum_{i=1}^N(w_{P,i}-w_{B,i})R_i$$

• Contribution of an asset (or factor) to portfolio variance:
  $$CV_i=\sum_j w_i w_j C_{ij}$$

• Maximum position size under liquidity constraint:
  $$\text{Max position}=\text{ADV}\times\text{liquidity limit %}$$

Days to build a maximum position = (AUM × position-size limit) / (ADV × daily-volume limit).

[FORMULAS IN MOCKS]

• Contribution of asset $i$ to total portfolio variance: $C_i = \sum_{j=1}^n x_i x_j C_{ij}$ where $x$ = weights and $C_{ij}$ = covariance

• Portfolio duration = $\sum w_i \times D_i$; solve for target weight

• Future value of home = Current value $\times (1 + g)^n$

• Required monthly rate solved from PV, PMT, FV, n; annualized = monthly $\times 12$ or $(1+\text{monthly})^{12}-1$

• Mortgage net proceeds = Asset value $\times$ LTV

• Sale-and-leaseback net proceeds = Sale price − Tax on (Sale price − Cost basis)

Liability-Driven and Index-Based Strategies

• Number of futures contracts to close the duration / money-duration gap:
  $N_f = \dfrac{\text{BPV}_L - \text{BPV}_A}{\text{BPV}_f}$
  (positive = buy/long; negative = sell/short)

• Futures BPV = $\dfrac{\text{BPV of cheapest-to-deliver}}{\text{Conversion factor of CTD}}$

• Money duration (dollar duration / BPV) = Modified duration $\times$ Portfolio market value
  (or the change in value for a 1 bp yield change)

[FORMULAS IN MOCKS]

• Maximum spending rate that preserves real portfolio value:
  multiplicative form $spending = \frac{1 + r}{(1 + i)(1 + f)} - 1$;
  additive approximation $spending \approx r - i - f$
  where $r$ = expected nominal return, $i$ = inflation, $f$ = management fee.

• Dollar duration = Duration × Market value × 0.01.
  Rebalancing ratio = Desired dollar duration / New dollar duration.
  Cash required = (Rebalancing ratio − 1) × Market value of positions.

• Basis-point value $BPV$ = Money duration × 0.0001.
  Number of futures contracts $N_f = \frac{BPV_{liability} - BPV_{portfolio}}{BPV_{futures}}$.

• Money duration = Modified duration × Market value

• Basis-point value $BPV$ = Money duration × 0.0001

• Number of futures contracts to close duration gap: $N_f = \frac{\text{BPV}{\text{liability}} - \text{BPV}{\text{portfolio}}}{\text{BPV}_{\text{futures}}}$ $Q206, Q232, Q241$

• Futures BPV ≈ CTD BPV / Conversion factor $Q232, Q241$

• PVBP = (Market value × Duration) / 10 000

• Return requirement (arithmetic) = (Annual subscription expense / Investable base) + Management fees + Inflation components

• Number of interest-rate futures contracts = $\dfrac{(D_T - D_I) \times P_I}{D_{\text{CTD}} \times P_{\text{CTD}}} \times \text{CF}$

• Swap notional to close duration gap:
  $\text{Asset BPV} + \left(\text{NP} \times \dfrac{\text{Swap BPV}}{100}\right) = \text{Liability BPV}$
  $\Rightarrow \text{NP} = $\text{Liability BPV} - \text{Asset BPV}$ \times \dfrac{100}{\text{Swap BPV}}$

Yield Curve Strategies

• Expected total return for a zero-coupon government bond portfolio:
  $E(R) \approx %$ Rolldown return $+ E(% \Delta$ Price due to investor’s view of benchmark yield$) + E(% \Delta$ Price due to investor’s view of currency value changes$)$

• Rolldown return:
  $\text{Rolldown return} = \dfrac{PV_1 - PV_0}{PV_0}$

• Price change due to yield view:
  $E(% \Delta$ Price due to investor’s view of benchmark yield$) = $-\text{Duration} \times \Delta \text{Yield}$ + $\bigl[0.5 \times \text{Convexity} \times (\Delta \text{Yield})^2\bigr]$

