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Formula Sheet

CAIA Level II — all formulas by Reading

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CAIA Level II Formula Sheet

16 September 2026

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Reading 2.1 — Asset Class Return

Asset class return\text{Asset class return}
Short-term real risk-free rate+Expected inflation+Risk premium\text{Short-term real risk-free rate} + \text{Expected inflation} + \text{Risk premium}

Reading 2.2 — Endowment Model

Change in endowment value\text{Change in endowment value}
Income from gifts−Spending+Net investment returns\text{Income from gifts} - \text{Spending} + \text{Net investment returns}
Total return\text{Total return}
Returns from SAA+Security selection+Market timing / TAA\text{Returns from SAA} + \text{Security selection} + \text{Market timing / TAA}

Reading 2.3 — Pension & Liability Modelling

Market value of equity (Eq)\text{Market value of equity } (E_q)
(OAt−OLt)+(At−Lt)(OA_t - OL_t) + (A_t - L_t)
% Change in liabilities\text{\% Change in liabilities}
−Modified duration×Change in yield-\text{Modified duration} \times \text{Change in yield}
Economic life (EL) in years\text{Economic life } (EL) \text{ in years}
EL=1ln⁡(1+R)×ln⁡[PaymentPayment−(R×Assets)]EL = \frac{1}{\ln(1+R)} \times \ln\left[\frac{\text{Payment}}{\text{Payment} - (R \times \text{Assets})}\right]
PV of growth annuity (ordinary, g=0)\text{PV of growth annuity (ordinary, } g = 0\text{)}
Initial paymentr−g[1−(1+g1+r)n]\frac{\text{Initial payment}}{r - g}\left[1 - \left(\frac{1+g}{1+r}\right)^{n}\right]

Reading 2.4 — Sovereign Wealth Funds

Balance of payments (reserve account change)\text{Balance of payments (reserve account change)}
ΔReserve Account=ΔCurrent Account+ΔCapital Account\displaystyle \Delta\text{Reserve Account} = \Delta\text{Current Account} + \Delta\text{Capital Account}
Norway GPFG spending rule\text{Norway GPFG spending rule}
Annual Government Transfer=3%×Current SWF Value\text{Annual Government Transfer} = 3\% \times \text{Current SWF Value}
Five drivers of currency appreciation\text{Five drivers of currency appreciation}
Lower inflation+Higher real rates+Capital inflows+Slower income growth+Export advantage\text{Lower inflation} + \text{Higher real rates} + \text{Capital inflows} + \text{Slower income growth} + \text{Export advantage}
Three commodity concerns\text{Three commodity concerns}
Volatility+Depletion+Diversification\text{Volatility} + \text{Depletion} + \text{Diversification}

Reading 2.5 — Family Offices & Tax

After-tax profit (non-Section 1256)\text{After-tax profit (non-Section 1256)}
Pre-Tax Profit×(1−TOrdinary)\text{Pre-Tax Profit} \times (1 - T_{\text{Ordinary}})
Blended tax rate (Section 1256)\text{Blended tax rate (Section 1256)}
T1256=(0.40×TOrdinary)+(0.60×TLTCG)T_{1256} = (0.40 \times T_{\text{Ordinary}}) + (0.60 \times T_{\text{LTCG}})
Sustainable spending rate (multi-gen trust)\text{Sustainable spending rate (multi-gen trust)}
2%–4%→perpetuity      6%–10%→depletion in ∼2 gen.2\%\text{–}4\% \to \text{perpetuity} \;\;\; 6\%\text{–}10\% \to \text{depletion in } \sim 2 \text{ gen.}

Reading 3.1 — Expected Utility & MVO

Probability-weighted expected return & SD\text{Probability-weighted expected return \& SD}
E(R)=∑iProbi×Ri    σ=∑inProbi×(Ri−μ)2\displaystyle E(R)=\sum_{i} \text{Prob}_{i} \times R_{i} \;\; \sigma = \sqrt{\sum_{i}^{n} \text{Prob}_{i} \times (R_{i} - \mu)^{2}}
Expected utility\text{Expected utility}
E[U(W)]=π1U(W1)+π2U(W2)    π=prob. of WE[U(W)]=\pi_{1} U(W_{1})+\pi_{2} U(W_{2}) \;\; \pi = \text{prob. of } W
Expected utility (return & variance)\text{Expected utility (return \& variance)}
μ−λ2σ2    λ=risk aversion\mu - \frac{\lambda}{2} \sigma^{2} \;\; \lambda = \text{risk aversion}
With higher moments\text{With higher moments}
μ−λ12σ2+λ2S−λ3K    VaR: μ−λ2VaRα\mu - \frac{\lambda_{1}}{2} \sigma^{2} + \lambda_{2} S - \lambda_{3} K \;\; \text{VaR: } \mu - \frac{\lambda}{2} \text{VaR}_{\alpha}
With liability growth  (G)\text{With liability growth} \; (G)
V×E(Rp)−λ2Var[V×Rp−L×G]\displaystyle V \times E(R_{p}) - \frac{\lambda}{2} \text{Var}[V \times R_{p} - L \times G]
Optimal risky asset weight\text{Optimal risky asset weight}
w=1λE[R−R0]σ2+Lδσ2\displaystyle w = \frac{1}{\lambda} \frac{E[R - R_{0}]}{\sigma^{2}} + L \frac{\delta}{\sigma^{2}}
Degree of risk aversion\text{Degree of risk aversion}
λ=E(Rp)−RfσP2\displaystyle \lambda = \frac{E(R_{p}) - R_{f}}{\sigma_{P}^{2}}
Portfolio return (Rp)\text{Portfolio return } (R_{p})
∑i=1Nwi(Ri−R0)+R0    w0=1−∑i=1Nwi\displaystyle \sum_{i=1}^{N} w_{i} (R_{i} - R_{0}) + R_{0} \;\; w_{0} = 1 - \sum_{i=1}^{N} w_{i}
Variance-covariance matrix  (G)\text{Variance-covariance matrix} \; (G)
Σ=[σ11⋯σ1N⋮σii⋮σN1⋯σNN]\displaystyle \Sigma = \begin{bmatrix} \sigma_{11} & \cdots & \sigma_{1N} \\ \vdots & \sigma_{ii} & \vdots \\ \sigma_{N1} & \cdots & \sigma_{NN} \end{bmatrix}
MVO objective function  (G)\text{MVO objective function} \; (G)
max⁡Rˉp−λ2σP2 s.t. wi≥0;  i=1,…,N\displaystyle \max \bar{R}_{p} - \frac{\lambda}{2} \sigma_{P}^{2} \text{ s.t. } w_{i} \ge 0; \; i=1,\dots,N
MVO objective (expanded)  (G)\text{MVO objective (expanded)} \; (G)
max⁡E ⁣[∑i=1Nwi(Ri−R0)+R0]−λ2Var ⁣[∑i=1Nwi(Ri−R0)+R0]\displaystyle \max E\!\left[\sum_{i=1}^{N} w_{i} (R_{i} - R_{0}) + R_{0}\right] - \frac{\lambda}{2} \text{Var}\!\left[\sum_{i=1}^{N} w_{i} (R_{i} - R_{0}) + R_{0}\right]

