Intensive Course on
Investments
GEC Academy, FB 4914
Instructor: Alexei Chekhlov
Real and Nominal Interest Rates
• Nominal Interest Rate: a risk-free rate for a particular currency, a
particular period. This is a growth rate of your money, R.
• Consumer Price Index in the U.S. measures the average purchasing
power change (i.e., rate of inflation, i) by averaging the price of
goods and services in the consumption basket of an average urban
family of four.
• Real Interest Rate: growth rate of your money reduced by inflation
(inflation-adjusted), r.
• Even though the nominal interest rate is risk-free, the real interest
rate is risky, as the inflation rate is uncertain!
approximately : r ≈ R − i,
or more precisely, 1 + r =
Intensive Course on
Investments, Chapter 5
1+ R
.
1+ i
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on Bloomberg: CPI Index Help
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on Bloomberg: CPI YOY Index GPO from 12/31/69 to 9/29/20
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Equilibrium Real Rate of Interest
• Variables: quantity of funds and real rate of interest.
• Demand curve is downward-sloping: the lower the real
interest rate is, the more businesses will want to invest in
projects and borrow.
• Supply curve is upward sloping: the higher the real interest
rates are, the greater the supply of household savings is.
• The government (Federal Reserve) can shift these supply
and demand curves through their policy changes.
• The increase in government’s budget deficit (fiscal policy,
such as higher spending of tax revenues) shifts the demand
curve (and equilibrium point E) to the right (to point E’).
• Expansionary monetary policy (some form of central bank
money printing) shifts supply curve to the right to offset the
previous shift.
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Investments, Chapter 5
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12
Real Interest Rates
Supply
E'
Equilibrium
Real Rate of
Interest
E
Demand
Curves
Equilibrium Funds
Lent
Funds
Figure 5.1: Determination of the equilibrium real rate of interest
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13
Irving Fisher Hypothesis
If the real rates are stable, then the nominal
rates are predicting the inflation rates, for
example: increase in nominal rates predicts
higher inflation.
R = r + E (i ).
Let us assume that expected inflation rate rises
from 8% to 10%, real interest rate is 3%, but real
rate is unchanged, what happens to the nominal
interest rates?
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Investments, Chapter 5
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14
Effect of Taxes
• Tax liabilities are based on nominal income, and the tax
rate (t) is based on the investor’s tax bracket.
• Bracket creep: nominal income grows due to inflation
and pushes tax payers into higher brackets.
• The tax system was not inflation-protected.
• Tax Reform Act of 1986 mandated inflation-linked tax
brackets.
Real After Tax Rate = R × (1 − t ) − i = r × (1 − t ) − i × t.
Example:
Your tax bracket:
Your investment yield:
Inflation rate:
Before tax real rate of interest:
Ideal net after tax real return:
Actual net after tax real return:
Intensive Course on
Investments, Chapter 5
30%
12%
8%
4%
2.8%
0.4%
almost wipes out by taxes!
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15
Rates of Return for Different
Holding Periods
• Zero-coupon Treasury securities for various
maturities. T-Bills can serve this up to 1 year.
Between 1 and 30 years, we can use Treasury
STRIPS (T-Notes and T-Bonds with coupons
“stripped” off).
Example 5.2: Rates of Return for Various Maturities
Prices of zero-coupon Treasuries with face value $100 and various maturities
Horizon, T
0.5 yr
1 yr
25 yrs
Price, P(T)
$
97.36
$
95.52
$
23.30
Risk-Free Return for Given Horizon
2.71%
4.69%
329.18%
100
r f (T ) =
− 1.
P (T )
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21
Annualized Rates of Return
• Effective Annual Rate
(EAR): for shorter than 1
year, compound perperiod return for a full
year; for longer than 1
year, ERA is annual rate
that would compound to
the same rate.
• Annual Percentage Rates
(APRs): annual rates on
short-term (T<1 yr)
investments which are
annualized by simple
interest.
Intensive Course on
Investments, Chapter 5
Example 5.2: Rates of Return for Various Maturities
Prices of zero-coupon Treasuries with face value $100 and various maturities
Horizon, T
0.5 yr
1 yr
25 yrs
$
$
$
Price, P(T)
97.36
95.52
23.30
