CFA Level I
Portfolio Management
Formula Reference Sheet
Returns
Holding Period Return (HPR)
HPR = Final Price + IncomeInitial Price − 1
Return of Foreign Asset (Domestic Currency)
RD = (1 + RLC) · (1 + RFX) − 1
Nominal vs. Real Return
Fisher Equation
(1 + rnominal) = (1 + rf) · (1 + π) · (1 + RP)
(nominal) = (risk-free) · (inflation) · (risk premium)
(1 + rreal) = (1 + rnominal) ÷ (1 + π)
(nominal) = (risk-free) · (inflation) · (risk premium)
(1 + rreal) = (1 + rnominal) ÷ (1 + π)
Diversification
Diversification Ratio
Risk of equally weighted portfolio of n securities
Risk of a single random security
Lower ratio = better diversification
Portfolio Stdev — Equally Weighted
σp = √σ̄2N + (N−1)N · C̄ov
σ̄² = avg variance of assets · C̄ov = avg covariance of assets
Portfolio Variance & Covariance
2-Stock Portfolio Variance
wA2σA2 + wB2σB2 + 2wAwB · CovAB
Covariance of x & y
Cov(xy) = σy · σx · ρxy
Expected Portfolio Return (2 assets)
E(Rp) = wiRf + (1−wi)E(Ri)
σp = (1−wi)σi
σp = (1−wi)σi
Capital Allocation Line (CAL) & Capital Market Line (CML)
Capital Allocation Line (CAL) — any risky portfolio
E(r)p = (WRf)(Rf) + (WRp)[E(Rp)]
σp = (WRp)(σRp)
Capital Market Line (CML) — market portfolio
E(r)p = (WRf)(Rf) + (Wm)[E(Rm)]
σp = (Wm)(σm)
CML — Solve for E(Rp) given σp
E(Rp) = [E(Rm) − Rfσm] σp + Rf
Beta
Beta of Stock i
βi = Covi,mktσ2mkt
=
ρi,mkt · σiσmkt
Annualized Rate
(1 + rate)n − 1
where n = number of periods per year
CAPM, SML & Market Model
CAPM
E(r) = Rf + β[E(Rm) − Rf]
Security Market Line (SML)
E(r) = [E(Rm) − Rf] · β + Rf
Market Model
Ri = β(Rm) + αi + ei
Performance Ratios
Sharpe Ratio
Rp − Rfσp
Uses total risk (σp). Best for undiversified portfolios.
Treynor Ratio
Rp − Rfβ
Uses systematic risk (β). Best for well-diversified portfolios.
Jensen’s Alpha
α = Rp − [Rf + β(Rm − Rf)]
Excess return above CAPM prediction. Positive = outperformance.
M² Ratio
M2 = (Rp − Rf) · σmσp + Rf
Equivalent to Sharpe Ratio scaled to market risk. Higher = better risk-adjusted return.
Utility & Optimal Weights
Utility Function
U = E(r) − ½ · A · σ2
A = risk aversion coefficient (higher A = more risk averse)
Optimal Weight in Risky Asset
w* = ασ2
Single-Index Model
Single-Index Model Return
Ri = (1−βi)Rf + βi · Rm + ei
also written: Ri = αi + βi(Rm) + ei
also written: Ri = αi + βi(Rm) + ei
Covariance (Single-Index)
Cov(Ri, Rm) = βi · σ2m
Beta (Single-Index)
βi = ρi,m · σiσm
Expected Return (Single-Index)
E(Ri) = Rf + βi · [E(Rm) − Rf]
Variance Decomposition
Total Variance
σ2i = βi2 · σ2m + σ2ei
= Systematic variance + Non-systematic variance
Non-Systematic (Residual) Variance
σ2ei = σ2i − βi2 · σ2m
Return Generating Models
Multi-Factor Model
Ri = αi + βi · Rm + ei
αi = stock-specific return · βi = factor sensitivity · ei = error term