Reading 3.1 (cont.) — Optimal Weights & Constraints

Optimal weights of risky assets  (G)\text{Optimal weights of risky assets} \; (G)
[w1⋮wN]=1λΣ−1[E[R1−R0]⋮E[RN−R0]]\displaystyle \begin{bmatrix} w_1 \\ \vdots \\ w_N \end{bmatrix} = \frac{1}{\lambda} \Sigma^{-1} \begin{bmatrix} E[R_1 - R_0] \\ \vdots \\ E[R_N - R_0] \end{bmatrix}
Optimal weight of risky asset (w)\text{Optimal weight of risky asset } (w)
1λE[R−R0]σ2\displaystyle \frac{1}{\lambda} \frac{E[R - R_0]}{\sigma^2}
Hurdle rate criterion\text{Hurdle rate criterion}
E(Rnew)>Rf+βnew[E(Rp)−Rf]    βnew=CovVar(Rp)\displaystyle E(R_{\text{new}}) > R_f + \beta_{\text{new}}[E(R_p) - R_f] \;\; \beta_{\text{new}} = \frac{\text{Cov}}{\text{Var}(R_p)}
MV adjusted for illiquidity\text{MV adjusted for illiquidity}
max⁡Rˉp−λ2σp2−ϕLp    Rˉi→Rˉi−ϕLi\displaystyle \max \bar{R}_p - \frac{\lambda}{2}\sigma_p^2 - \phi L_p \;\; \bar{R}_i \to \bar{R}_i - \phi L_i
MV adjusted for factor exposure\text{MV adjusted for factor exposure}
Rit=ai+biFt+εit    bp=∑i=1Nwibi    bp≤bˉ\displaystyle R_{it} = a_i + b_i F_t + \varepsilon_{it} \;\; b_p = \sum_{i=1}^{N} w_i b_i \;\; b_p \le \bar{b}

Reading 3.3 — Portfolio Risk Decomposition

Portfolio variance\text{Portfolio variance}
σP2=Var ⁣[∑i=1NwiRi]=∑i=1N∑j=1Nwiwjσij\displaystyle \sigma_{P}^{2} = \text{Var}\!\left[\sum_{i=1}^{N} w_{i} R_{i}\right] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_{i} w_{j} \sigma_{ij}
2-asset portfolio variance\text{2-asset portfolio variance}
σP2=wi2σi2+wj2σj2+2wiwjσi,j\sigma_{P}^{2} = w_{i}^{2} \sigma_{i}^{2} + w_{j}^{2} \sigma_{j}^{2} + 2w_{i} w_{j} \sigma_{i,j}
Marg. contribution of i to risk\text{Marg. contribution of } i \text{ to risk}
∂σP∂wi=σiPσPwi=ρiσiwi\displaystyle \frac{\partial \sigma_{P}}{\partial w_{i}} = \frac{\sigma_{iP}}{\sigma_{P}} w_{i} = \rho_{i} \sigma_{i} w_{i}
Portfolio total risk\text{Portfolio total risk}
σP=∑marginal contributions\sigma_{P} = \sum \text{marginal contributions}
Total risk (in risk factors)\text{Total risk (in risk factors)}
σP=(ρF1σF1b1)+(ρF2σF2b2)+(ρεσε)\sigma_{P} = (\rho_{F_{1}} \sigma_{F_{1}} b_{1}) + (\rho_{F_{2}} \sigma_{F_{2}} b_{2}) + (\rho_{\varepsilon} \sigma_{\varepsilon})
Sharpe ratio\text{Sharpe ratio}
R−Rfσ\displaystyle \frac{R - R_{f}}{\sigma}
Volatility-weighted weight (wi)\text{Volatility-weighted weight } (w_{i})
1/σi∑j=1N1/σj\displaystyle \frac{1 / \sigma_{i}}{\sum_{j=1}^{N} 1 / \sigma_{j}}

Reading 3.4 — Private Equity Cash Flows

Overcommitment ratio\text{Overcommitment ratio}
Total commitmentsResources available\displaystyle \frac{\text{Total commitments}}{\text{Resources available}}
Total CF\text{Total CF}
Investment×Equity multiple\text{Investment} \times \text{Equity multiple}
Management fee\text{Management fee}
Investment×Mgmt fee (%)×Holding period\text{Investment} \times \text{Mgmt fee (\%)} \times \text{Holding period}
Carried interest\text{Carried interest}
(Gross profit−Mgmt fee)×Carried interest (%)↔Total CF−Investment(\text{Gross profit} - \text{Mgmt fee}) \times \text{Carried interest (\%)} \leftrightarrow \text{Total CF} - \text{Investment}
Net profit\text{Net profit}
Gross profit−Mgmt fee−Carried interest\text{Gross profit} - \text{Mgmt fee} - \text{Carried interest}
Secondary price (P0)\text{Secondary price } (P_0)
P0=∑t=0TCFt(1+IRRBuyer)t    Discount=NAV−PNAV\displaystyle P_0 = \sum_{t=0}^{T} \frac{CF_t}{(1+\text{IRR}_{\text{Buyer}})^t} \;\; \text{Discount} = \frac{\text{NAV} - P}{\text{NAV}}

Reading 3.5 — Dynamic Strategies (Buy-&-Hold, CM, CPPI)