• Approximate percentage change in full portfolio value:
  $%\Delta PV_{\text{Full}} \approx -(\text{ModDur} \times \Delta \text{Yield}) + \bigl[\tfrac12 \times \text{Convexity} \times (\Delta \text{Yield})^2\bigr]$

• Butterfly spread:
  $\text{Butterfly spread} = -(\text{short-term yield}) + $2 \times \text{medium-term yield}$ - (\text{long-term yield})$

• Domestic-currency return on foreign bond:
  $R_{\text{DC}} = (1 + R_{\text{FC}})(1 + R_{\text{FX}}) - 1$

• Key-rate duration contribution:
  $\text{Key rate duration} = \text{Modified duration} \times \dfrac{\text{bond position}}{\text{portfolio position}}$
  Sum of key-rate durations equals effective portfolio duration.

[FORMULAS IN MOCKS]

• Fixed-income expected return ≈ Yield income + Rolldown return + E(Δ price from yield/spread views) − E(credit losses) + E(currency gains/losses).
  Yield income = Annual coupon / Current price.
  Rolldown return = (Price at horizon − Beginning price) / Beginning price.

• Expected fixed-income return: $E(R) \approx$ Yield income + Rolldown return + $E(\Delta\text{price})$ − $E(\text{credit losses})$ + $E(\text{currency})$

• Yield income = Annual coupon / Current bond price $Q208, Q220, Q224, Q240$

• Rolldown return = (Price at horizon − Beginning price) / Beginning price $Q208, Q220, Q224, Q240$

• Butterfly spread = −(Short-term yield) + (2 × Medium-term yield) − Long-term yield

• Price-change approximation: $-\text{MD} \times \Delta y + 0.5 \times \text{Convexity} \times (\Delta y)^2$

• Foreign-bond duration contribution = Country beta × Local-currency duration

• Unhedged foreign-bond return $\approx r_l + e$ where $e = (S_{t+1}-S_t)/S_t$

• Hedged foreign-bond return $\approx r_l + (i_d - i_f)$ or $i_d + (r_l - i_f)$

[MOCK: _MOCK_PMP_2_1.md]

• Key-rate relative performance approximation:
  $\Delta\text{Relative value} \approx $\text{KR}{port} - \text{KR}{bench}$ \times \Delta y$

Fixed-Income Active Management: Credit Strategies

$$ $\text{YTM}(i) = \text{YTM(Treasury)} + \text{Yield Spread}(i)$ $$

Expected excess return / excess spread (relative value analysis)
EXR ≈ (s × t) − (Δs × SD) − (t × p × L)
or
Expected excess spread ≈ Spread₀ − (EffSpreadDur × ΔSpread) − (POD × LGD)
where s/Spread₀ = credit spread (Z-spread or OAS), t = holding period (years), SD/EffSpreadDur = spread duration, Δs/ΔSpread = expected change in spread, p/POD = probability of default, L/LGD = loss severity/given default.

Active share
Active share = ½ × Σ |w_{portfolio,i} − w_{benchmark,i}| for i = 1 to N

Fundamental law of active management
E(R_A) = IC × √BR × σ_RA × TC
where IC = information coefficient, BR = breadth (# independent investment decisions/year), σ_RA = active risk (tracking error), TC = transfer coefficient.

Approximate bond price change
ΔP/P ≈ −MD × Δy or −EffSpreadDur × Δs

G-spread
G-spread = bond YTM − linearly interpolated on-the-run government YTM (weights based on maturity distance to bracketing gov bonds)

I-spread
I-spread = bond YTM − linearly interpolated swap rate (swap rate = gov YTM + swap spread for same tenor)

Asset swap spread $ASW$
ASW ≈ bond coupon − interpolated swap rate (same maturity)

CDS upfront payment / initial price
Upfront (% notional) ≈ EffSpreadDur_CDS × (market spread − standard coupon)
CDS price (per 100 notional) ≈ 100 − upfront

VaR / tail risk scaling
Horizon yield change (z-score) ≈ daily yield volatility × z × √trading days
Position loss ≈ notional × (price/100) × (−MD × Δy)

[FORMULAS IN MOCKS]

• Expected excess return (credit): $$\text{EXR} \approx s \times t - \Delta s \times \text{SD} - (t \times p \times L)$$ where $s$ = spread, $t$ = horizon, SD = spread duration, $p$ = default probability, $L$ = loss severity $Q211, Q226, Q236$