Risk-Free Return for Given Horizon
2.71%
4.69%
329.18%
100
r f (T ) =
− 1.
P(T )
1 + EAR = [1 + rf (T )]
1/ T
(1 + EAR) − 1
APR =
T
GEC Academy, FB 4914
T
22
Continuous Compounding
• Difference between APR and EAR grows
with frequency of compounding T.
• What is the limit of [1+T *APR]1/T as T→0+?
• This limit is called continuous compounding
(CC): 1+EAR=exp(rcc).
• For CC rates the following equation is exact:
rcc(real)=rcc(nominal)-icc.
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Investments, Chapter 5
GEC Academy, FB 4914
23
EAR=0.058
Compounding Period
1 year
6 months
1 quarter
1 month
1 week
1 day
Continuous
T
1.0000
0.5000
0.2500
0.0833
0.0192
0.0027
r f (T )
0.0580
0.0286
0.0142
0.0047
0.0011
0.0002
APR
0.05800
0.05718
0.05678
0.05651
0.05641
0.05638
0.05638
APR=0.058
r f (T )
0.0580
0.0290
0.0145
0.0048
0.0011
0.0002
EAR
0.05800
0.05884
0.05927
0.05957
0.05968
0.05971
0.05971
Table 5.1: Annual percentage rates (APR) and effective annual rates (EAR).
In the first set of columns we hold the EAR fixed at 5.8% and solve for APR.
In the second set we hold APR fixed at 5.8% and solve for EAR.
• What is a better choice: to invest $100,000 for 3
years with
– (a) monthly rate of 1%; EAR=(1+0.01)12-1=12.68%
– (b) annual CC rate of 12%? EAR=e0.12-1=12.75%.
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GEC Academy, FB 4914
24
Historical Data Other Than Bloomberg
Federal Reserve Bank of St. Louis or FRED
(Federal Reserve Economic Data):
https://fred.stlouisfed.org/
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Investments, Chapter 5
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25
Intensive Course on
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GEC Academy, FB 4914
26
Treasury Bills and Inflation
Since January 1, 1926, historical returns for T-Bills, Inflation, and
real rates are available in CRSP database (U. of Chicago).
Table 5.2: Statistics for T-bill rates, inflation rates, and real rates, 1926–2019
All 94 years, 1926-2016
Earlier 26 yrs, 1926-1951
Recent 68 yrs, 1952-2016
Average Annual Rate
T-Bills
Inflation Real T-Bill
3.37
2.96
0.50
1.04
1.68
-0.29
4.26
3.45
0.80
Correlation of inflation with nominal T-Bills
Correlation of inflation with real T-Bills
Intensive Course on
Investments, Chapter 5
Standard Deviation
T-Bills
Inflation Real T-Bill
3.09
3.98
3.74
1.29
5.95
6.27
3.11
2.82
2.11
Fisher Hyp.
1926-2016 1926-1951 1952-2016
41.7%
-30.2%
0.73
-70.1%
-97.7%
-0.25
GEC Academy, FB 4914
27
U.S. Rates History
20
15
10
5
0
1920
1930
1940
1950
1960
1970
T-Bills
Inflation
1980
1990
2000
2010
2020
-5
-10
-15
-20
Intensive Course on
Investments, Chapter 5
Real T-Bills
GEC Academy, FB 4914
28
U.S. Wealth Index
25
20
15
10
5
0
1920
1930
1940
1950
1960
T-Bills
Intensive Course on
Investments, Chapter 5
1970
Inflation
1980
1990
2000
2010
2020
Real T-Bills
GEC Academy, FB 4914
29
Expected Return and Standard Deviation
• Holding Period Return
(HPR) in an equity
investment:
HPR =
Ending Share Price − Beginning Price + Cash Dividend
.
Beginning Price
• We can quantify
uncertainty of expected
return in terms of
scenarios s and their
probabilities p(s):
E (r ) =  p( s ) × r ( s ).
s
• The standard deviation σ
of the rate of expected
return as a measure of
risk:
Intensive Course on
Investments, Chapter 5
σ =  p( s ) × [r ( s ) − E (r )] .
2
2
s
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30
A
1
2
3
4
5
6
7
8
9
10
11
12
13
B
C
D
E
F
G
H
I
Excess
Returns
Squared
Deviations
from Mean
Spreadsheet 5.1: D istribution of HPR on the Stock Index Fund
Rates of return expressed as decimals
Purchase Price =
State of the
Econom y
Excellent
Good
Poor
Crash
T-bill Rate =
$100
Year-end
Price
Probability
0.25
0.45
0.25
0.05
Cash
Divid end s
126.50
110.00
89.75
46.00
4.50
4.00
3.50
2.00
0.04
Deviations
from Mean
H PR
0.31
0.14
-0.07
-0.52
14 Expected Value (m ean)
SUMPRODUCT(b9:b12,e9:e12) =
0.0976
15 Variance of H PR
SUMPRODUCT(b9:b12,g9:g12)=
0.0380
0.212
0.042
-0.165
-0.618
16 Stand ard Deviation of H PR
SQRT(E15) =
17 Risk Prem ium
SUMPRODUCT(b9:b12,h9:h12) =
18 Stand ard Deviation of Excess Return
Intensive Course on
Investments, Chapter 5
Squared
Deviations
from Mean
0.045
0.002
0.027
0.381
0.27
0.10
-0.11
-0.56
0.045
0.002
0.027
0.381
0.1949
0.0576
SQRT(SUMPRODUCT(b9:b12,i9:i12)) =
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0.1949
31
Excess Return and Risk Premium
• Risk premium: the difference between the expected HPR
and the risk-free rate.
• Excess return is the risk premium for a particular holding
period (for example, per year).