Initial portfolio value\text{Initial portfolio value}
V0=N0S0+M0B0V_0 = N_0 S_0 + M_0 B_0
Portfolio weights\text{Portfolio weights}
wt=NtStVt    1−wt=MtBtVt\displaystyle w_t = \frac{N_t S_t}{V_t} \;\; 1 - w_t = \frac{M_t B_t}{V_t}
Portfolio positions\text{Portfolio positions}
Nt=wtVtSt    Mt=(1−wt)VtBt\displaystyle N_t = \frac{w_t V_t}{S_t} \;\; M_t = \frac{(1-w_t) V_t}{B_t}
Buy-&-hold portfolio value\text{Buy-\&-hold portfolio value}
Vt=N0St+M0Bt    VtV0=w0StS0+(1−w0)BtB0\displaystyle V_t = N_0 S_t + M_0 B_t \;\; \frac{V_t}{V_0} = w_0 \frac{S_t}{S_0} + (1-w_0)\frac{B_t}{B_0}
Constant-mix (CM) weights at t=1\text{Constant-mix (CM) weights at } t{=}1
w0 and (1−w0)w_0 \text{ and } (1 - w_0)
CM portfolio return volatility\text{CM portfolio return volatility}
wequity×volequityw_{\text{equity}} \times \text{vol}_{\text{equity}}

Reading 3.5 (cont.) — CPPI, Options & Futures Overlay

Geometric mean return\text{Geometric mean return}
RC≈Rˉ−σ22R_C \approx \bar{R} - \frac{\sigma^2}{2}
CPPI equity & treas. investments\text{CPPI equity \& treas. investments}
St=mCt=m(Vt−Ft)    Ft=Ate−r(T−t)    Bt=Vt−StS_t = m C_t = m(V_t - F_t) \;\; F_t = A_t e^{-r(T-t)} \;\; B_t = V_t - S_t
CPPI variations — reset floor to\text{CPPI variations — reset floor to}
x% of V ⁣:  St=mVt(1−x)    x% of H ⁣:  St=m(Vt−xHt)\text{x\% of } V\!:\; S_t = m V_t (1-x) \;\; \text{x\% of } H\!:\; S_t = m(V_t - x H_t)
Stop-loss\text{Stop-loss}
St=mVt    m=1 if Vt>F    m=0 if Vt≤FS_t = m V_t \;\; m{=}1 \text{ if } V_t > F \;\; m{=}0 \text{ if } V_t \le F
Long put + risky asset\text{Long put + risky asset}
VT=ST+max⁡(K−ST,0)={STif ST>KKif ST≤K\displaystyle V_T = S_T + \max(K - S_T, 0) = \begin{cases} S_T & \text{if } S_T > K \\ K & \text{if } S_T \le K \end{cases}
Put-call parity\text{Put-call parity}
c+Ke−rT=p+Sc + K e^{-rT} = p + S
Long call + cash\text{Long call + cash}
V0=Nc+NKerT+D    VT={NST+DerTif ST>KNK+DerTif ST≤K\displaystyle V_0 = Nc + NK e^{rT} + D \;\; V_T = \begin{cases} NS_T + De^{rT} & \text{if } S_T > K \\ NK + De^{rT} & \text{if } S_T \le K \end{cases}
Replicate put option  (G)\text{Replicate put option} \; (G)
p=Mt−ΔtSt    =Long riskless+Short Δt stocks    Put delta: −Δtp = M_t - \Delta_t S_t \;\; = \text{Long riskless} + \text{Short } \Delta_t \text{ stocks} \;\; \text{Put delta: } -\Delta_t
Return: illiquid (S) + riskfree\text{Return: illiquid } (S) \text{ + riskfree}
Rt+1=wtrS,t+1+(1−wt)rf    rs,t=rf+α+βrq,t+εtR_{t+1} = w_t r_{S,t+1} + (1-w_t) r_f \;\; r_{s,t} = r_f + \alpha + \beta r_{q,t} + \varepsilon_t
Target return\text{Target return}
Rt+1T=ktrS,t+1+(1−kt)rfR^T_{t+1} = k_t r_{S,t+1} + (1-k_t) r_f
Return on portfolio with futures\text{Return on portfolio with futures}
Rt+1=[wtrS,t+1+(1−wt)rf]+Ftrq,t+1R_{t+1} = [w_t r_{S,t+1} + (1-w_t) r_f] + F_t r_{q,t+1}
Optimal futures position\text{Optimal futures position}
Ft≈(αrq,t+β)(kt−wt)\displaystyle F_t \approx \left(\frac{\alpha}{r_{q,t}} + \beta\right)(k_t - w_t)

Reading 4.1 — Futures & Beta Management

Beta of new portfolio\text{Beta of new portfolio}
βNew=βPort+(F/P)βFutures\beta_{\text{New}} = \beta_{\text{Port}} + (F/P)\beta_{\text{Futures}}
# of Futures for portable alpha\text{\# of Futures for portable alpha}
Value of position hedged×BetaIndex value×Multiplier\displaystyle \frac{\text{Value of position hedged} \times \text{Beta}}{\text{Index value} \times \text{Multiplier}}
Put-call parity (Hedged portfolio)\text{Put-call parity (Hedged portfolio)}
Bond=+Stock+Put−Call\text{Bond} = +\text{Stock} + \text{Put} - \text{Call}
Delta of put & call option\text{Delta of put \& call option}
ΔOptionPriceΔStockPrice\displaystyle \frac{\Delta \text{OptionPrice}}{\Delta \text{StockPrice}}

Reading 4.2 — CAPM, Market & Benchmark Models

CAPM market model\text{CAPM market model}
Rit−Rf=βi(Rmt−Rf)+εitR_{it} - R_f = \beta_i(R_{mt} - R_f) + \varepsilon_{it}
Market model (Ex-post)\text{Market model (Ex-post)}
Rit−Rf=ai+Bi(Rmt−Rf)+eitR_{it} - R_f = a_i + B_i(R_{mt} - R_f) + e_{it}
Benchmark model (Ex-post)\text{Benchmark model (Ex-post)}
Rt−Rf=a+B(RBenchmark,t−Rf)+etR_t - R_f = a + B(R_{\text{Benchmark},t} - R_f) + e_t