• Spread duration ≈ DTS / OAS

• Expected annual credit loss = Probability of default × Loss severity

• Breakeven spread change (quarterly) = Quarterly yield advantage / Duration

• Probability of default approximation:
  $\text{POD} = \dfrac{\text{Spread}}{\text{LGD}},\quad \text{LGD} = 1 - \text{RR}$

• Credit spread from POD:
  $\text{Credit spread} = \text{POD} \times \text{LGD}$

• CDS P&L approximation (protection):
  $\text{P&L} \approx \text{Notional} \times \Delta\text{Spread} \times \text{Effective spread duration}$
  (sign flips for short protection)

• Duration-matching notional for CDS:
  $\text{Notional}{5y} = \dfrac{\text{EffSpreadDur}{10y}}{\text{EffSpreadDur}{5y}} \times \text{Notional}{10y}$

• Expected excess spread return:
  $$E[\text{ExcessSpread}] = \text{Spread}_0 - \text{EffSpreadDur} \times \Delta\text{Spread} - (\text{POD} \times \text{LGD})$$

Trade Strategies and Execution

• Delay cost = Number of shares executed $\times$ (Arrival price – Decision price)
  (signs adjust for sell orders)

• Arrival cost (bps) = Side $\times$ $\dfrac{\text{Average price} - \text{Arrival price}}{\text{Arrival price}} \times 10{,}000$
  where Side = $+1$ for buy, $-1$ for sell

• Implementation shortfall $IS$ = Execution cost + Opportunity cost + Fees
  Execution cost = $\sum s_j p_j - \sum s_j p_d$
  Opportunity cost = $(S - \sum s_j)(p_n - p_d)$

• Expanded implementation shortfall = Delay cost + Trading cost + Opportunity cost + Fees
  Trading cost = $\sum s_j p_j - $\sum s_j$ p_0$
  where $p_0$ = arrival price

• Market-adjusted cost (bps) = Arrival cost (bps) – $\beta \times$ Index cost (bps)
  Index cost (bps) = Side $\times$ $\dfrac{\text{Index VWAP} - \text{Index arrival price}}{\text{Index arrival price}} \times 10{,}000$

• VWAP cost (bps) = Side $\times$ $\dfrac{P - \text{VWAP}}{\text{VWAP}} \times 10{,}000$

Liquidity aggregation (bottom-up): $$ \text{Liquid} = \sum $w_i \times %\text{Liquid}_i$ $$ $$ \text{Illiquid} = \sum $w_i \times %\text{Illiquid}_i$ $$ Example: $$ \text{Liquid} = $1% \times 100%$ + $24% \times 100%$ + $39% \times 50%$ = 44.5% $$ $$ \text{Illiquid} = $21% \times 100%$ + $15% \times 50%$ = 28.5% $$

Average composite fee (for net-of-fees comparison): $$ \frac{(0.0100 \times 5) + [0.0060 \times (60 - 5)]}{60} = 0.0063\ $0.63%$ $$

Levered cost of obtaining leverage (3× factor = 66.67% borrowed): $$ \frac{\text{Notional} \times 0.6667 \times \text{Borrowing rate}}{\text{Notional}} $$ ETF borrowing rate = 3-month MRR + 35 bps = 2.15%; futures opportunity cost = 3-month MRR = 1.80%.

Total levered cost = unlevered cost + additional leverage cost.

Geometric smoothing spending rule: $$ \text{Spending}t = $60% \times \text{Spending}{t-1}$ + [40% \times (5% \times MV_{t-1})] $$

[FORMULAS IN MOCKS]

• Implementation shortfall components: #sunday $\text{Execution cost} = \sum S_j P_j - \sum S_j P_d$
  $\text{Opportunity cost} = (S - \sum S_j)(P_n - P_d)$
  $\text{Fixed fees} = \text{Shares traded} \times \text{fee per share}$
  $\text{IS} = \text{Execution cost} + \text{Opportunity cost} + \text{Fixed fees}$
  $\text{IS (bps)} = \dfrac{\text{IS ($)}}{\text{Total shares} \times P_d} \times 10{,}000$