CFA - Asset Allocation - Math

market value of equity =

nominal GDP x earnings/GDP x P/E ratio

Real GDP growth (in terms of labor)

Labor input growth + labor productivity growth, TFP is already captured in labor product

Long-term capital gains in equity markets

% change in nominal GDP + % change in profits/GDP + % change in P/E

Long term domestic market equity return

nominal gdp growth + dividend yields

Maco - net exports =

net private saving + government surplus

Macaulay Duration =

YTM * Duration

Gordon Growth Model Equation

PV = dividend in period 1/(dividend growth - long term growth)

Grinold and Kroner Model For Equity Returns

Dividend yield + (expected % change in total earnings - expected percent change in shares outstanding) + expected percent change in the P/E Ratio + expected inflation (if real earnings is given)

Income Return Component of Grinold and Kroner

dividend yield - increase in shares outstanding

Repricing Return Component of Grinold and Kroner

Change In P/E

Singer-Terhaar Model - Risk Premium Assuming Full Integration

risk premium = correlation with global market x local market standard deviation x global market sharpe ratio + local market illiquidity premium

Sharpe Ratio

Excess return or risk premium/standard deviation

Singer-Terhaar Model - Risk Premium Assuming Full Segmentation

Risk premium = local market standard deviation * local market sharpe ratio + illiquidity premium, if local sharpe not given, use global sharp

Risk Premium for Semi-segmented markets using Singer-Terhaar Model

degree of integration x full segmented + (1-degree of integration) x full integration

Cap Rate =

Net operating Income/property value

Infinite Period Return On Real estate =

cap rate + net operating income growth

Finite Period Return On Real estate =

cap rate + net operating income growth rate - percentage change in cap rate

(Dornbusch overshooting mechanism) expected percent change in exchange rate using capital flows =

(1+ (domestic - foreign short term rates and risk premiums)) * spot exchange rate domestic/foreign

Utility maximization of portfolio =

expected market return - 0.5* lambda * variance of portfolio

Roy’s safety first ratio =

(return of portfolio - MAR)/standard deviation of portfolio

Pension plan surplus =

market value of investable portfolio assets - PV of pension liabilities

Pension funding ratio =

market value of assets/present value of liabilities

Surplus optimization - surplus return =

(change in asset value - change in liability value)/Initial asset value

Rule for optimal risk budgeting

all asset classes should have ratio of excess return to MCTR = Sharpe ratio

MCTR =

beta of asset classes * portfolio standard deviation

ACTR =

weight of asset in portfolio * MCTR

Portfolio standard deviation (in terms of ACTR)

sum of all asset class ACTR

Check if asset allocation are optimal

Does excess return/MCTR = portfolio sharpe ratio? Or are each asset classes excess return/MCTR equal?

Questions with EF corner portfolios and client portfolios

the given portfolio’s expected return, standard deviation, and asset allocation will be the weighted average of two corner portfolios closest to it

Determining investor’s minimum required return

treat like a bond: PV = today’s savings, N = investment period, PMT = required withdrawal each period, FV = future value required, find 1/Y

What constraints need to bee applied when including non-tradable asset in asset allocation

Its share need to = share of the current value of total tradable + nontradable assets that are being included

Pre-tax rebalancing range =

after tax rebalancing range * (1-tax rate)

taylor rule (nominal) =

neutral rate + expected inflation + 0.5* output cap + 0.5* inflation deviation from target

taylor rule (real) =

real neutral rate + 0.5* output cap + 0.5* inflation deviation from target

sharpe ratio after-tax (sharpe ratio of an taxable portfolio)

[asset return ( 1-tax rate on asset) - risk free rate (1-tax rate on interest)]/(standard deviation * (1-tax rate on asset)

Roy’s safety first ratio should be used when

investor concerned to downside risk and wants to minimize probably of loss exceeding X%

Given corner portfolios, target return, and risk free rate, how to find which corner portfolio to target and weight?

Find the corner portfolio with the highest sharpe ratio using the risk free rate, then w corner portfolio + (1-w) x risk free rate = target return, find w; w can be negative

Historical equity risk premium using bond yield plus risk premium method

historical equity return - historical 10-year bond yield

Shrinkage estimation calcullation:

parameter value x(recent historical average) + (1-parameter value)x current estimate

long run expected returnn on property =

long run NOI growth + cap rate; long run NOI growth should be close to GDP growth

Interpreting monte carlo simulation final portfolio value projection

X% probability that portfolio value would be lower than the projected value

trick when asked to meet a real return spending X in Y number of years but given portfolio return is nominal, what’s today’s required contribution?

geometricc difference of portfolio nominal return and expected annual inflation, then do PMT = X (real), N = Y, FV = 0, 1/Y = the geometricc difference, compute PV!

if excess return/MCTR is higher than portfolio sharpe ratio, that asset class allocation should be

increased