Chapter 5
Zero-coupon bond: pays its owner only one cash flow, on the maturity date.
100
r(T) = −1, where P(T) is the price paid today, T is the maturity date and R is
P (T )
the percentage increase in the value of the investment over this holding period.
You can write this for the holding period as:
Price increase+ Income 100−P ( T )+ 0
Holding period return = r(T) = =
P(T ) P (T )
Ending price of a share−beginning price of a share+ cash dividend
HPR =
beginning price
Lower price and lower present value, but a higher total return if it is for a longer
period.
Effective annual rate (EAR): the percentage increase in funds per year.
Annual percentage rates (APR): simple interest that ignores compounding.
1 + EAR = ¿
APR = n x [(1+ EAR ¿ ¿ 1/ n−1
Continues compounding (CC)
1 + EAR = exp(rCC) = eRcc
If you want to find the RCC you need to do: ln (1 + EAR) = RCC
The level of interest rates is determined by:
1) The supply of funds from savers (primarily households)
2) The demand for funds from businesses to be used to finance investments
in plant, equipment and inventories
3) The government’s net demand for funds as modified by actions of the
Federal Reserve Bank
4) The expected rate of inflation
Interest rate: a promised rate of return, usually in a specific currency.
Nominal interest rate: the growth rate of your money.
Real interest rate: the growth rate of your purchasing power.
1+ rnom rnom−i
1 + rreal = rreal ≈ rnom – i rreal =
1+i 1+i
Consumer price index (CPI): measures purchasing power by averaging the
prices of goods and services in the consumption basket of an average urban
family of four.
Fisher hypothesis: rnom = rreal + E(i)
Rreal = (1-t) – it (inflation x tax)
Actual behavior
Realized return or the holding period of return: Change in the value of a security,
today versus the past.
HPR: holding period of
return or dividend yield +
capital gains
Dit = cash dividend at t
Pit = share price at t
Pi,t-1 = share price at t – 1
Dividend yield: the percent return form dividends.
Excess return = Ri – Rf
Ri comes from the realized return answer.
, o It is the difference between the actual rate of return and the actual risk-
free rate
o For example, if the one-year Treasury has returned 2.0% and the
technology stock Facebook has returned 15% then the excess return
achieved for investing in Facebook is 13%
15% - 2% = 13%.
Expected behavior
(Ps1 x Rs1) + (Ps2 x Rs2) etc.
You are never sure what the return is going to be at some point in the future
o Uncertainty around HPR Rs
o Quantify beliefs about the state of the market
o Different scenarios s
o Each with a probability Ps this should be together equal to 1
o Define the expected return
Risk Premium = E[Ri] – rf
o Return an asset is expected to yield in excess of the risk-free rate
o Form of compensation for investors
o Payment to investors for tolerating the extra risk in a given investment
over that of a risk-free asset
Measures of dispersion
When thinking about risk, we are interested in the
likelihood of deviations from the expected return. We
don’t know if it is
going to happen, but it
is an expectation.
Variation is the probability x (HPR – expected return) + for the rest the same.
The wider the figure is, the more variations in the values can be, the more risk.
The blue one is smaller, higher mean, so more likely to happen.
Larger values:
o Less likely to observe expected outcome
o Higher risk
Risk aversion is really important in how much investors are going to invest. If the
risk-premium is zero, they would not invest any money in stocks. Must be a
positive risk-premium.
Other statics
1
Arithmetic average of historic rates of return somteken r(s)
n
Terminal value = (1 + r1) x (1 +
r2) …. X (1 + rn) (1 + g)n g
= terminal value1/n – 1
, a) Skewness: characterizes the degree of asymmetry of a distribution around
its means. It is a pure number that characterizes only the shape of the
distribution.
b) Kurtosis: measures the size of a distribution’s tails. For a heavy-tailed
distribution, probability mass shifts from the intermediate parts of the
distribution to both the tails and the middle.
Kurtosis is a non-dimensional measure.
Why normal distribution?
1) Very well-behaving distribution
2) Stability: stable distribution
3) Additivity
Intuition: know very well confidence
intervals.
The annualized Sharpe ratio is obtained
when multiplying the Sharpe ratio times
√ 12
Sharp ratio = the reward to risk ratio.
Lognormal distribution: lognormal means that the log of the final portfolio
value, ln(WT) is normally distributed.