Reading 4.3 — PME & Real Assets

Hypothetical PME NAV\text{Hypothetical PME NAV}
Final hypoth. market accumulation (HMA)+PE fund NAV\text{Final hypoth. market accumulation (HMA)} + \text{PE fund NAV}
HMA\text{HMA}
Total index units×Market index value\text{Total index units} \times \text{Market index value}
KS-PME\text{KS-PME}
HMA DistributionsHMA Capital calls\displaystyle \frac{\text{HMA Distributions}}{\text{HMA Capital calls}}
Cap rate\text{Cap rate}
NOIProperty value\displaystyle \frac{\text{NOI}}{\text{Property value}}
Expected return with risk premium\text{Expected return with risk premium}
E(Ri)=Rf+Risk premiumiE(R_i) = R_f + \text{Risk premium}_i

Reading 4.4 — Risk Measures & Smoothing

Trading level\text{Trading level}
Funding level+Notional funding\text{Funding level} + \text{Notional funding}
Margin-to-equity ratio\text{Margin-to-equity ratio}
Margin requirementNAV\displaystyle \frac{\text{Margin requirement}}{NAV}
Capital at risk\text{Capital at risk}
Loss if hit stop-loss priceNAV\displaystyle \frac{\text{Loss if hit stop-loss price}}{NAV}
VaR\text{VaR}
(α×σ)+μ    α=−1.645 (95%)    α=−1.96 (97.5%)    Pr⁡(z≤α)=1−c(\alpha \times \sigma) + \mu \;\; \alpha = -1.645 \text{ (95\%)} \;\; \alpha = -1.96 \text{ (97.5\%)} \;\; \Pr(z \le \alpha) = 1 - c
Mean return & Variance\text{Mean return \& Variance}
μ=1T∑t=1TRt    σt2=1T−1∑t=1T(Rt−μ)2\displaystyle \mu = \frac{1}{T} \sum_{t=1}^{T} R_t \;\; \sigma_t^2 = \frac{1}{T-1} \sum_{t=1}^{T} (R_t - \mu)^2
Exponentially smoothed mean\text{Exponentially smoothed mean}
μt−1=(1−λ)μt−2+λRt−1\mu_{t-1} = (1 - \lambda)\mu_{t-2} + \lambda R_{t-1}
Exponentially smoothed variance\text{Exponentially smoothed variance}
σt2=(1−λ)σt−12+λ(Rt−μt−1)2\sigma_t^2 = (1 - \lambda)\sigma_{t-1}^2 + \lambda(R_t - \mu_{t-1})^2
Omega (Ω)\text{Omega } (\Omega)
∑i=1Nmax⁡(Ri−T,0)∑i=1Nmax⁡(T−Ri,0)\displaystyle \frac{\sum_{i=1}^{N} \max(R_i - T, 0)}{\sum_{i=1}^{N} \max(T - R_i, 0)}
Reported price (Pt,reported)\text{Reported price } (P_{t,\text{reported}})
αPt,true+α(1−α)Pt−1,true+α(1−α)2Pt−2,true+…\alpha P_{t,\text{true}} + \alpha(1-\alpha)P_{t-1,\text{true}} + \alpha(1-\alpha)^2 P_{t-2,\text{true}} + \dots
True price (Pt,true)\text{True price } (P_{t,\text{true}})
Pt−1,reported+1α(Pt,reported−Pt−1,reported)P_{t-1,\text{reported}} + \frac{1}{\alpha}(P_{t,\text{reported}} - P_{t-1,\text{reported}})
True & reported returns\text{True \& reported returns}
Rt,true=Rt,reported−ρRt−1,reported1−ρ    Rt,reported=ρRt−1,reported+(1−ρ)Rt,trueR_{t,\text{true}} = \frac{R_{t,\text{reported}} - \rho R_{t-1,\text{reported}}}{1 - \rho} \;\; R_{t,\text{reported}} = \rho R_{t-1,\text{reported}} + (1-\rho) R_{t,\text{true}}
Correlation coefficient\text{Correlation coefficient}
ρi,j=σi,jσiσj\rho_{i,j} = \frac{\sigma_{i,j}}{\sigma_i \sigma_j}
Variance & volatility of true returns\text{Variance \& volatility of true returns}
σtrue2=σreported21+ρ1−ρ    σtrue=σtrue2\sigma_{\text{true}}^2 = \sigma_{\text{reported}}^2 \frac{1+\rho}{1-\rho} \;\; \sigma_{\text{true}} = \sqrt{\sigma_{\text{true}}^2}
Beta\text{Beta}
βtrue=βreported1−ρ    [β=ρimσiσm]\beta_{\text{true}} = \frac{\beta_{\text{reported}}}{1-\rho} \;\; \left[\beta = \frac{\rho_{im}\sigma_i}{\sigma_m}\right]

Reading 5.1 — Interest Rate & Credit Models

Vasicek model\text{Vasicek model}
r~t+1=rt+κ(μ−rt)+σε~t+1    E[rt+1]=rt+κ(μ−rt)\tilde{r}_{t+1} = r_t + \kappa(\mu - r_t) + \sigma\tilde{\varepsilon}_{t+1} \;\; E[r_{t+1}] = r_t + \kappa(\mu - r_t)
CIR model\text{CIR model}
rt+1=rt+κ(μ−rt)+rt σε~t+1r_{t+1} = r_t + \kappa(\mu - r_t) + \sqrt{r_t}\,\sigma\tilde{\varepsilon}_{t+1}
Ho and Lee model\text{Ho and Lee model}
rt+1=rt+θt+σε~t+1r_{t+1} = r_t + \theta_t + \sigma\tilde{\varepsilon}_{t+1}
Black, Derman & Toy (BDT) model\text{Black, Derman \& Toy (BDT) model}
0.5(1+r0)[(1+ru)+(1+rd)]−1    ru=rde2σ0.5(1+r_0)[(1+r_u)+(1+r_d)]-1 \;\; r_u = r_d e^{2\sigma}
Recovery rate (RR)\text{Recovery rate (RR)}
PV of sum to be recoveredEAD\displaystyle \frac{\text{PV of sum to be recovered}}{EAD}
Loss given default (LGD)\text{Loss given default (LGD)}
EAD(1−RR)=EAD−PV of sum recoveredEAD(1-RR) = EAD - \text{PV of sum recovered}
Expected loss\text{Expected loss}
LGD×PD=EAD(1−RR)×PDLGD \times PD = EAD(1-RR) \times PD
Merton’s levered firm’s value\text{Merton's levered firm's value}
Assets=Equity+Risky debt\text{Assets} = \text{Equity} + \text{Risky debt}
Put-call parity\text{Put-call parity}
Assets=Call+(Riskless Bond−Put)\text{Assets} = \text{Call} + (\text{Riskless Bond} - \text{Put})
Merton’s probability of default\text{Merton's probability of default}
Pr⁡(AT≤K)=1−N(d−σAτ)\Pr(A_T \le K) = 1 - N(d - \sigma_A\sqrt{\tau})
Firm’s asset volatility\text{Firm's asset volatility}
σassets≈σequity(EquityAssets)\sigma_{\text{assets}} \approx \sigma_{\text{equity}}\left(\frac{\text{Equity}}{\text{Assets}}\right)
Black-Scholes call value\text{Black-Scholes call value}
Et=AtN(d)−Ke−rτN(d−σAτ)    d=ln⁡(At/K)+(r+0.5σA2)τσAτE_t = A_t N(d) - Ke^{-r\tau}N(d-\sigma_A\sqrt{\tau}) \;\; d = \frac{\ln(A_t/K)+(r+0.5\sigma_A^2)\tau}{\sigma_A\sqrt{\tau}}

