CAIA L2 - Equations

Question 1

Formula

Expected Real Return
of Private Equity
and Public Equities

1.5 - Demystifying Illiquid Assets

Answer 1

ER = ( y’u’ + g’u’ ) + [ (D/E) * (r’u’ - k’d’) ] + m - f

y’u’ = Dividend yield
g’u’ = real earnings per share growth rate
(D/E) = leverage (Debt to Equity)
unlevered ER => r’u’ = y’u’ + g’u’
m = multiple expansion
f = PE fees

ER = ( y’pub’ + g’pub’ ) + m’pub’ - f’pub’
m’pub’ = ~0
f’pub’ = ~10 bps

1.5 - Demystifying Illiquid Assets

Question 2

Formula

Target equities investment of the CPPI re-balance

1.6 - An Introduction to Portfolio Rebalancing Strategies

Answer 2

Target equities investment = M *(Portfolio Value - Floor Value)

M= multiplier (constant proportion, higher than 1)

1.6 - An Introduction to Portfolio Rebalancing Strategies

Question 3

Formula

Return of Illiquid asset
(in dinamic rebalancing strategy,
using a liquid financial instrument, like future)

1.6 - An Introduction to Portfolio Rebalancing Strategies

Answer 3

return on illiquid asset ‘t’ = R’f’ + α + (β × Futures’t’ ) + ε

R’f’ = risk-free rate of return

α = alpha of the illiquid asset portfolio

β = beta of the illiquid asset relative to futures

ε = tracking error

1.6 - An Introduction to Portfolio Rebalancing Strategies

Question 4

Formula

Weight of Futures in a
dynamic rebalancing using futures

1.6 - An Introduction to Portfolio Rebalancing Strategies

Answer 4

F’t’ =[ (α / R’F,t’) + β ] * ( k’t’ - w’t’ )

F’t’ = weight of futures position (i.e., the proxy)

R’F,t’ = expected return on the futures

k’t’ = optimal weight of risky asset

w’t’ = current weight of the illiquid risky asset

1.6 - An Introduction to Portfolio Rebalancing Strategies

Question 5

Formula

Vacicek Model

3.1 - Modeling Overview and Interest Rate Models

Answer 5

r’t+1’ = r’t’ + k(μ−r’t’) + σ ε’t+1’
Assumes:
constant volatility
mean reversion

3.1 - Modeling Overview and Interest Rate Models

Question 6

Formula

CIR Model
(Cox, Ingersoll, and Ross model)

3.1 - Modeling Overview and Interest Rate Models

Answer 6

r’t+1’ = r’t’ + k(μ−r’t’) + √(r’t’) σ ε’t+1’
“corrects” Vacicek => disallows negative rates

3.1 - Modeling Overview and Interest Rate Models

Question 7

Formula

Ho-Lee Model

(one of the arbitrage-free models)

3.1 - Modeling Overview and Interest Rate Models

Answer 7

r’t+1’ = r’t’ + θ’t’ + σε’t+1’

3.1 - Modeling Overview and Interest Rate Models

Question 8

Formula

Up rate = f(r’d’)
r’u’ = ?

BDT model
(one of the arbitrage-free models)

3.1 - Modeling Overview and Interest Rate Models

Answer 8

r’u’ = r’d’ e^(2σ)

3.1 - Modeling Overview and Interest Rate Models

Question 9

Formula

Expected Loss E[Loss]
Loss Given Default LGD
Recovery Rate RR

3.2 - Credit Risk Models

Answer 9

E[Loss] = LGD × PD
LGD = EAD × (1 – RR)
RR = present value of recovered sum / EAD

3.2 - Credit Risk Models

Question 10

Formula

Equity Value of
Merton Model

3.2 - Credit Risk Models

Answer 10

E’t’ = A’t’×N(d) − K×e^(−r×t) × N(d−σ’A’√τ)

r = annualized short-term interest on risk-free debt
τ = T – t (i.e., time left to debt maturity)
σ’A’ = annualized volatility of the asset rate of return
N (⋅) = cumulative probability function
d = standard normal distribution

3.2 - Credit Risk Models

Question 11

Formula

Debt Value of
Merton Model

3.2 - Credit Risk Models

Answer 11

D’t’ = K × e^(–(r + s’t’)×τ)
s’t’ = annual spread due to credit risk
s’t’ = −(1/τ) × ln[ N(d−σ’A’√τ) + (A’t’/K) × e^(r×τ) × N(−d) ]

3.2 - Credit Risk Models

Question 12

Formula

KMV credit risk model

3.2 - Credit Risk Models

Answer 12

σ’E’ = (A’t’ / E’t’) Δ σ’A’

