stuff from lesson 1 Flashcards
the fundamental 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 (real assets or capital formation)
- The government’s net supply demand for funds as modified by actions of the monetary authority
An interest rate
a promised rate of return denominated in some unit of account over some time period
risk-free rate
no default risk
The consumer price index (CPI)
measures purchasing power
nominal interest rate
the growth rate of your money
real interest rate
the growth rate of your purchasing power
r = R - i or r = (R - i)/(1 + i)
R: the nominal rate
r: the real rate
i: the inflation rate
is the future real rate known?
nah bro
what determines the real interest rate?
supply
demand
government actions
expected inflation
why do economists frequently talk as if there were a single representative rate economy wide?
because the many different interest rates economywide move together
the nominal rate equation
The so-called Fisher equation
R = r + E(i)
R: the nominal rate
r: the real rate
E(i): expected inflation
what does the Fisher equation imply when real rates are reasonably stable?
changes in nominal rates ought to predict changes in inflation rates
Tax liabilities are based on nominal income or real income?
nominal income
the after-tax interest rate
R(1 - t)
The real after-tax rate
what does this mean?
r = (R(1 - t) - i + i · t) / (1 - t)
the after-tax real rate of return falls as inflation rises
Zero-coupon bonds
bonds that are sold at a discount from par value
provide their entire return from the difference between the purchase price and the ultimate repayment of par value
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
how can we find effective annual rate (EAR)?
what is the formula?
we compound the periodic interest
EAR = (1 + nominal rate)^(payments per year) - 1
annual percentage rates
formula
simple interest
APR = ((1 + EAR)^r - ) / T
what is the formula to find the EAR when we, theoretically, reach the point of continuous compounding?
EAR = e^rcc - 1
rcc: The continuously compounded annual percentage rate
for the past 56 years (except since like 2001), what has been the driving force for the nominal rate of interest?
why?
inflation
because of the fisher equation
average rate of inflation in Canada his increased or decreased?
decreased
a wealth index
The progression of the value of a $1 investment
through time, has inflation fucked up our purchasing power even if we include compound interest?
ye bruv
when will the correlation between inflation and nominal T-bill rates will be close to perfect (1.0)?
how does it affect the correlation between inflation and the realized real rate
When the expected real rate is stable and realized inflation matches initial expectations
the correlation between inflation and the realized real rate will be close to 0
when will the correlation between inflation and nominal T-bill rates will be close to perfectly negative (-1.0)?
why?
when investors either ignored or were very poor at predicting inflation
because the real rate would then fall one-for-one with any increase in inflation
dividend yield
dividends earned per dollar invested
total holding-period return (HPR)
depends on the price at which you check it in the future (at the end of the period)
also depends on the amount of dividend you received
HPR = capital gain (loss) + dividends yield
expected or mean return E(r)
probability-weighted average of the rates of return in all scenarios
The standard deviation of the rate of return (σ)
a measure of risk
measure the uncertainty of outcomes
the square root of the variance
risk free rate
the rate you can earn by leaving money in risk-free assets such as T-bills, money market funds, or the bank
how do we measure the expected reward for the risk involved in investing in an asset?
the risk premium
the difference between the expected return (expected HPR) and the risk free rate
ER - RF
excess return
The difference between the actual rate of return on a risky asset and the risk free rate
the risk premium
the expected excess return
risk aversion
with no risk premium, no one would invest
only invest if the reward is enough to compensate for the risk undertaken
In theory then, there must always be a positive risk premium on stocks in order to induce risk-averse investors to hold the existing supply of stocks instead of placing all their money in risk-free asset
forward-looking scenario analysis
we determine a set of relevant scenarios and associated investment outcomes (rates of return)
–> we assign probabilities to each
–> conclude by computing the risk premium (the reward) and standard deviation (the risk) of the proposed investment
in which form do asset and portfolio return histories come from?
in the form of time series of past realized returns
–> they disregard past assessment of probabilities
–> we only observe dates and associated HPRs
arithmetic average for rates of return
(E of past returns) / n
n: number of periods
basically, the average of the past rates of return because each past scenarios are equal scenarios
geometric mean
[(1 + past return 1) · (1 + past return 2) · (1 + past return 3) · … · (1 + past return n)]^(1/n) - 1
when does the arithmetic average return form a historical period provide a good forecast of expected HPR?
If the time series of historical returns fairly represent the true underlying probability distribution
when is the discrepancy between the arithmetic and geometric averages larger?
when the swings in rates of return are larger
–> between the compound rate earned over the sample period and the average of the annual returns
what are we interested in when thinking about risk?
we are interested in the likelihood of deviations from the expected return
the two types of standard deviation
ex ante (expected standard deviation)
ex post (standard deviation looking at past results)
ex ante (expected standard deviation)
σ = (E(probi) · (ri - ER)^2)^(1/2)
future or expected
ex post (standard deviation looking at past results)
past or historical
σ = ((E(ri - r_)^2) / (n - 1))^(1/2)
r_: the average return (using the arithmetic average)
ri: return in year I
n: number of observations
do observation frequencies have any impact on the accuracy of mean estimates?
what else could improve the accuracy of mean estimates?
nah boy
the duration of sample time series is what improves the accuracy of mean estimates
how is the monthly average return (r_M) annualized (r_A)?
with compounding
r_A = (1 + r_M)^12 - 1
–> or just do it with financial calculator
how can the accuracy of estimates of the standard deviation and higher moments be made more precise?
by increasing the number of observations
how do we annualize monthly standard deviation (σM) when monthly returns are uncorrelated to one another?
