Class Business

Groups Upcoming Homework Stock-Trak Accounts

Returns 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

Returns are the correct measuring stick to compare investments.

Gross return: – A: 101/100 = 1.01 = 101%– B: 101/95 = 1.06 = 106%

– In general: GRt+1=Gett+1/Payt

• 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 Net return:

– A: 101/100 --1 = 0.01 = 1%– B: 101/95 – 1 = 0.06 = 6%

– In general: NRt+1=Gett+1/ Payt - 1

• 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

Suppose you pay $30 in some investment and get back $33.

– What is the gross return?• GR=33/30=110%

– What is the net return?• NR = 1.10 - 1 =10%

Returns to a Portfolio Suppose you invest $10,000 in 3 stocks.

• 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

Net return:

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

(.20)(1.10) (.50)(1.05) (.30)(.95) 1

.20(.10) .50(.05) .30( .05)

NR

Returns to a Portfolio: Example

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

Why are asset returns on average positive?

– Time value of money• Would you rather have a car now or wait five

years?

– Present values are not equal to future values

Future Value 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?

– 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

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

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

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 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:

• Each dollar invested grows to 1.05

– Next six months:• 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

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

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)

Example:– Suppose 1-month rate is 1%– APR=12 x .01=12%

Example:– Suppose 1-week rate is 0.3%– APR=52 x .003 = 15.6%

APR is found by multiplying rate by number of time periods there are in 1 year.

Example: Effective Rates & APR

Suppose the current 4 month rate is 3%.

What is the effective annual rate?– (1.03)3=1.0927– Effective annual rate is 9.27%

What is the APR?– APR=.03 x 3 = .09

Characterizing Risk

We can construct returns, now we introduce uncertainty about possible outcomes and how to integrate this notion into investment characterization and evaluation

Use Probability models from Statistics

Probability Models

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

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?

– How much would you pay to play this game (price)?

Probability Models

Suppose price to play = $0.85 We can draw a model of net returns:

Two-state probability model– Two states– Two returns– Two probabilities