σ’A’ = asset volatility

3.2 - Credit Risk Models

Question 13

Formula

Distance to Default (DD)
(KMV Model)

3.2 - Credit Risk Models

Answer 13

DD’t’ = (A’t’ − K) / (A’t’ × σ’A’)

K = default trigger
σ’A’ = asset volatility

3.2 - Credit Risk Models

Question 14

Formula

Expected Default Frequency (EDF)
(KMV Model)

3.2 - Credit Risk Models

Answer 14

EDF = DD’n,default’ / DD’n,total’

DD’n,default’ = the percentage of firms that defaulted within a one-year time horizon when their asset values were within n standard deviations away from default
DD’n,total’ = the percentage of total firms with n standard deviations away from default this represents

3.2 - Credit Risk Models

Question 15

Formula

P’t’ - probability that a firm can survive for t years
(Reduced-form models)

3.2 - Credit Risk Models

Answer 15

p’t’ = e^(–λ×t)

λ = default intensity
(1/λ) = expected time until default (usually in years)

Probability that default could occur between time s and t, and assuming no default until time s:
p(s) – p(t) = e^(–λ ×s) – e^(–λ ×t)

3.2 - Credit Risk Models

Question 16

Formula

D’0’ - Risky Debt With Default Intensity
and
D’0’ - Risky Debt With Default Intensity + Recovery Rates
(Reduced-form models)

3.2 - Credit Risk Models

Answer 16

D’0’ = e^(−(r+λ)×T) × K

λ = default intensity
(1/λ) = expected time until default (usually in years)

D0≈ e^{−[r+λ×(1−RR)]×T} × K

s’t’ = λ × (1 – RR)

3.2 - Credit Risk Models

Question 17

Formula

Altman’s Z-Score Model

3.2 - Credit Risk Models

Answer 17

Z =
(1.2 × X1) + (1.4 × X2) + (3.3 × X3) + (0.6 × X4) + (1 × X5)

• Default (or distressed) group: Z < 1.81
• Gray zone: 1.81 ≤ Z ≤ 2.99
• Nondefault (or safe) group: Z > 2.99
• X1: working capital/total assets. The ratio is a measure of liquid assets to total capitalization and is an indicator of liquidity. A firm with operating losses will see declining working capital.
• X2: retained earnings/total assets. The ratio is a measure of cumulative profitability and measures the relative size of the amount of a firm’s reinvested earnings, losses, or both. The ratio also indirectly measures leverage because high ratios may be indicative of more profit retention and less use of debt.
• X3: earnings before interest and taxes/total assets. The ratio is a measure of productivity independent of leverage and taxes. It is most appropriate for measuring corporate failure.
• X4: market value of equity/book value of total liabilities. Solvency. The ratio measures the magnitude of decline in a firm’s assets before they are exceeded by the value of liabilities and the firm becomes insolvent.
• X5: sales/total assets. The ratio is a measure of activity (turnover) by looking at the ability of a firm’s assets to generate sales.

3.2 - Credit Risk Models

Question 18

Formula

Fama-French Model

Fama-French-Carhart Model

Fama-French five-factor model

3.3 - Multi-Factor Equity Pricing Models

Answer 18

3 factors = Market Size Value
E’Ri’ – R’f’ = β’i’[E(R’m’) – R’f’] + β’1i’[E(R’s’ – R’b’)] + β’2i’[E(R’h’ – Rl’)]

4 factors = Market Size Value Momentum
E’Ri’ – R’f’ = β’i’[E(R’m’) – R’f’] + β’1i’[E(R’s’ – R’b’)] + β’2i’[E(R’h’ – Rl’)] + β’3i’[E(R’w’ – R’d’)]

5 factors = Market Size Value Robustness Conservativeness

Alternative assets do not usually have factor exposures that mirror traditional assets

3.3 - Multi-Factor Equity Pricing Models

Question 19

Formula

The Present Value of Stochastic Discount Factors

3.3 - Multi-Factor Equity Pricing Models

Answer 19

PV = E[(π’u’ × m’u’ × x’u’) + (π’d’ × m’d’ × x’d’)]

π’u’ = the probability of the up state
π’d’ = the probability of the down state
m = stochastic discount factor

3.3 - Multi-Factor Equity Pricing Models

Question 20

Formula

Utility Functions
in Terms of:
- Expected Returns and Variance
- Value at Risk

3.4 - Asset Allocation Processes and the Mean-Variance Model

Answer 20

Utility Functions in Terms of Expected Returns and Variance:
E[u(w)] = μ − [ (λ/2) × σ^2]

