CFA III Formulas

Aggregate MV of equity (V_t^e)

GDP_t × S_t^k × P/E_t

level of nominal GDP

share of profit in economy

PE Ratio

E(R_e) expected return on equity

V_t^e + D_t

MV of equity

Dividend Yield

Taylor Rule( r_target )

r_neutral + π_e + 0.5 ( GDP_e − GDP_trend ) + 0.5 ( π_e − π_target )

nominal Target Rate

Neutral Target Rate

Expected Inflation

Output Gap

Inflation Gap

Trade Deficits (X - M)

(S - I) + (T - G)

Cap Rates Positively Related to

Vacancy Rates

Cap Rates Inversely related to

Availability of Debt Financing

Utility

E/R - (0.005 × λ × σ²)
λ = Risk Aversion Coefficient

E/R = Expected Return

σ² = Variance (Square of Standard Deviation)

Grinold Kroner E(R_e)

(D/P − %ΔS) + %ΔE + %ΔP/E

Size Factor

A zero-dollar portfolio long in small stocks and short in large stocks

value-growth factor

A zero-dollar portfolio long in value (high book-to-market) stocks and short growth (low book-to-market) stocks

two-portfolio approach

The hedging portfolio could be constructed using various techniques such as cash flow matching, duration matching, and immunization.

Name few Risk Factors

volatility, liquidity, inflation, interest rates, duration, foreign exchange, and default risk

Strategic asset allocation

establishes a portfolio's long-term asset class exposures by integrating each element of investment policy with capital market expectations.It affords an investor the ability to control systematic risk exposures by aligning their risk and return objectives with the actual portfolio of investments.

Tactical asset allocation

involves adjustments away from the strategic mix to take advantage of short-term projections of relative asset class performance, also manages exposure to different assets classes by specifying a target allocation percentage range.

narrower rebalancing corridor

High asset class volatility

wider rebalancing corridor

Higher investor risk tolerance, Higher transaction costs

intertemporal consistency

Consistency over various time horizons

Cross-sectional consistency

refers to consistency across asset classes

financial stages that form the critical planning phase

1. Career development (age 35–50)

2. Peak accumulation (age 50–60)

3. Preretirement (age 60–65)

conditional VaR

The average loss in the tail of a distribution that lies beyond a VaR estimate,also known as expected tail loss and expected shortfall.

Empty voting

occurs when an investor borrows stock for the explicit purpose of exercising the voting rights attached to the stock.

In a stock loan

voting rights are transferred to the stock borrower.

Volatility of Equity σE

√M²σ²_A+ (M−1)²σ²_L−2×M×(M−1)×σ_A×σ_L×corr<sub>A,L

Multiplier = Assets / Equity</sub>

Duration of Equity Capital (D_E)

( D_A×M ) − (D_L×(M−1) × (△i / △y) )

Leverage multiplier = 1 / equity capitalization ratio

Rolling yield

coupon yield + rolldown return

Coupon Yield / Income

annual coupon / current average bond price

Rolldown return

( Expected bond price in one year / Current average bond price ) - 1

BHB Allocation Effect

( W_portfolio - W_BM ) × Return_BM

BHB Selection Effect

W_BM × [ Return_portfolio - Return_BM ]

BHB Interaction Effect

( W_portfolio - W_BM ) × [ Return_portfolio - Return_BM ]

Paper Return

Total Shares × (Decision Price - Closing Price)

Actual Return

Shares Sold × (Execution Price - Closing Price) - Commissions

Information Shortfall

Paper Return - Actual Return

Information Shortfall in Basis Points (BPS)

( (Paper Return - Actual Return) / (Total Shares × Decision Price) ) × 100²

Market-adjusted cost (bps)

arrival cost (bps) − β × index cost (bps)

arrival cost (bps)

side × ((execution price – arrival price) / arrival price ) × 100²

index cost (bps)

side× ( (index VWAP – index arrival price) / index arrival price ) × 100²

volume-weighted average price (VWAP)

approach waits until the end of a trading day, but it runs the risk that the price has moved in the wrong direction and the trade remains unfilled.