• Investors are risk-averse: if risk premium is zero or negative,
investors are not willing to commit funds to the risky
investment, preferring the risk-free one.
Example:
You can invest $27,000 in corporate bond with the price of $900 per $1000 par value
and paying $75 in coupons p.a. Alternatively, you can invest in T-Bills that yield 5%.
Interest Rates
High
Unchanged
Low
Probability
20%
50%
30%
Year-End Bond Price
$850
915
985
HPR
2.78%
10.00%
17.78%
HPR
Excess HPR
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Weighted HPR $$ at Year-End
0.56%
$
27,750
5.00%
$
29,700
5.33%
$
31,800
10.89%
5.89%
$
29,940
32
Time Series vs Scenario Analysis
• Asset return history comes in the form of time series of past realized
returns, and not explicit original assessments of the probabilities of
those returns.
• We must infer the probability distributions from that limited historical
data.
• Geometric mean is used to produce time-weighted average return,
as opposed to dollar- or asset-weighted.
n
1
1 n
For p ( s ) = , we get E (r ) =  p ( s) × r ( s ) = ×  r ( s ).
n
n s =1
s =1
If returns are Gaussian, E (Geometric Average) = E (Arithmetic Average) −
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Investments, Chapter 5
GEC Academy, FB 4914
σ2
2
33
.
A
1
2
3
4
5
6
7
8
9
B
C
D
E
F
Spreadsheet 5.2: Time Series of HPR for the S&P 500
Period
2001
2002
2003
2004
2005
Implicitly Assumed
Probability = 1/5
0.2
0.2
0.2
0.2
0.2
HPR (decimal)
-0 .1 1 8 9
-0 .2 2 1 0
0 .2 8 6 9
0 .1 0 8 8
0 .0 4 9 1
10 Arithmetic average AVERAGE(c5:c9) =
11 Expected HPR
SUMPRODUCT(b5:b9) =
12
Standard D eviation
13
0.0196
0.0586
0.0707
0.0077
0.0008
Gross HPR =
1+HPR
0.8811
0.7790
1.2869
1.1088
1.0491
Wealth
Index*
0.8811
0.6864
0.8833
0.9794
1.0275
0.0210
0.0210
SUMPRODUCT(b5:b9,c5:c9)^ .5 =
0.1774
Check:
STDEV(c5:c9) =
0.1983
1.0054^ 5=
14
Geometric average return
15 * The valu e of $1 invested at the beginning of the samp le p eriod (1/ 1/ 2001).
Intensive Course on
Investments, Chapter 5
Squared
D eviation
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GEOMEAN (e5:e9) – 1 =
0.0054
1.0275
34
Recovering from a Drawdown
• You invested $1,000,000 into the S&P 500
index at the very beginning of the 2008.
Given the rate of return for 2008 of -38.6%,
what rate of return in 2009 would have been
necessary for your portfolio to fully recover?
• Answer: (1-.386)*(1+R)=1 has a solution
R=1/(1-.386)-1=62.9%!
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35
Variance and Standard Deviation
• We are interested in deviations from the expected return, as a
measure of risk. Variance=Expected Value of Squared Deviations.
• Historical (biased) estimate leads to an absurd result (=0) for n=1, a
correction of the bias leads to an un-biased estimate.
σ 2 =  p( s) × [r ( s ) − E (r )]2 .
2
1 n
2
Biased esimate : σˆ = ×  [r ( s) − r ] .
n s =1
n
1
2
Unbiased estimate : σˆ =
×  [r ( s ) − r ] .
n − 1 s =1
2
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Sharpe Ratio
• Not only the investors are interested in expected excess return to be
over the T-Bills rate, but also in risk premium being commensurate
with the risk they take.
• Thus, the reward-to-volatility ratio, where risk is measured as
standard deviation of the excess returns, is used to evaluate the
performance of investments and investment managers.
Risk Premium
.
Sharpe Ratio =
Standard Deviation of Excess Return
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37
A
1
2
3
4
5
6
7
8
9
10
11
12
13
B
C
D
E
F
G
H
I
Excess
Returns
Squared
Deviations
from Mean
Spreadsheet 5.1: D istribution of HPR on the Stock Index Fund
Rates of return expressed as decimals
Purchase Price =
State of the
Econom y
Excellent
Good
Poor
Crash
T-bill Rate =
$100
Year-end
Price
Probability
0.25
0.45
0.25
0.05
Cash
Divid end s
126.50
110.00
89.75
46.00
4.50
4.00
3.50
2.00
0.04
Deviations
from Mean
H PR
0.31
0.14
-0.07
-0.52
14 Expected Value (m ean)
SUMPRODUCT(b9:b12,e9:e12) =
0.0976
15 Variance of H PR
SUMPRODUCT(b9:b12,g9:g12)=
0.0380
0.212
0.042
-0.165
-0.618
SQRT(E15) =
17 Risk Prem ium
SUMPRODUCT(b9:b12,h9:h12) =
Intensive Course on
Investments, Chapter 5
0.045
0.002
0.027
0.381
0.27
0.10
-0.11
-0.56
Sharpe Ratio:
16 Stand ard Deviation of H PR
18 Stand ard Deviation of Excess Return
Squared
Deviations
from Mean
0.045
0.002
0.027
0.381
0.30
0.1949
5.76%
SQRT(SUMPRODUCT(b9:b12,i9:i12)) =
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19.49%
38
Normal (Gaussian) Distribution
• A bell-shaped continuous
distribution, given by the
2