µ= mean return on an investment
λ= constant for the level of risk aversion
λ = [ E(R’p’)−R’f’ ] / (σ’p’^2)
σ^2= variance of an investment

Utility Functions With Value at Risk

E[u(w)] = μ − [ (λ’VaR’/2) × VaRα]

3.4 - Asset Allocation Processes and the Mean-Variance Model

Question 21

Formula

risky asset’s weight “w” of
Mean-Variance Optimization With Risky and Riskless Assets

and

with growing liabilities

3.4 - Asset Allocation Processes and the Mean-Variance Model

Answer 21

w = (1/λ) * [E(R−R’0’)] / (σ^2)

λ= constant for the level of risk aversion
λ = [ E(R’p’)−R’f’ ] / (σ’p’^2)
σ^2= variance of an investment
__–__–__
with growing liabilities:
w = { (1/λ) * [E(R−R’0’)] / (σ^2) } + L [δ/(σ^2]

δ= covariance between the growth rates in the liabilities and assets

L = size of the liabilities relative to the assets (e.g., if the value of the liabilities is 30% higher than the value of the assets, then L =1.3)

3.4 - Asset Allocation Processes and the Mean-Variance Model

Question 22

Formula

Hurdle Rates

3.4 - Asset Allocation Processes and the Mean-Variance Model

Answer 22

E[R’New’] – Rf > [E(R’p’) – R’f’] × β’New’

E[R’New’] = new asset’s expected return
E[R’p’] = optimal portfolio’s expected return
R’f’ = riskless rate
β’New’ = new asset’s beta in relation to the optimal portfolio

A new asset “New” is included to a portfolio “p” if the expression above is true

3.4 - Asset Allocation Processes and the Mean-Variance Model

Question 23

Formula

Return adjustment for illiquidity
(Mean-variance optimization Framework with
Liquidity Penalty Function)

3.4 - Asset Allocation Processes and the Mean-Variance Model

Answer 23

max R’p’−[(λ/2) × σ’p’^2] − ϕL’p’
L’p’ = ∑w’i’ × L’i’

ϕ = a positive number that represents investor preference for liquidity

Most illiquid assets have a penalty of 1 (i.e., L’i’ = 1), which reduces expected returns by ϕ.

3.4 - Asset Allocation Processes and the Mean-Variance Model

Question 24

Formula

Total risk of a portfolio

3.5 - Other Asset Allocation Approaches

Answer 24

σp = (ρF1 × σF1 × b1) + (ρF2 × σF2 × b2 ) + (ρε × σε)

3.5 - Other Asset Allocation Approaches

Question 25

Formula

Expected return
(for various asset classes)

4.1 - Types of Asset Owners and the Investment Policy Statement

Answer 25

expected return = short-term real riskless rate + expected inflation + risk premium

Short-term real riskless rate
- Stable, minimum 0%
- Lower than real growth rate

Expected inflation
- Much less stable given its dependence on central bank policies and long-term growth

Risk premium per asset class
- Assume historical amounts if past estimates of volatilities, correlations, and risk exposures are unchanged

4.1 - Types of Asset Owners and the Investment Policy Statement

Question 26

Formula

Equity value
(in terms of plan’s funded status)

4.3 - Pension Fund Portfolio Management

Answer 26

E’t’ = OA’t’ – OL’t’ + (A’t’ – L’t’)

E’t’ = Equity
OA’t’ = operating assets
OL’t’ = operating liabilities
A’t’ – L’t’ = funded status of the DB plan

4.3 - Pension Fund Portfolio Management

Question 27

Formula

Expected economic life (EL)

4.3 - Pension Fund Portfolio Management

Answer 27

EL =
– [1/ln(1+R) ] × ln[ (payment – R×assets) / payment]

R = expected after-fee investment return after inflation (i.e., the real return after fees)

payment = annual spending in the first year of retirement (expected to increase at the rate of inflation)

assets = value of the portfolio

Dica: “P-rassets sobre P”
Dica 2: Tem muitos itens duplicados (R, payment, ln, “-“). Os 3 primeiros tem tanto no numerador quanto no denominador
Dica 3: se tiver alíquota, PMT = bruto/(1-alíquota)

4.3 - Pension Fund Portfolio Management

Question 28

Formula

present value of a growth (growing) annuity
or “cost of a annuity”

4.3 - Pension Fund Portfolio Management

Answer 28

PV’ordinary annuity’ =
[initial payment/(r-g)] × {1–[(1+g)/(1+r)]^n}

payment = fixed annual annuity payment
r= discount rate
g= growth rate of annuity
n= number of periods involved

4.3 - Pension Fund Portfolio Management