Algorithmic Trading Strategy

• Relatively small trade (maximum size of 15% of the daily volume)

• Low bid-ask spread

• High urgency

time-weighted average price (TWAP)

breaks up the trade into specific time slots during the day and also eliminates trade outliers

dark strategy

recommended for trades with high bid-ask spreads or large trade sizes to market volumes.

Semi-Liquid

Private investments requiring one year or less to liquidate(greater than one quarter but less than one year)

Liquid

Investments requiring less than one quarter to liquidate

highly liquid

time to cash of less than one week

Illiquidity premium (%)

expected return on illiquid asset (%) − expected return on marketable asset (%)

Higher Tracking Error

Higher Sampling, Higher Expense Ratio

Risk-oriented factor-weighted

focuses on minimizing portfolio variance, which is an historically focused approach.There is no effort to forecast future volatility, which is one drawback to this method.

equally weighted index

as soon as stock prices change, portfolio weights are no longer equal and the portfolio will need rebalancing. have greater risk and have a greater weighting in small-cap, not large-cap, stocks.

packeting

when a large-cap company's capitalization decreases so that it qualifies as mid-cap stock, a portion of the stock's market capitalization is moved to the mid-cap index on the reconstitution date. If the stock still meets the criteria for inclusion in the mid-cap index at the next reconstitution date, the remainder of the stock is moved from the large-cap to the mid-cap index.

Buffering

practice of establishing a threshold level for a change in a firm's capitalization that must be met before moving it from one index to another on a reconstitution date. A stock is not actually moved into a new index until it moves beyond the buffer zone.

Cash drag

refers to the low return from the cash that equity funds hold as a portion of their portfolio. Cash balances reduce returns in rising equity markets, but they typically increase returns in falling equity markets.

expected compound return of an asset (R_g)

R_a − σ² / 2

Confirmation bias

is the tendency by investors to only look at investment information that validates their beliefs and ignore information that may contradict it. This bias results in poorly diversified portfolios with unnecessary risk exposures and large holdings of underperforming securities.

Regret aversion bias

causes investors to be overly cautious for fear of making bad decisions. This bias results in managers holding on to positions for too long and missing out on potentially much more profitable investments.

The illusion of control Bias

causes investors to believe they can influence investment outcomes, and as a result, select outperforming securities. This may result in excessive numbers of trades or concentrated positions.

overconfidence bias

is an emotional bias that causes investors to have too much faith in their own investment selection abilities, causing managers to underestimate risks and overestimate returns.

BPV_L (Basis Point Values of Liabilities)

MV_L × Bond Duration × 0.0001

BPV_A (Basis Point Values of Assets)

MV_A × Bond Duration × 0.0001

Duration Gap

BPV_L- BPV_A

BPV_Futures

BPV_CTD ÷ Conversion Factor

Number of Contracts to Hedge

Duration Gap / BPV_Futures

Opportunity cost(bps)

[(Closing price – decision price) / benchmark portfolio] × shares missed

benchmark portfolio = original number of shares × decision price

Delay cost(bps)

[(Arrival price – decision price) / benchmark portfolio] × shares purchased

benchmark portfolio = original number of shares × decision price

Trading Cost (bps)

[(Execution price – arrival price) / benchmark portfolio] × shares purchased

benchmark portfolio = original number of shares × decision price

Implementation Shortfall (bps)

(Paper Portfolio Return - Actual Return) / (original number of shares × decision price) * 100²

Paper Portfolio Return

original number of shares × ( Closing Price - Decision Price)

Actual Gain

[ Shares Purchased × (Closing Price - Actual Execution Price) ] - [Shares × Commission price ]

%ΔPV^full (Percentage Change in Price)

(ModDur × Δyield) + [½ × convexity × (Δyield)²]

Asset BPV is below liability BPV

When do we Purchase bond contracts

price weighting

replicates holding an equal number of shares in each security

fundamental weighting

allows the portfolio to tilt toward the factors expected to outperform and away from the ones expected to underperform. It is a form of smart beta.