following Gaussian
(
1
x − µ) 
.
f ( x; µ , σ ) =
× exp −
2
function:

2
σ
σ 2π


• Here: μ is the mean, σ2 is
the variance, and
correspondingly, σ is
standard deviation.
Intensive Course on
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x = µ;
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x 2 = σ 2.
39
Normal Distribution in Excel
250
200
150
100
50
Bin
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40
3.53
3.08
2.64
2.19
1.75
1.30
0.85
0.41
-0.04
-0.48
-0.93
-1.37
-1.82
0
-2.26
-0.00949
1.01356
-3.71252
3.94815
300
-2.71
0.497
0.291
0.000
1.000
350
-3.15
Average
StdDev
Min
Max
400
-3.60
=rand() =normsinv()
0.988
2.25867
0.136
-1.09873
0.178
-0.92232
0.801
0.84516
0.823
0.92683
0.472
-0.06994
0.240
-0.70625
0.262
-0.63864
0.567
0.16777
0.063
-1.52726
0.555
0.13786
0.770
0.73767
0.977
1.99811
0.301
-0.52285
0.705
0.53949
0.270
-0.61190
0.978
2.00624
0.058
-1.57109
0.386
-0.28972
0.439
-0.15422
0.439
-0.15273
Frequency
#
1
2
3
4
5
6
7
8
9
10
9,990
9,991
9,992
9,993
9,994
9,995
9,996
9,997
9,998
9,999
10,000
Constructing Normal Curve
• A newspaper stand that turns a profit of $100 on a good
day and breaks even ($0) on a bad day. After 3 days:
two good days, profit $200
one good and one bad day, profit $100
one good and one bad day, profit $100
two bad days, profit $0
• Pascal triangle:
Pascal Triangle
1
1
1
1
1
1
1
Intensive Course on
Investments, Chapter 5
1
2
3
4
5
6
7
8
1
3
6
10
15
21
28
36
GEC Academy, FB 4914
1
4
10
20
35
56
84
120
1
5
15
35
70
126
210
330
1
1
6
7
21
28
56
84
126 210
252 462
462 924
792 1,716
1
8
36
120
330
792
1,716
3,432
=K6+J7
41
Binomial Coefficients
(x + y )
n
C =C
n
k
Intensive Course on
Investments, Chapter 5
n
= C × x
n
k
k =0
n −1
k −1
+C
n −1
k
n−k
× y , where :
k
n!
, and C =
.
k!×(n − k )!
n
k
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42
Gaussian with μ=0.1 and σ=0.2
2.5
 ( x − µ )2
× exp  −
f ( x; µ , σ ) =
2σ 2
σ 2π

