Checkpoint 3 (Alts, Portfolio Management)

Commodity Basis

Spot - Future

Commodity Calendar Spread

Futures Price of nearer maturity - Future of more distant

Futures Price Commodity

Spot + Storage Cost - Convenience yield

Roll Return

$ of expiring futures - $ new futures contract / $ expiring futures contract

Total Return Commodity

Collateral Return + Price Return + Roll Return

• Collateral return = interest on the T-bills

• Price return = change in the spot price

• Roll return = the part of the futures return that comes from the futures price converging to spot (positive in backwardation, negative in contango)

Net Operating Income (NOI)

Rental Income + Other Income - vacancy/collection loss - operating expense

• Rental Income @ full capacity

• Effective Gross Income

  • Rental Income + Other Income - vacancy/collection

Debt Service Coverage Ratio

NOI/Debt Service

• Debt Service

  • interest + principal

Loan to Value

loan amount / appraised value

max loan amount

LOWEST OF DSCR & LTV

Equity Dividend Rate

First year CF / Equity

• First year CF = NOI - debt service

After Tax Return (ROE)

(NOI - Interest - taxes) / equity investments

• taxes = t * ( NOI - DepN - Interest)

Income approach

NOI / Cap Rate

NPV(NOI) + PV(TV)

• cap rate

  • dr - g

  • direct cap

Appraised Based Index Return

BASED OF PROFESSIONAL VALUATIONS

• (NOI - CAPEX + (END MV - BEG MV)) / BEG MV

Appraisal lag Return

HISTORICAL DATA INPUTS - UNSMOOTH THE LAG

Appraisal lag can be mitigated by unsmoothing the data. To unsmooth an index, a modified autoregressive model can be applied:

R*_t = αR_t + (1 – α) R*_t–1

REITS requirement in RE and 5/50 rule

• 75% in RE

• no fewer than 5 can hold 50%

Cap Rate

NOI comps / Transaction Price comps

• NOI comps

  • subtract out noncash rent

  • turns into true cash figure

income yield on a property, ignoring leverage and financing

Funds From Operations FFO

Accounting net earnings + DepN/AmortN/Writeoffs/Impairments - G/L

AFFO

FFO - noncash S/L rent adj - recurring maintenance capex & leasing commission

NAV

Value of operating real estate + Cash and accounts receivable − Debt and other

Inter-Temporal Rate of Substitution

High = Save

Low = Spend

Risk Free Rate from Utility

(1/E(m1)) - 1

Price of asset with ITRS and rf

Cov() = risk premium

Interest rate short and long term

Rshort = R + inflation

RLong = R + inflation + theta

• theta = uncertainty about inflation

Taylor’s Rule

Break Even Inflation Rate

• Yield on non-inflation index bond - yield on inflation index bond

• inflation + theta

Required Rate of Return for Credit Risky Bond

Discount Rate for Equities

discount rate for commercial real estate

• R + π + θ + γ

  • γ = additional risk premium for credit risk = credit spread

• discount rate for equity = R + π + θ + γ + κ

  • κ = additional risk premium relative to risky debt for an investment in equities

  • λ = equity risk premium = γ + κ

• discount rate for commercial real estate = R + π + θ + γ + κ + φ

  • κ = risk premium for uncertainty about terminal value of property (similar to the equity risk premium)

  • φ = risk premium for illiquidity

Active Return /w Active Weights

• Active Return

  • E(R_A) = E(R_P) − E(R_B)

• Active Weight

  • E(R_A) = ΣΔw_iE(R_i)

    • Δw_i = active weight of security i = w_Pi  ​− ​w_Bi

Active Return Decomposition

Return in Benchmark + Active Return in Portfolio

Risk Adjusted Active Return (alpha)

Rp - β * Rb

• R_P is the portfolio's return.

• R_B is the benchmark's return.

