Transcript Returns and Statistics

Class Business
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Groups
Upcoming Homework
Stock-Trak Accounts
Returns
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How can we compare investment results?
Suppose you invest
– $100 in asset A
– $95 in asset B
At the end of the year, your investment in
– A has grown to $101
– B has grown to $101
So would you be indifferent between these two investments?
Returns
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Returns are the correct measuring stick to compare
investments.
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Gross return:
– A: 101/100 = 1.01 = 101%
– B: 101/95 = 1.06 = 106%
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In general: GRt+1=Gett+1/Payt
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GRt+1=the gross return realized at time t+1
Gett+1=the amount of money you get at time t+1
Payt = the amount of money you invested at time t
Returns
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Net return:
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A: 101/100 --1 = 0.01 = 1%
B: 101/95 – 1 = 0.06 = 6%
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In general: NRt+1=Gett+1/ Payt - 1
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NRt+1=the net return realized at time t+1
Gett+1=the amount of money you get at time t+1
Payt = the amount of money you invested at time t
Note: net return = gross return – 1
In general when I say “return” I mean the net return.
Returns: Example
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Suppose you pay $30 in some investment
and get back $33.
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What is the gross return?
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GR=33/30=110%
What is the net return?
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NR = 1.10 - 1 =10%
Returns to a Portfolio
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Suppose you invest $10,000 in 3 stocks.
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1:GE 2: NKE
3: OMX
How much do you put in each?
Suppose you divide money accordingly:
– GE: 2,000 NKE: 5,000 OMX: 3,000
Suppose over a period, returns are:
– r1=10%, r2=5%, r3=-5%
What is the return on your portfolio?
Net Returns to a Portfolio
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Net return:
NR  (.20)(1.10)  (.50)(1.05)  (.30)(.95)  1
 .20(.10)  .50(.05)  .30(.05)
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So the net return on the portfolio is just the weighted
average of the net returns of the individual assets in
the portfolio
Weights are the fraction of investment in each asset
This works no matter what your initial investment is,
or how much you put in each asset.
This extends to gross return
Returns to a Portfolio: Example
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Suppose your investment equity is $1000
– 30% is in IBM
– 10% is in Microsoft
– 60% is in Dell
– rIBM=8%, rMSFT=10%, rDELL=-5%
What is the return on your portfolio?
rp=.30(0.08)+0.10(0.10)+0.60(-0.05)
=0.40%
Time Value of Money
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Why are asset returns on average positive?
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Time value of money
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Would you rather have a car now or wait five
years?
Present values are not equal to future
values
Future Value
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Suppose the interest rate is 10%.
If you invest $100 now,
– How much do you have in 1 year?
– How much do you have in two years?
– How much do you have in three years?
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In general: Sn=P0(1+r)n
Sn= the value you have at the end of year n
P0= initial principal invested
r = return on investment
n = number of time periods
Present Value
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Suppose you can borrow and lend at 10%
Suppose you are offered
– $100 in 10 years
– $X now
What amount of cash (X) would make you indifferent
between the two deals?
X = 100/(1.1010) = $38.55
In general: PV(Sn)=Sn/(1+r)n
– Sn= the value you will receive at the end of year n
– r = return associated with risk of receiving Sn
– n = number of time periods until money is
received
Example: Present and Future
Values
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Suppose you invest $100 now, and earn a
return of 7% every year for 50 years. How
much do you have at the end of the 50
years?
– Sn=100(1.07)50 = 2945.70
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What is the present value of 1 million
received in 50 years if you can borrow and
lend at 7%?
– PV($1M)=$1M/(1.07)50 = 33,947.76
Effective Rate of Return
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Rates of return are time specific.
The effective rate is the actual rate earned over an alternative time
period
Example:
– 6-month rate is 5%
– What is effective annual rate?
– First six months:
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Next six months:
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Each dollar invested grows to 1.05
Each dollar invested grows to (1.05)(1.05)=(1.05)2
Effective net annual rate is (1.05)2-1 = 10.25%
Effective Rates of Return
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In general, effective rates satisfy the
following equation:
(1+rA)n=(1+rB)
Where
– rA is the effective A-period rate (e.g. a
month)
– rB is the effective B-period rate (e.g. a
year)
– n is the number of A time periods in B
(e.g. 12)
Example - Effective Rate
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The annual rate is 15%
What is effective 1-month rate?
Invest $1
– In one year you have 1.15
Invest $1 at 1-month rate rm
– In one year you have (1+ rm)12
– (1+ rm)12 = 1.15, solve for rm
– rm = 1.012%
– 1.2% monthly return
Annual Percentage Rates
(APR)
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Example:
– Suppose 1-month rate is 1%
– APR=12 x .01=12%
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Example:
– Suppose 1-week rate is 0.3%
– APR=52 x .003 = 15.6%
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APR is found by multiplying rate by number
of time periods there are in 1 year.
Example: Effective Rates &
APR
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Suppose the current 4 month rate is 3%.
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What is the effective annual rate?
– (1.03)3=1.0927
– Effective annual rate is 9.27%
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What is the APR?
– APR=.03 x 3 = .09
Characterizing Risk
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We can construct returns, now we introduce
uncertainty about possible outcomes and
how to integrate this notion into investment
characterization and evaluation
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Use Probability models from Statistics
Probability Models
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Think about flipping a coin.
Only two possible outcomes:
– Heads
– Tails
Suppose if heads is flipped, we get $1
If tails is flipped, we win $0.80
Probability Models
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What do we expect to get on average?
Game: The coin is flipped 100 times.
– What would be your forecast of the
payoff from playing this game?
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How much would you pay to play this
game (price)?
Probability Models
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Suppose price to play = $0.85
We can draw a model of net returns:
Two-state probability model
– Two states
– Two returns
– Two probabilities