1
2.0




1.5
1.0
0.5
0.0
-0.6
-0.4
x = µ − 2σ
-0.2
x = µ −σ
0.0
x=µ
0.2
x = µ +σ
0.4
x = µ + 2σ
0.6
0.8
-0.5
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Some Gaussian Properties
µ + 3σ
f ( x; µ , σ ) × dx = 99.74%;

µ σ
−3
µ + 2σ
f ( x; µ , σ ) × dx = 95.44%;

µ σ
−2
µ +σ
f ( x; µ , σ ) × dx = 68.26%.

µ σ
−
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Gaussian Distribution
• Is completely characterized by two parameters,
μ and σ.
• Belongs to a family of stable distributions: they
preserve the additivity property, i.e. when
several assets returns, which are normal are
added in a portfolio, the portfolio returns are also
normal.
• If investment returns are Gaussian, standard
deviation is a complete measure of risk and
Sharpe Ratio is a complete measure of riskadjusted performance.
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Gaussian Moments
Central moments :
(x − µ )
p
if p is odd,
0,
= p
σ × ( p − 1)!!, if p is even.
Central absolute moments :
x−µ
Intensive Course on
Investments, Chapter 5
p
 2
, if p is odd,

p
= σ × ( p − 1)!!× π
1,
if p is even.

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46
Deviations from Normal
• Deviations from Normal are too
significant to ignore.
• A measure of asymmetry called
skew(ness): ratio of average cubed
deviations from the average (3rd
moment) to the standard deviation
cubed:
• A measure of the degree of fat tails,
or kurtosis: a ratio of the 4th moment
to the standard deviation to the 4th
power with 3 subtracted:
Intensive Course on
Investments, Chapter 5
(x − µ )
3
Skewness =
σ
3
(x − µ )
.
4
Kurtosis =
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4
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Negative Skewness
2.5
2.0
1.5
1.0
0.5
0.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
-0.5
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Positive Kurtosis
2.5
2.0
1.5
1.0
0.5
0.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
-0.5
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Value at Risk (VaR)
• n% VaR: A certain amount of money that will be lost
with a probability of n% of a specific portfolio of
financial assets and over a specific period of time.
• For example, -$820K will be lost over one day for a
particular portfolio of financial assets with a
probability of 5%.
• It is a n-percentile of a financial loss after a certain
holding time.
• VaR for Normal distribution:
5% VaR = µ − 1.65 × σ .
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Conditional VaR (CVaR) or Expected Shortfall
• An alternative to VaR which is more sensitive to
the shape of the tail of the PDF.
• n% CVaR is the expected loss conditionally
averaged over n% of worst possible outcomes:
−L
 x × P(x,τ )× dx, where :
X=
x = −∞
-L
L:
 P(x,τ )× dx = n.
x =-∞
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Lower Partial Standard Deviation
and Sortino Ratio
• Look at negative outcomes separately, and the
downside deviation of them as a measure of
risk, DD;
• We can look at returns beating some benchmark
r0 (for example, risk-free rate).
R − r0
S=
, where :
DD
1/ 2