Reading 5.1 (cont.) — Bond Pricing & Default

Black-Scholes put value (Pt)\text{Black-Scholes put value } (P_t)
Ke−rτN(−d+σAτ)−AtN(−d)Ke^{-r\tau}N(-d+\sigma_A\sqrt{\tau}) - A_t N(-d)
Risky debt price (Dt)\text{Risky debt price } (D_t)
Ke−rτ−Put pricet=Risk-free debt−Put valueKe^{-r\tau} - \text{Put price}_t = \text{Risk-free debt} - \text{Put value}
Risky debt price (Dt)\text{Risky debt price } (D_t)
Ke−(r+s)τ    s=credit spreadKe^{-(r+s)\tau} \;\; s = \text{credit spread}
Credit spread (s)\text{Credit spread } (s)
−1τln⁡[N(d−σAτ)+AtKerτN(−d)]\displaystyle -\frac{1}{\tau}\ln\left[N(d-\sigma_A\sqrt{\tau}) + \frac{A_t}{K}e^{r\tau}N(-d)\right]
Firm’s equity volatility (σE)\text{Firm's equity volatility } (\sigma_E)
AtEt×Δ×σA\frac{A_t}{E_t} \times \Delta \times \sigma_A
Distance to default (DD)\text{Distance to default (DD)}
At−KAtσA    K=default trigger\displaystyle \frac{A_t - K}{A_t \sigma_A} \;\; K = \text{default trigger}
Expected default frequency (EDF)\text{Expected default frequency (EDF)}
# firms defaulted in 1yr with DD=nTotal # firms with DD=n\displaystyle \frac{\text{\# firms defaulted in 1yr with DD}=n}{\text{Total \# firms with DD}=n}
Probability of surviving t years\text{Probability of surviving } t \text{ years}
p(t)=e−λt    Probability of default=1−pp(t) = e^{-\lambda t} \;\; \text{Probability of default} = 1 - p
Prob. of default between t & Δt\text{Prob. of default between } t \text{ \& } \Delta t
Conditional: λΔt    Unconditional: e−λtλΔt\text{Conditional: } \lambda\Delta t \;\; \text{Unconditional: } e^{-\lambda t}\lambda\Delta t
Prob. of default between s & t\text{Prob. of default between } s \text{ \& } t
p(s)−p(t)=e−λs−e−λtp(s) - p(t) = e^{-\lambda s} - e^{-\lambda t}
Bond price (D0)\text{Bond price } (D_0)
Ke−(r+λ)T    λ=default intensityKe^{-(r+\lambda)T} \;\; \lambda = \text{default intensity}
Bond price (D0) w/ recovery RR\text{Bond price } (D_0) \text{ w/ recovery } RR
e−rT[Ke−λT+RR×K(1−e−λT)]≈Ke−[r+λ(1−RR)]T    Credit spread=λ(1−RR)e^{-rT}\left[Ke^{-\lambda T} + RR \times K(1-e^{-\lambda T})\right] \approx Ke^{-[r+\lambda(1-RR)]T} \;\; \text{Credit spread} = \lambda(1-RR)
Altman model\text{Altman model}
Z=1.2X1+1.4X2+3.3X3+0.6X4+X5Z = 1.2X_1 + 1.4X_2 + 3.3X_3 + 0.6X_4 + X_5

Reading 5.2 — Options & Binomial Trees

Defaultable zero-cpn bond value\text{Defaultable zero-cpn bond value}
V0=$1(1−p)    V0=$1e−(δ+π)t    p=prob. of defaultV_0 = \$1(1-p) \;\; V_0 = \$1e^{-(\delta+\pi)t} \;\; p=\text{prob. of default}
Binomial tree up (u) & down (d) factors\text{Binomial tree up (u) \& down (d) factors}
u=eσΔt    d=1/uu = e^{\sigma\sqrt{\Delta t}} \;\; d = 1/u
Up risk-neutral probability\text{Up risk-neutral probability}
p=(r−d)/(u−d)    r=1+risk-free ratep = (r-d)/(u-d) \;\; r = 1 + \text{risk-free rate}
Current stock price S\text{Current stock price } S
[p×Su]+[(1−p)×Sd]r\displaystyle \frac{[p \times Su] + [(1-p) \times Sd]}{r}
Payoff at expiration (Call)\text{Payoff at expiration (Call)}
Top: max⁡(Su−K,0)    Bottom: max⁡(Sd−K,0)\text{Top: } \max(Su-K,0) \;\; \text{Bottom: } \max(Sd-K,0)
Payoff at expiration (Put)\text{Payoff at expiration (Put)}
Top: max⁡(K−Su,0)    Bottom: max⁡(K−Sd,0)\text{Top: } \max(K-Su,0) \;\; \text{Bottom: } \max(K-Sd,0)
Option value f\text{Option value } f
pfu+(1−p)fdr\displaystyle \frac{pf_u + (1-p)f_d}{r}
Parity\text{Parity}
Stock price×Conversion ratio\text{Stock price} \times \text{Conversion ratio}
Convertible bond price\text{Convertible bond price}
max⁡(Parity,  Par+Coupon)\max(\text{Parity},\; \text{Par} + \text{Coupon})
Interest rate in BDT tree\text{Interest rate in BDT tree}
iU=iLe2σi_U = i_L e^{2\sigma}
Call option value\text{Call option value}
Non-callable bond value−Callable bond value\text{Non-callable bond value} - \text{Callable bond value}

