CFA- Portfolio Construction - Math

HHI =

sum of squared weights of individual stocks in a portfolio

If stocks are equal weighted, then 1/HHI =

the number of stocks in a portfolio

If stocks are market cap weighted, then 1/HHI

is less than the number of stocks in the portfolio

Interpreting the 1/HHI number

The effective number of shares has the same concentration risks as an equally weighted 1/HHI-stock portfolio

Portfolio Leverage =

portfolio return/portfolio equity

Coupon income (of the 5 steps of FI return) in %

coupon payment/current bond price

Leveraged portfolio return =

borrowing rate + (leverage/equity)*(return on asset - borrowing rate)

rolldown return (in the five steps of FI return)

Projected ending price if no change in yield curve/begning price-1

Rolling yield =

coupon return + rolldown return

Rebate rate (of securities lending) =

collateral earnings rate - security lending rate

Macaulay duration

PV weighted average of times to cash flows

Modified duration

Macaulay duration * (1+ periodic discount rate)

Spread duration

change in relative value given a spread change

Leverage induced by future contracts

(notional value of contract - margin amount)/margin amount

Capital contribution =

rate of contribution* (committed capital - capital previously called)

Distribution in a year =

distribution rate * (net asset value x (1+ expected growth rate)

(alternatives) Net asset value in year t =

net asset value in previous year * (1+ growth rate) + capital committed - distribution

Multiple on invested capital (MOIC)

(Value of investments + distributions) / total invested capital

human capital =

PV of projected nominal income * % of survival, discounted by nominal risk free rate + risk premium for income volatility

Accrual tax FVIF=

(1+Return* (1-tax rate))^T

deferred tax FVIF=

[(1+growth rate)^T] - [(1+R)^T-1)]*tax rate

percentage of investment return lost to tax =

tax paid/gross gain

tax drag=

different between pretax gross returns and after tax future value

basis in tax calculations denotes

the initial purchase price

Future value with basis

FV of initial investment - (1-Basis)*tax rate

adjust for inflation when calculating future value, FV =

divide by (1+inflation rate)^t

funded ratio of DB plan =

fair value of plan assets/PV of DB obligations

spending of endowments in each period =

weight of spending previous year x [spending previous year (1+inflation)] + (1-weight) x (spending rate average AUM), weight can range from 1 (constant) to 0 (market value)

To preserve return, endowment nominal rate of return =

spending rate + higher education price index (HEPI)

% change in equity of banks and insurers =

change in assets x leverage multiplier - change in liabilities x (leverage multiplier-1)

leverage multiplier = (or if given capital equity ratio of X%)

asset/equity, (or 100/X)

Expected volatility of changes in market value of equity capital of banks (standard deviation) =

leverage multiplier² x variance of asset + (leverage multiplier -1)x liabilities variance - 2 x leverage multiplier x (leverage multiplier-1) x asset standard deviation x liabilities standard deviation x correlation of changes in asset and liabilities, then square root

modified duration of equities for banks and insurers=

(duration of asset x leverage multiplier )- (duration of liability x (leverage multiplier-1) x change in yield of liabilities relative to a 1% change in yield of assets)

Modified duration of equities for banks (what does it mean)

estimated fall in equity value for each 1% increase in yields on asset portfolio

market inside spread =

best ask - best bid

Interpreting/using the limit order book

Given quantity to buy, buy from the highest ask price first, then go down the list

Effective spread =

2 x side x (transaction price - midquote price), if buying, side = 1, if selling, side = -1

To compare effective spread with quoted bid-ask spread

If effective spread < quoted spread (basically transacted within market offer-bid), then price improvement

quoted spread =

market offer - bid

VWAP transaction cost =

trade size x side x (trade VWAP - benchmark VWAP), side = 1 if buy, side = -1 if sell

Interpreting a X% VaR estimate of $Y

assuming X% of the time, the maximum loss will be Y, or (1-X) of the time the loss will be larger than Y

market or benchmark VWAP

same math as VWAP, just for all trades that happened in the day

How do you determine whether there’s sufficient capital based on survival probability and monte carlo similations giving you portfolio value in different market conditions (different percentails)?

as long as at each given horizon, portfolio survival ratio (# of positive portfolio value trials/total trials ran on monte carlo simulation) is larger than combined survival ratio of all beneficiaries, then sufficient

annual return after tax (when used in calculations) =

annual return * (1-tax rate), then do (1+%)

average quoted spread

simple average of all given quoted spreads