- we find the monthly variance (σ^2M)
- we multiply the monthly variance by 12
–> 12σ^2M = σ^2A
- we find the annualized standard deviation
–> σA
The Reward-to-Variability (Sharpe) Ratio
we measure the attraction of an investment portfolio by the ratio of its risk premium to the Standard Deviation of its excess returns
–> shows the importance of the tradeoff between reward (the risk premium) and risk
Sharpe ratio (for portfolios) = (risk premium) / (SD of excess returns)
widely used to evaluate the perfjoamcne of investment managers
how do we annualize the Sharpe ratio from monthly rates?
we multiply the risk premium (numerator) by 12
we multiply the SD (denominator) by (12)^(1/2)
the four factors of the normal distribution we need to know
- the normal distribution is symmetric
- the normal distribution belongs to a unique family of distributions characterized as “stable,” because of the following property: When assets with normally distributed returns are mixed to construct a portfolio, the portfolio return is also normally distributed
- scenario analysis is greatly simplified when only mean and SD need to be estimated to obtain probabilities of future scenarios
- when constructing portfolios of securities, we must account for the statistical dependence of returns across securities
–> when securities are normally distributed, the statistical relationships between returns can be summarized with a correlation coefficient
what does normality assure?
normality of excess returns hugely simplify portfolio selection
assures us that standard deviation is a complete measure of risk
assures us that the sharpe ratio is a complete measure of portfolio performance
Moments of a distribution
can define a distribution
- mean
- volatility
- skewness
- Kurtosis
what does the number of moments needed to define a distribution depend on?
on how well it behaves
the mean (arithmetic Average)
he first moment of the normal distribution
the point where we observe the highest number of observations
Volatility (standard deviation)
Volatility is the square root of the Variance
–> both are the second moment of the distribution
measures dispersion of the distribution
measures the uncertainty of the outcome
Skewness
the third moment of the distribution
measures asymmetry of the distribution
the ratio of the average cubed deviation from the mean to the cubed standard deviation
what happens if If skewness is positive?
the Standard Deviation (Volatility) overestimates the risk
what happens if skewness is negative?
the Standard Deviation underestimates the risk
Kurtosis
the fourth moment of the distribution
measures the degree of fat tails in the distribution
what does a higher kurtosis mean?
the fatter the tails of the distribution, the more the standard deviation will underestimate the risk
the values of skewness and kurtosis in a normal distribution
they are both 0
Factors Affecting Stock Prices Fluctuations
Expectations about future CFs (the numerator)
Expectation of Discount Rates
–> RF (Changes by Central Bank Rates)
–> RP (Changes in the amount of risk and Changes in the compensation required by investors to bear risk (price of risk))
how does inflation affect bonds?
Discount Rates may rise
Bonds future CFs are known in advance
–> Based on the PV pricing formula, Bond Price may Fall
–> bond will sell at discount if interest rate goes above coupon rate
how does inflation affect stocks?
Discount Rates may rise
Stock future CFs are not known in advance
Over the long run, the two effect cancel each other and stock prices remain unchanged
Over the short run, stock prices tend to fall (until the market adjust)
when do we use the geometric average?
when we want to estimate the average one period return
achieved on our investment
when do we use the arithmetic average?
when want to predict future expected next period return
true or false
Geometric Average is always less than Arithmetic Average
true
the most important
macroeconomic measure
Interest rates
Other Measures of Risk
Value at Risk (VaR)
Lower Conditional Tale Expectation (LCTE) or Expected Shortfall (ES)
Lower Partial Standard Deviation (LPSD)
Sortino Ratio
Value at Risk (VaR)
Is the quintile of a distribution
Usually we define VaR by the 1% or 5% quintile
A 1% VaR means that there is a 1% probability that our loss will be below the VaR value
However we have no idea about the magnitude of our loss
Lower Conditional Tale Expectation (LCTE) or Expected Shortfall (ES)
LCTE answer the question that VaR does not answer
LCTE adds to VaR
LCTE gives us a rough idea about what should be the expected magnitude of our loss
If VaR tell us that there is a 5% probability that our loss be equal or below the VaR value then LCTE continues and say that the expected (average) loss would be of …
Lower Partial Standard Deviation (LPSD)
Similar to usual standard deviation
Uses only negative deviations from the risk-free return
Addresses the asymmetry in returns issue
Sortino Ratio
The ratio of average excess returns to LPSD
true or false
discount rate = RF + Risk Premium (or excess returns)?
true
RF: compensate investors for the time value for money
Risk Premium (or excess returns): compensate investors for bearing the risk of future CFs