Reading 5.3 — Multi-Factor Models

Multi-factor asset pricing (Ex-ante)\text{Multi-factor asset pricing (Ex-ante)}
E(Ri)−Rf=∑j=1Jβij[E(Rj)−Rf]E(R_i) - R_f = \sum_{j=1}^{J} \beta_{ij}[E(R_j) - R_f]
Multi-factor (Ex-post)\text{Multi-factor (Ex-post)}
Rit−Rf=∑j=1Jβij[Rjt−Rf]+εitR_{it} - R_f = \sum_{j=1}^{J} \beta_{ij}[R_{jt} - R_f] + \varepsilon_{it}

Reading 5.3 (cont.) — Fama-French & Carhart

Fama-French (Ex-ante)\text{Fama-French (Ex-ante)}
E(Ri)−Rf=βi[E(Rm)−Rf]+β1i[E(Rs−Rb)]+β2i[E(Rh−Rl)]E(R_i) - R_f = \beta_i[E(R_m)-R_f] + \beta_{1i}[E(R_s-R_b)] + \beta_{2i}[E(R_h-R_l)]
Fama-French-Carhart (Ex-ante)\text{Fama-French-Carhart (Ex-ante)}
E(Ri)−Rf=βi[E(Rm)−Rf]+β1i[E(Rs−Rb)]+β2i[E(Rh−Rl)]+β3i[E(Rw−Rd)]E(R_i)-R_f = \beta_i[E(R_m)-R_f] + \beta_{1i}[E(R_s-R_b)] + \beta_{2i}[E(R_h-R_l)] + \beta_{3i}[E(R_w-R_d)]

Reading 5.4 — Momentum & Digital Assets

Profit/loss (Momentum trade)\text{Profit/loss (Momentum trade)}
{St+1−Stif St>St−1St−St+1if St<St−1\begin{cases} S_{t+1} - S_t & \text{if } S_t > S_{t-1} \\ S_t - S_{t+1} & \text{if } S_t < S_{t-1} \end{cases}
Signal-to-noise ratio (SNR)\text{Signal-to-noise ratio (SNR)}
∣Pt−Pt−n∣∑i=0n−1∣Pt−i−Pt−i−1∣\displaystyle \frac{|P_t - P_{t-n}|}{\sum_{i=0}^{n-1} |P_{t-i} - P_{t-i-1}|}
Market divergence index (MDI)\text{Market divergence index (MDI)}
1M∑i=1MSNRi(n)\displaystyle \frac{1}{M}\sum_{i=1}^{M} SNR_i(n)
Stock-to-flow ratio\text{Stock-to-flow ratio}
Supply of coinsAnnualized issuance of coin\displaystyle \frac{\text{Supply of coins}}{\text{Annualized issuance of coin}}
Metcalfe’s law\text{Metcalfe's law}
n(n−1)2\displaystyle \frac{n(n-1)}{2}

Reading 5.5 — PCA & Market-Timing Tests

Marginal %age of variance explained by PC\text{Marginal \%age of variance explained by PC}
PC’s eigenvalueSum of eigenvalues\displaystyle \frac{\text{PC's eigenvalue}}{\text{Sum of eigenvalues}}
Cumulative %age of var. explained by a PC\text{Cumulative \%age of var. explained by a PC}
PC’s marginal %age+Sum of previous marginal %ages\text{PC's marginal \%age} + \text{Sum of previous marginal \%ages}
Empirical (ex-post) Fama-French\text{Empirical (ex-post) Fama-French}
Rit−Rf=ai+bmi(Rmt−Rf)+b1i(Rst−Rbt)+b2i(Rht−Rlt)+eitR_{it} - R_f = a_i + b_{mi}(R_{mt}-R_f) + b_{1i}(R_{st}-R_{bt}) + b_{2i}(R_{ht}-R_{lt}) + e_{it}
2nd-order partial autocorrelation\text{2nd-order partial autocorrelation}
ρ2−ρ121−ρ12\displaystyle \frac{\rho_2 - \rho_1^2}{1 - \rho_1^2}
Dummy variable (D) model [Market-timing test]\text{Dummy variable (D) model [Market-timing test]}
Rit−Rf=ai+[bi,d+(Dt×bi,diff)](Rmt−Rf)+eitR_{it} - R_f = a_i + \big[b_{i,d} + (D_t \times b_{i,diff})\big](R_{mt}-R_f) + e_{it}
Quadratic model [Market-timing test]\text{Quadratic model [Market-timing test]}
Rit−Rf=ai+bim(Rmt−Rf)2+eitR_{it} - R_f = a_i + b_{im}(R_{mt}-R_f)^2 + e_{it}
# of subsamples for rolling windows\text{\# of subsamples for rolling windows}
N=T−m+1N = T - m + 1

Reading 5.6 — Calendar Spreads

Co-integrated prices\text{Co-integrated prices}
ln⁡(pt)−aln⁡(st)=ut    [stationary process]\ln(p_t) - a\ln(s_t) = u_t \;\; \text{[stationary process]}
Calendar spread profit/loss per unit\text{Calendar spread profit/loss per unit}
Profit/Loss×Units per Contract×Position size\text{Profit/Loss} \times \text{Units per Contract} \times \text{Position size}
Substitute test statistic (SS)\text{Substitute test statistic (SS)}
ln⁡(Closing price of AClosing price of B)\displaystyle \ln\left(\frac{\text{Closing price of } A}{\text{Closing price of } B}\right)
Test statistic\text{Test statistic}
SS−Moving average of SSSD of SS\displaystyle \frac{\text{SS} - \text{Moving average of SS}}{\text{SD of SS}}
Covered interest rate parity\text{Covered interest rate parity}
FtS0=(1+rFCU)t(1+rDCU)t\displaystyle \frac{F_t}{S_0} = \frac{(1+r_{FCU})^t}{(1+r_{DCU})^t}

Reading 5.7 — Tax Effects (Tax Shields)