2
DD =   ( x − r0 ) × f ( x,τ ) × dx  .


 x = −∞

r0
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Historical Returns Analysis
• 3-mo Treasury Bills Returns : the least risky of all assets –
there is essentially no risk that U.S. government will fail to
honor its commitments to investors, their short maturities
mean that their prices are relatively stable.
• U.S. Treasury Long-Term Bonds Returns : are also
certain to be repaid, but the prices of these bonds
fluctuate as interest rates vary, so they carry meaningful
risk.
• U.S. Equity Market Index Returns : the broadest possible
U.S. equity portfolio, including all stocks listed on the
NYSE, AMEX (now part of NYSE), and NASDAQ.
Common stocks are the riskiest of the three groups of
securities. As a part-owner of the corporation, your
success will depend on the success or failure of the firm.
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3-mo Treasury Bills Returns Histogram
μ=
3.37%
σ=
3.10%
T-Bills
16%
14%
12%
10%
8%
6%
4%
2%
57.5%
52.5%
47.5%
42.5%
37.5%
32.5%
27.5%
22.5%
17.5%
12.5%
7.5%
2.5%
-2.5%
-7.5%
-12.5%
-17.5%
-22.5%
-27.5%
-32.5%
-37.5%
-42.5%
-47.5%
0%
Annual Return %
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3-mo Treasury Bills Returns vs. Gaussian
μ=
3.37%
3.10%
σ=
100.00%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
10.00%
1.00%
0.10%
0.01%
Gaussian
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T-Bills
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U.S. Treasury Long-Term Bonds Returns Histogram
μ=
5.88%
σ=
9.85%
T-Bonds
7%
6%
5%
4%
3%
2%
1%
57.5%
52.5%
47.5%
42.5%
37.5%
32.5%
27.5%
22.5%
17.5%
12.5%
7.5%
2.5%
-2.5%
-7.5%
-12.5%
-17.5%
-22.5%
-27.5%
-32.5%
-37.5%
-42.5%
-47.5%
0%
Annual Return %
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U.S. Treasury Long-Term Bonds Returns vs. Gaussian
μ=
5.88%
σ=
9.85%
100.00%
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
10.00%
1.00%
0.10%
0.01%
Gaussian
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U.S. Equity Market Index Returns Histogram
μ= 11.90%
σ= 20.03%
T-Bonds
3%
2%
2%
1%
1%
57.5%
52.5%
47.5%
42.5%
37.5%
32.5%
27.5%
22.5%
17.5%
12.5%
7.5%
2.5%
-2.5%
-7.5%
-12.5%
-17.5%
-22.5%
-27.5%
-32.5%
-37.5%
-42.5%
-47.5%
0%
Annual Return %
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U.S. Equity Market Index Returns vs. Gaussian
μ= 11.90%
σ= 20.03%
100.00%
-60%
-40%
-20%
0%
20%
40%
60%
10.00%
1.00%
0.10%
0.01%
Gaussian
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Historical Risk Estimates
80%
70%
60%
50%
40%
30%
20%
10%
0%
1927
1940
1954
1968
1982
1995
2009
Annualized StdDev of the Market Index
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Risk and Return of Investments in Major Asset Classes 1026-2019
Average
Risk Premium
Standard Deviation
Max
Min
T-Bills
3.37%
N/A
3.10%
14.71%
-0.02%
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T-Bonds
5.88%
2.51%
9.85%
40.36%
-14.90%
Stocks
11.90%
8.53%
20.03%
57.35%
-44.04%
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U.S. Equity Market Index Monthly Returns Histogram
μ=
0.67%
σ=
5.35%
Stock Index Monthly
10%
9%
8%
7%
6%
5%
4%
3%
2%
1%
37.5%
32.5%
27.5%
22.5%
17.5%
12.5%
7.5%
2.5%
-2.5%
-7.5%
-12.5%
-17.5%
-22.5%
-27.5%
-32.5%
-37.5%
-42.5%
0%
Monthy Return %
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U.S. Equity Market Index Monthly Returns vs. Gaussian
μ=
0.67%
σ=
5.35%
100.00%
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
10.00%
1.00%
0.10%
0.01%
Gaussian
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Global View
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Log-Normal Distribution Over the Long Haul
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