• β is the portfolio's beta relative to the benchmark

Sharpe and Information Ratio

SR = Rp - Rf / stdp

market neutral funds SR = IR

Optimal Active Risk

SR w/ optimal active risk & Portfolio total risk

Information Coefficient

Corr(forecasted, realized returns)

• measurement of skill

IC	Meaning
+1	Perfect forecasting skill
0	No skill (forecasts unrelated to outcomes)
−1	Forecasts perfectly wrong (consistently backwards)

Transfer Coefficient

Corr(Active Weight, Optimal Weight)

• For an unconstrained active portfolio, the active weights will be equal to the optimal weights and TC = 1.

• For a constrained portfolio (e.g., constraints on short positions or active risk), TC may be less than 1.

Breath (BR)

number of independent active bets each year

• if a manager takes active positions in 10 securities each month, then BR = 10 × 12 = 120.

Grinold Rule

μᵢ = IC × σᵢ × Sᵢ

IC = skill, σᵢ = risk, Sᵢ = score of security (standardized, variance of 1)

Expected active return

E(R_A) = Σ Δwᵢ μᵢ

Sum of weight differences × expected alphas

E(R_A) = TC × IC × √BR × σ_A

IR × σ_A

Optimal information ratio (unconstrained + constrained)

IR* = IC × √BR

Unconstrained portfolio

IR = TC × IC × √BR

TC = transfer coefficient

Optimal expected active return (unconstrained)

E(R_A)* = IC × √BR × σ_A

IR* × σ_A

Optimal active risk (constrained portfolio)

σ_A* = TC × (IR*/SR_B) × σ_B	IR* = unconstrained IR; SR_B = benchmark Sharpe ratio

Sharpe ratio of constrained portfolio

SR_PC = √(SR_B² + TC² × IR*²)

Benchmark Sharpe squared plus TC² × IR*²

Expected return given IC, and TC

Expected active return given IC	E(R_A | IC) = TC × IC × √BR × σ_A	Same as the constrained expected active return from the first table
TC²	TC² = share of active return from skill	Fraction of active return explained by the forecasts
Noise	1 − TC² = noise	Fraction of active return unrelated to skill

IC from market timing

IC = 2(% correct) − 1

If correct 50% of the time, IC = 0

Std. dev. of return differences (R_x − R_y)

$$\sigma_{C} = \sqrt{\sigma_{x}^{2} - 2\,\sigma_{x}\,\sigma_{y}\,r_{x,y} + \sigma_{y}^{2}}$$

Active annualized risk and return

Tilts

Risk Budget Active / Full Active Risk

breadth adjustment for correlated decisions

maximum spread

Total Cost / Round Trip trading cost

total cost = round-trip trading cost + management fees

round-trip trading cost = round-trip commission + spread

spread = bid ask spread

APT Equation

E(R_P) = R_F + β_P,1(λ₁) + β_P,2(λ₂) + ... + β_P,k(λ_k)

Macro 2 Factor Bond returns

R_i = E(R_i) + b_i1F_GDP + b_i2F_QS + ε_i

Fundamental Equity Model

R_i = a_i + b_i1F_P/E + b_i2F_SIZE + ε_i

Active Return

Active Risk / TE

Active risk = tracking error = σ_{(RP – RB)}

active risk squared

active factor risk + active specific risk

Cahart Model

E(R) = R_F + β₁RMRF + β₂SMB + β₃HML + β₄WML

where:   

• E(R) = expected return      

• RF = risk-free rate of return

• RMRF = return on value-weighted equity index − the risk-free rate   

• SMB = average return on small cap stocks − average return on large cap stocks   

• HML = average return on high book-to-market stocks − average return on low book-to-market stocks  

• WML = average returns on past winners − average returns on past losers

Portfolio std

VaR

[E(r ) - alpha *(port std)](-1)

annualized mean/ std

mean * n

std * sqrt(n)

change in call price

$$\Delta V \approx \text{delta}\,\Delta P + \frac{1}{2}\,\text{gamma}\,(\Delta P)^{2} + \text{vega}\,\Delta\sigma$$