1.1 Rates of Return and 1.2 Components and Measures of Return

Financial instruments

standardized means of trading financial assets in markets

• Equity and debt securities

• Tradable standardized contracts (i.e derivatives)

Financial indicators

observable measures of value that do not generate cash flows, but impact returns

• interest rates

• currency exchange rates

• market indexes

Single Period Return

Price return, capital distribution return and total return

Price return

(P1-P0)/Po

Dividend Yield (Capital distribution return)

R distribution= Income (dividend amount)/P0

Only include dividend yield in total return!

Asset’s holding period return (total return)

• Also return of unlevered portfolio

= Return Price + return distribution

= (P1-P0+Div Income)/P0

Actual Return vs Expected Return

Actual Return = Ex post return

Expected return = ex ante return

Realized vs unrealized returns

Realized returns = capital distributions received + proceeds from selling

Unrealized Returns= returns based on current market price

Risk

Difference between expected returns (ex ante return) and actual returns (ex post return)

Returns Across Multiple Periods

Arithmetic + Geometric returns

Arithmetic mean return

(R1+R2+… Rn)/n

• Do not reflect multi-period compounding

• Better for single periods)

Geometric Mean return

Where n= number of periods

((1+r1)*(1+r2)*(1+rn))^(1/n) — 1

• Always <= arithmetic mean (as n increases)

• Reflects compounding; Better for multiple periods

Annualized returns (messy periods)

= (1+ r)^n -1

n days= (365/days) -1

n weeks=(52/weeks) -1

For months= (12/months)-1

Annualized Returns

(1 + return during the period)^c – 1

Where c: monthly=12, quarterly=4, semiannual=2\

Continuous Compounding

ln(end value/beginning value) NOT Ln of %s

Summary of 1.1 LOS 1a

For single period returns

r= (P1-Po+ income) / Po

For multiple period returns (geometric return)

rg = ((1+r1)(1+r2)…*(1+rn)) *(1/n) - 1, where n = number of periods

Annualized Return

= (1+ return during period) ^ c -1, where c= number of periods per year

Continuous Compounding

= Ln(end value/beg value)

Interest Rates

RF Rate (nominal risk-free rate) + Risk premium rate

Risk Free Rate (Treasury Bill)

This is also referred to as the nominal risk-free rate)

Includes real risk free rate and expected inflation rate

Real Risk-free rate

(1+ normal risk - free rate)/(1+inflation rate)

Or approximation: nominal risk free rate - expected inflation risk

Real Rate

Real rates are adjusted for inflation (inflation has been subtracted, but this does not mean the interest rate is risk free)

(1+nominal rate)/(1+inflation rate) —1

Risk Premiums

Risk premiums= Nominal Rate - RF

of risk, all increasing the required rate of return.

• Bond risk premiums (default, liquidity and maturity risk)

• Stock risk premiums (size, value)

Capital Gains

(Sold Price - Initial Price) * number of shares, excluding dividends!

Gross Return

Total Return - Transaction costs

Net Return

Gross return - management fees + admin costs

After-tax return

after‐tax return = (price return)(1 − capital gains tax rate) +

(capital distribution return)(1 − capital distribution tax rate)

Note * capital gains is once shares are sold

Leveraged Return

(rp*(VD+VE) - rd*VD)/VE

rp= rate of return on that portfolio (unlevered)

rd= rate on leveraged portfolio

and VE= amount can invest without leverage)