PV of depreciation tax shield\text{PV of depreciation tax shield}
∑t=1TDeprec. tax shield(1+Rd)t=∑t=1TDeprec.t×Tax rate(1+Rd)t\displaystyle \sum_{t=1}^{T} \frac{\text{Deprec. tax shield}}{(1+R_d)^t} = \sum_{t=1}^{T} \frac{\text{Deprec.}_t \times \text{Tax rate}}{(1+R_d)^t}
Without tax deferral After-tax rate\text{Without tax deferral After-tax rate}
r(1−Tax rate)r(1 - \text{Tax rate})
After-tax FV\text{After-tax FV}
PV[1+r(1−Tax)]TPV[1+r(1-\text{Tax})]^T

Reading 5.7 (cont.) — Tax Effects

With tax deferral After-tax rate\text{With tax deferral After-tax rate}
{1+[(1+r)T−1](1−Tax rate)}1/T−1=(FV/PV)1/T−1\left\{1 + \left[(1+r)^T - 1\right](1-\text{Tax rate})\right\}^{1/T} - 1 = (FV/PV)^{1/T} - 1
After-tax FV\text{After-tax FV}
PV{1+[(1+r)T−1](1−Tax rate)}=PV{(1+r)T(1−Tax rate)+Tax rate}PV\left\{1 + \left[(1+r)^T - 1\right](1-\text{Tax rate})\right\} = PV\left\{(1+r)^T(1-\text{Tax rate}) + \text{Tax rate}\right\}
Profit\text{Profit}
FV−PVFV - PV
Operating CF\text{Operating CF}
EBIT+Depreciation\text{EBIT} + \text{Depreciation}
Accuracy of estimate\text{Accuracy of estimate}
1n    n=# of observations\displaystyle \frac{1}{\sqrt{n}} \;\; n = \text{\# of observations}

Reading 6.1 — Hedge Fund Replication

Linear replication model\text{Linear replication model}
Rt,HF−rf=β1(F1r−rf)+⋯+βK(FKt−rf)+εtR_{t,HF} - r_f = \beta_1(F_{1r} - r_f) + \dots + \beta_K(F_{Kt} - r_f) + \varepsilon_t
Cash weight of replic. portfolio; R2\text{Cash weight of replic. portfolio; } R^2
βCash=1−∑i=1Kβi\beta_{\text{Cash}} = 1 - \sum_{i=1}^{K} \beta_i
Return on replicating portfolio\text{Return on replicating portfolio}
RRe,T+1=β^1,TF1,T+1+⋯+β^K,TFK,T+1R_{\text{Re},T+1} = \hat{\beta}_{1,T}F_{1,T+1} + \dots + \hat{\beta}_{K,T}F_{K,T+1}
# of shares to short (convertible arb.)\text{\# of shares to short (convertible arb.)}
Delta×Convertible bond priceStock price\displaystyle \frac{\text{Delta} \times \text{Convertible bond price}}{\text{Stock price}}

Reading 6.2 — Equal Risk Weight

Equal risk weight\text{Equal risk weight}
1/σi∑n=1N(1/σn)\displaystyle \frac{1/\sigma_i}{\sum_{n=1}^{N} (1/\sigma_n)}

Reading 6.3 — Basis

Basis\text{Basis}
Spot price−Futures price\text{Spot price} - \text{Futures price}

Reading 6.4 — IRR & PME Ratios

Interim IRR\text{Interim IRR}
∑t=0TDt(1+IIRRT)t+NAVT(1+IIRRT)T=∑t=0TCt(1+IIRRT)t    ∑t=0TCFt(1+IIRRT)t+NAVT(1+IIRRT)T=0\displaystyle \sum_{t=0}^{T} \frac{D_t}{(1+IIRR_T)^t} + \frac{NAV_T}{(1+IIRR_T)^T} = \sum_{t=0}^{T} \frac{C_t}{(1+IIRR_T)^t} \;\; \sum_{t=0}^{T} \frac{CF_t}{(1+IIRR_T)^t} + \frac{NAV_T}{(1+IIRR_T)^T} = 0
DPI ratio\text{DPI ratio}
∑t=0TDt∑t=0TCt\displaystyle \frac{\sum_{t=0}^{T} D_t}{\sum_{t=0}^{T} C_t}
RVPI ratio\text{RVPI ratio}
NAVT∑t=0TCt\displaystyle \frac{NAV_T}{\sum_{t=0}^{T} C_t}
TVPI ratio [=DPI+RVPI]\text{TVPI ratio } [= DPI + RVPI]
∑t=0TDt+NAVT∑t=0TCt\displaystyle \frac{\sum_{t=0}^{T} D_t + NAV_T}{\sum_{t=0}^{T} C_t}
FVs of all distributions & contributions\text{FVs of all distributions \& contributions}
FVD=∑t=0TDt(ITIt)    FVC=∑t=0TCt(ITIt)FV_D = \sum_{t=0}^{T} D_t\left(\frac{I_T}{I_t}\right) \;\; FV_C = \sum_{t=0}^{T} C_t\left(\frac{I_T}{I_t}\right)
PME ratio\text{PME ratio}
FVD+NAVFVC\displaystyle \frac{FV_D + NAV}{FV_C}
Simple average IIRR\text{Simple average IIRR}
1N∑i=1NIIRRi,T\displaystyle \frac{1}{N}\sum_{i=1}^{N} IIRR_{i,T}
Commitment-weighted IIRR\text{Commitment-weighted IIRR}
1∑i=1NCCi∑i=1NCCi×IIRRi,T\displaystyle \frac{1}{\sum_{i=1}^{N} CC_i} \sum_{i=1}^{N} CC_i \times IIRR_{i,T}
Pooled average IIRR\text{Pooled average IIRR}
∑i=1N∑t=0TCFi,t(1+IIRRP,T)t+∑i=1NNAVi,T(1+IIRRP,T)T=0\displaystyle \sum_{i=1}^{N}\sum_{t=0}^{T} \frac{CF_{i,t}}{(1+IIRR_{P,T})^t} + \sum_{i=1}^{N} \frac{NAV_{i,T}}{(1+IIRR_{P,T})^T} = 0

Reading 7.1–7.2 — Volatility & ROE

Volatility (σ) over T periods\text{Volatility } (\sigma) \text{ over } T \text{ periods}
Perfect autocorr.: σT=σ1T    No autocorr.: σT=σ1T\text{Perfect autocorr.: } \sigma_T = \sigma_1 T \;\; \text{No autocorr.: } \sigma_T = \sigma_1 \sqrt{T}
Return on equity (ROE)\text{Return on equity (ROE)}
(ROA×L)−r(L−1)(ROA \times L) - r(L-1)