Event-driven strategies

focus on market inefficiencies that result from "events" like the proposed acquisition.

Pairs trading

takes long and short positions in two similar stocks, but the motivation is to profit from relative mispricing between the two stocks that have historically been highly correlated.

Market microstructure-based investing

seeks to benefit from extremely short-lived (few milliseconds) temporary mispricings that result from demand-supply order imbalances.

Singer-Terhaar model

(Degrees of Integration × Fully Integrated Market risk premium) + (Degrees of Segmentation × Fully Segmented Market risk premium)

Degrees of Segmentation = (1 - Degrees of Integration) if not provided

Fully Integrated Market Risk Premium

Correlation with global market × Volatility or Standard Deviation × Sharpe ratio for global and segmented markets

Fully Segmented Market risk premium

1 × Volatility or Standard Deviation × Sharpe ratio for global and segmented markets

A factor-based variance/covariance matrix

A matrix which is biased and inconsistent

Capitalization rates

positively related to vacancy rates and inversely related to the availability of credit

Dynamic Hedging

is a portfolio insurance technique used to increase or reduce exposure to underlying securities or asset classes.

Reverse optimization

improves the quality of expected returns that are inputted into a forward-looking optimization.

Resampling

uses Monte Carlo simulation to estimate a large number of potential capital market assumptions for MVO. These assumptions lead to an equal number of simulated frontiers.

Marginal Sharpe Ratio

( E(R)-R_f ) / MCTR_i

Marginal contribution to Portfolio Risk (MCTR_i)

asset beta relative to portfolio × portfolio standard deviation
β_i × δ_p

Absolution contribution to portfolio risk (ACTR_i)

W_i × MCTR_i

liquidity is higher for bonds

(1) issued at close to par or market rates, (2) new, (3) large in size, (4) issued by a frequent and well-known issuer, (5) simple in structure, and (6) of high credit quality.

Portfolio is optimized

if the excess return / MCTR of an allocation equals the portfolio Sharpe ratio

Chronic physical climate risks

the gradual impacts of climate change. Examples include rising sea levels and rising temperatures from global warming.

acute physical climate risks

are one-off weather events such as wildfires, hurricanes, storms, droughts, and heat waves.

Climate transition risk

is the risk of being too slow to transition to the new zero-carbon world, effectively left behind with an outdated business model.

Mitigation strategies

reduce the reliance on fossil fuels and carbon-intensive resources. Increases in carbon regulation, restrictions on usage, and carbon taxes will likely increase input costs.

Adaptation strategies

look to prosper in the zero-carbon world, investing in markets, companies, and technologies that are likely to benefit from the transition.

modified duration of an insurance firm's equity capital (D*_E)

(A/E )D*_A–((A/E)–1))D*_L × (Δi / Δy)

just transition

considers the impacts of climate change mitigation and adaptation on local communities and individuals in developing countries, providing communications and support to those impacted.

Five-step liquidity risk management process

1. Establish liquidity risk parameters.

2. Assess the liquidity of the current portfolio and monitor the evolution over time.

3. Develop a cash flow model and project future expected cash flows.

4. Stress test liquidity needs and cash flow projections.

5. Develop an emergency plan.

lectronic quote matchers

Who exploits the option values of orders waiting to be filled (a.k.a. standing orders)

Electronic front runners

are low-latency traders who use artificial intelligence methods to identify when other traders are trying to fill orders on the same side of the market.

Electronic arbitrageurs

look across markets for arbitrage opportunities to buy undervalued and sell overvalued securities.

market spread

lowest ask – highest bid

Treynor measure

(R_P − R_F) / β_P