Securities Analysis 3FB3 February 26 , 2014
Chapter 4: Return and Risk: Analyzing the Historical Record
Determinants of the Level of Interest Rates
Factors that determine the level of interest rates:
• The supply of funds from savers, primarily households
• The demand for funds from businesses to be used to finance physical investments in
plant, equipment, and inventories
• The government’s net supply and/or demand for funds as modified by actions of the
Real and Nominal Rates of Interest
The consumer price index CPI measures purchasing power.
Nominal interest rate: the growth rate of your money
Real interest rate: the growth rate of purchasing power
R – nominal rate, r – real rate, i – inflation rate
r = R – I ▯approximation rule
r = (Ri) / (1+i)
^ Shows that the approximation rule overstates the real rate by the factor 1/(1+i)
The approximation rule is more exact for small inflation rates and is perfectly exact for
continuously compounded rates.
Because future inflation is risky, the real rate of return is risky, even if the nominal rate is risk
The Equilibrium Real Rate of Interest
Supply, demand, and government actiths determine the real interest rate. Nominal int rate = real
rate + expected inflation. Thus, 4 factor affecting interest rate is the expected rate of inflation.
Equilibrium is at the point of intersection of the supply and demand curves of funds.
Gov and the central bank can shift supply and demand curves right or left through fiscal and
monetary policies. A forecast that indicates higher government borrowing increases expected
future interest rates. The central bank can offset such a rise through an expansionary monetary
policy, which will shift the supply curve to the right.
The Equilibrium Nominal Rate of Interest
Irving Fisher: nominal rate increase with inflation rate, Fisher equation:
R = r + E(i) ▯E(i) is the current expectation of the inflation rate that will prevail over the coming
Higher nominal rate predict higher inflation rates.
Nominal interest rates can be viewed as the sum of the required real rate on nominally risk free
assets, plus a “noisy” forecast of inflation.
Taxes and the Real Rate of Interest
Tax liabilities are based on nominal income and the tax rate determined by investor’s tax bracket.
“Bracket creep” is when nominal income grows due to inflation and pushes taxpayers into higher
Indexlined tax brackets do not provide relief from the effect of inflation on the taxation of
savings. [Type text] [Type text] [Type text]
t ▯tax rate
After tax interest rate: R(1t)
The real aftertax rate is approx. the aftertax nominal rate minus the inflation rate
R(1t) – i = (r+i)(1t) = r(1t) – it
Thus, aftertax real rate of return falls as the inflation rate rises.
The tax code does not recognize that part of your return is lost to inflation, so you are still taxed
4.2 Comparing Rates of Return for Different Holding Periods
Zerocoupon bond: sold at a discount from par value, return from the difference between the
purchase price and the ultimate repayment of par value.
P(T) ▯price, T years, i.e. $100 par value
Calculate total riskfree return: percentage increase in the value of the investment over the life of
rfT) = [100 / P(T)] – 1
For T = 1, ^ provides the risk free rate for an investment horizon of 1 year
To compare the returns on investments with differing horizons, reexpress each total return as a
rate of return for a common period. Usually use effective annual rate (EAR) ▯the percentage
increase in funds invested over a 1 year horizon.
For a 1 year investment, the EAR equals the total return, r (1),fand the gross return, (1+EAR), is
the terminal value of a $1 investment.
Annual Percentage Rates
Rates on ST investments are often annualized using simple rather than compound interest. These
are called annual percentage rates (APRs).
The difference between APR and EAR grows with the frequency of compounding. As T
approaches zero, we approach continuous compounding (CC), and the relation of EAR to the
APR, denoted by r foccthe continuously compounded case: 1+EAR = e rcc
ln (1+EAR) = r cc
With less frequent compounding, a higher APR is necessary to provide an equivalent effective
Bills and Inflation, 19572009
If the real rate remains approximately constant, the Fisher equation predicts a close connection
between inflation and the rate of returns on Tbills.
Wealth index – obtained for a current year by compounding the portfolio value from the end of
the previous year by 1+r, the gross rate of return in the current year.
4.4 Risk and Risk Premiums
Dividend yield dividends earned per dollar invested ▯$100/share, yearly div of $4, dividend
yield is 4% th
Securities Analysis 3FB3 February 26 , 2014
Holdingperiod return (HPR) will depend on the price that will prevail one year from now, for
above example. If it was $110, your capital gain would be $10 and HPR 14%
HPR is capital gain income plus dividend income per dollar invested
HPR = ending price of a share – beginning price + cash dividend / beginning price
This assumes dividend is paid at end of period, ignores reinvestment income if not.
Dividend yield + capital gains yield = HPR
Expected Return and Standard Deviation
The expected/mean rate of return is a probabilityweighted average of the rates of return in all
The standard deviation of the rate of return is a measure of risk. It is the square root of the
variance, which in turn is the expected value of the squared deviations from the expected return.
The higher the volatility in outcomes, the higher the average value of these squared deviations.
Variance and SD measure the uncertainty of outcomes.
As long as the probability distribution is symmetric about the mean, the standard deviation is an
adequate measure of risk.
Excess Returns and Risk Premiums
How much to invest in index fund? Must ask how much of an expected reward is offered.
Measure the reward as the difference between the expected HPR on the index stock fund and the
risk free rate (rate you can earn by leaving money in RF assets, such as Tbills, money market
funds, or the bank). This differen