Reading 7.3 — Volatility of Levered Fund

Volatility of levered fund\text{Volatility of levered fund}
σlev=L σunlev    L=Assets/Equity\sigma_{\text{lev}} = L\,\sigma_{\text{unlev}} \;\; L = \text{Assets/Equity}

Reading 8.1 — Greeks

Vega (ν) of put & call\text{Vega } (\nu) \text{ of put \& call}
∂p∂σ=SN′(d)T    per bp: ν/100    Δp≈νΔσ\frac{\partial p}{\partial \sigma} = SN'(d)\sqrt{T} \;\; \text{per bp: } \nu/100 \;\; \Delta p \approx \nu\Delta\sigma
Gamma (γ) of put & call\text{Gamma } (\gamma) \text{ of put \& call}
N′(d)SσT=νσS2T\displaystyle \frac{N'(d)}{S\sigma\sqrt{T}} = \frac{\nu}{\sigma S^2 T}
Process for change in vol.\text{Process for change in vol.}
σt+Δ−σt=γΔ+δΔY+ϕΔJ    (diffusion & jump)\sigma_{t+\Delta} - \sigma_t = \gamma\Delta + \delta\Delta Y + \phi\Delta J \;\; \text{(diffusion \& jump)}

Reading 8.2 — Variance Swaps & VIX

Theta (θ) of call & put\text{Theta } (\theta) \text{ of call \& put}
−SN′(d)σ2T\displaystyle -\frac{SN'(d)\sigma}{2\sqrt{T}}
Variance swap payoff\text{Variance swap payoff}
Notional value×(Realized variance−Strike var.)\text{Notional value} \times (\text{Realized variance} - \text{Strike var.})
30-Day VIX contract price\text{30-Day VIX contract price}
PS(TL−30TL−TS)+PL(30−TSTL−TS)P_S\left(\frac{T_L - 30}{T_L - T_S}\right) + P_L\left(\frac{30 - T_S}{T_L - T_S}\right)
Variance of portfolio\text{Variance of portfolio}
σp2=∑i=1Nwi2σi2+∑i>jN−1∑jNwiwjσiσjρij\sigma_p^2 = \sum_{i=1}^{N} w_i^2 \sigma_i^2 + \sum_{i>j}^{N-1}\sum_{j}^{N} w_i w_j \sigma_i \sigma_j \rho_{ij}
Average correlation\text{Average correlation}
ρaverage≈σp2σi2    σp2=portfolio variance\rho_{\text{average}} \approx \frac{\sigma_p^2}{\sigma_i^2} \;\; \sigma_p^2 = \text{portfolio variance}

Reading 8.3 — After-Tax Returns & Structured Products

After-tax rate\text{After-tax rate}
r(1−T)r(1-T)
With tax deferral\text{With tax deferral}
{1+[(1+r)N−1](1−T)}1/N−1\left\{1 + \left[(1+r)^N - 1\right](1-T)\right\}^{1/N} - 1
With tax deferral & tax deduction\text{With tax deferral \& tax deduction}
{(1+r)N[1−TN1−T0]}1/N−1\left\{(1+r)^N\left[\frac{1-T_N}{1-T_0}\right]\right\}^{1/N} - 1
Asian call payout\text{Asian call payout}
max⁡(X−K,0)    X=avg. price of underlying\max(X-K, 0) \;\; X = \text{avg. price of underlying}
Principal-protected barrier note (long straddle)\text{Principal-protected barrier note (long straddle)}
ATM Up-&-Out Call+ATM Down-&-Out Put\text{ATM Up-\&-Out Call} + \text{ATM Down-\&-Out Put}
Structured products with kinks\text{Structured products with kinks}
Asset+Bear Put Spread−OTM Call\text{Asset} + \text{Bear Put Spread} - \text{OTM Call}
Structured products with leverage\text{Structured products with leverage}
Asset+Bull Spread\text{Asset} + \text{Bull Spread}
Price & price change of zero-cpn bond\text{Price \& price change of zero-cpn bond}
P=Fe−r(T−t)    dPdt=rP    [Bond pays F at time T]P = Fe^{-r(T-t)} \;\; \frac{dP}{dt} = rP \;\; \text{[Bond pays } F \text{ at time } T\text{]}

Reading 8.4 — International Real Estate

Return (R) on international property\text{Return (R) on international property}
V0(1+r)(1+fx)−V0V0=(1+r)(1+fx)−1≈r+fx\displaystyle \frac{V_0(1+r)(1+fx) - V_0}{V_0} = (1+r)(1+fx) - 1 \approx r + fx
Variance & volatility of int. RE return\text{Variance \& volatility of int. RE return}
σd2=σfx2+σr2+2 Cov(fx,r)    ρ=0:σd=σfx2+σr2    ρ=1:σd=σfx+σr    ρ=−1:σd=∣σr−σfx∣\sigma_d^2 = \sigma_{fx}^2 + \sigma_r^2 + 2\,\text{Cov}(fx, r) \;\; \rho{=}0: \sigma_d{=}\sqrt{\sigma_{fx}^2{+}\sigma_r^2} \;\; \rho{=}1: \sigma_d{=}\sigma_{fx}{+}\sigma_r \;\; \rho{=}{-}1: \sigma_d{=}|\sigma_r{-}\sigma_{fx}|
Expected hedged return on global RE\text{Expected hedged return on global RE}
Real estate return−Expected hedging cost=Target country int. rate−Home int. rate\text{Real estate return} - \text{Expected hedging cost} = \text{Target country int. rate} - \text{Home int. rate}
After-tax net yield on RE investment\text{After-tax net yield on RE investment}
Initial NOI yieldTotal cost=Initial NOI yieldPurchase price(1+Acquisition cost)=Yield on total cost(1−Tax rate)\displaystyle \frac{\text{Initial NOI yield}}{\text{Total cost}} = \frac{\text{Initial NOI yield}}{\text{Purchase price}(1+\text{Acquisition cost})} = \text{Yield on total cost}(1-\text{Tax rate})

Reading 8.5 — Market Microstructure

Volume-weighted avg. price (VWAP)\text{Volume-weighted avg. price (VWAP)}
Close×VolumeTotal volume\displaystyle \frac{\text{Close} \times \text{Volume}}{\text{Total volume}}
Y

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CAIA Level II Preparation