Zero-coupon bond: pays its owner only one cash flow, on the maturity date.
100
r(T) = −1, where P(T) is the price paid today, T is the maturity date and R is
P (T )
the percentage increase in the value of the investment over this holding period.
You can write this for the holding period as:
Price increase+ Income 100−P ( T )+ 0
Holding period return = r(T) = =
P(T ) P (T )
Ending price of a share−beginning price of a share+ cash dividend
HPR =
beginning price
Lower price and lower present value, but a higher total return if it is for a longer
period.
Effective annual rate (EAR): the percentage increase in funds per year.
Annual percentage rates (APR): simple interest that ignores compounding.
1 + EAR = ¿
APR = n x [(1+ EAR ¿ ¿ 1/ n−1
Continues compounding (CC)
1 + EAR = exp(rCC) = eRcc
If you want to find the RCC you need to do: ln (1 + EAR) = RCC
The level of interest rates is determined by:
1) The supply of funds from savers (primarily households)
2) The demand for funds from businesses to be used to finance investments
in plant, equipment and inventories
3) The government’s net demand for funds as modified by actions of the
Federal Reserve Bank
4) The expected rate of inflation
Interest rate: a promised rate of return, usually in a specific currency.
Nominal interest rate: the growth rate of your money.
Real interest rate: the growth rate of your purchasing power.
1+ rnom rnom−i
1 + rreal = rreal ≈ rnom – i rreal =
1+i 1+i
Consumer price index (CPI): measures purchasing power by averaging the
prices of goods and services in the consumption basket of an average urban
family of four.
Fisher hypothesis: rnom = rreal + E(i)
Rreal = (1-t) – it (inflation x tax)
Actual behavior
Realized return or the holding period of return: Change in the value of a security,
today versus the past.
HPR: holding period of
return or dividend yield +
capital gains
Dit = cash dividend at t
Pit = share price at t
Pi,t-1 = share price at t – 1
Dividend yield: the percent return form dividends.
Excess return = Ri – Rf
Ri comes from the realized return answer.
, o It is the difference between the actual rate of return and the actual risk-
free rate
o For example, if the one-year Treasury has returned 2.0% and the
technology stock Facebook has returned 15% then the excess return
achieved for investing in Facebook is 13%
15% - 2% = 13%.
Expected behavior
(Ps1 x Rs1) + (Ps2 x Rs2) etc.
You are never sure what the return is going to be at some point in the future
o Uncertainty around HPR Rs
o Quantify beliefs about the state of the market
o Different scenarios s
o Each with a probability Ps this should be together equal to 1
o Define the expected return
Risk Premium = E[Ri] – rf
o Return an asset is expected to yield in excess of the risk-free rate
o Form of compensation for investors
o Payment to investors for tolerating the extra risk in a given investment
over that of a risk-free asset
Measures of dispersion
When thinking about risk, we are interested in the
likelihood of deviations from the expected return. We
don’t know if it is
going to happen, but it
is an expectation.
Variation is the probability x (HPR – expected return) + for the rest the same.
The wider the figure is, the more variations in the values can be, the more risk.
The blue one is smaller, higher mean, so more likely to happen.
Larger values:
o Less likely to observe expected outcome
o Higher risk
Risk aversion is really important in how much investors are going to invest. If the
risk-premium is zero, they would not invest any money in stocks. Must be a
positive risk-premium.
Other statics
1
Arithmetic average of historic rates of return somteken r(s)
n
Terminal value = (1 + r1) x (1 +
r2) …. X (1 + rn) (1 + g)n g
= terminal value1/n – 1
, a) Skewness: characterizes the degree of asymmetry of a distribution around
its means. It is a pure number that characterizes only the shape of the
distribution.
b) Kurtosis: measures the size of a distribution’s tails. For a heavy-tailed
distribution, probability mass shifts from the intermediate parts of the
distribution to both the tails and the middle.
Kurtosis is a non-dimensional measure.
Why normal distribution?
1) Very well-behaving distribution
2) Stability: stable distribution
3) Additivity
Intuition: know very well confidence
intervals.
The annualized Sharpe ratio is obtained
when multiplying the Sharpe ratio times
√ 12
Sharp ratio = the reward to risk ratio.
Lognormal distribution: lognormal means that the log of the final portfolio
value, ln(WT) is normally distributed.
