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FINS Investments and Portfolio Selection University of New South Wales Semester 1 2012 Week 2 Russell Jame
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Buying on Margin & Short Selling
BKM 3.6-3.7
Buying on Margin
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Buying on Margin
Borrow cash to buy more of the asset Incur liability to pay back cash Increase expected return Increase standard deviation
Buying on Margin
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Price of MSFT=50 Suppose you have $10,000 and you expect Microsoft to perform well in the near future. You borrow $5000 from broker at 10%
effective annual rate
Buy $15,000 worth of MSFT stock (300 shares) Assets=15000, Liabilities=5000, Equity=10000
Return of Buying on Margin
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MSFT increases 25% Assets Pay Back Loan (Liability) Equity Return
15,000*1.25=18,750
MSFT decreases 25%
15,000*.75=11,250
5,000 * 1.10 =-5,500 18,750 5,500 =13,250
5,000 * 1.10 =-5,500 11,250 5,500 =5,750
13,250/10,000 -1 =32.5% 5,750/10,000-1=-42.5%
Portfolio Weights
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Portfolio weights allow us to find returns from buying on margin more quickly. Let's see how this works. Payoff= IE(1+rs) + L(1+rs) L(1+rf)
IE = Investment Equity L = Loan amount rs= return on asset rf = risk-free rate
Portfolio Weights
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return = payoff/IE 1
IE(1+rs ) + L(rs - rf ) IE return= - IE IE Do some algebra to get . . . (IE+L) L return = (rs ) - (rf ) IE IE
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Returns and Buying on Margin return = wr + (1-w)r
s f
IE+L w= IE
Again, the return is a weighted sum of the returns of the assets in your portfolio. Weights are a fraction of your investment equity in each asset. The asset you have borrowed money to invest in will always have a weight that is greater than one. The weight on the risk-free return (at which you are borrowing) will have a negative weight.
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Returns and Buying on Margin
Assets = 15000 Equity=10000 Liabilities=5000 Weight in MSFT = 15000/10000=1.5 Weight in bonds = -5000/10000=-0.5 If Microsoft returns 25%
Your return: 1.5*.25+(-.5)*.1=32.5% Your return: 1.5*(-.25)+(-.5)*.1=-42.5%
If Miscrosoft returns -25%
Margin and Losses
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Lower bound for any stock: 0
Assets = Liabilities + Equity Equity = Assets - Liabilities
Lower bound for assets = 0
Lower bound for equity = - Liabilities
Margin Limits
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SEC mandates limitations on borrowing. Limit is defined in terms of "the margin".
equity P(S)-L margin = = Value of Stock P(S)
P=Price of security S=Shares owned L=Value of Loan
Margin Limits
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Initial margin must exceed 50% Maintenance margin set by broker MSFT example:
Buy 300 shares of MSFT at $50 per share Use 10,000 of equity and 5,000 borrowed
Initial margin = (300*50-5000)/15000= 67%
Margin Call
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Assume MSFT drops within a year by 40%
Price=.60*50=30
Margin=[300*30-5000*1.10]/ (3000*30)=39% Assume maintenance margin=40% You will get a margin call Broker will mandate either that you close
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Three Possible Options to Satisfy Margin Call
Close out position
Sell the MSFT stocks for $9,000 Pay back the loan of $5000+interest
Reduce your loan how much?
The margin increases to [300(30)-X]/[300(30)] = 50%
X=300*30-.5*300*30=4500 Must pay off 5000*1.1-4500=1000
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Three Possible Options to Satisfy Margin Call
Increase your position in MSFT
How much? [S*30-5500]/S*30=.50 S=5500/(30-.5*30)=366.67 You already own 300 shares so must buy 67 more shares
Example
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How far can price fall immediately before getting a margin call? (Ignore interest on loan.)
(P*300-5000)/(P*300)=.40 P*300-5000=.40*P*300 P*(300-.4*300)=5000 P=5000/(300-.4*300)=27.78
Short Selling
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Securities are sold by someone who does not own them.
1. 2. 3.
Borrow the securities from somebody. Sell the securities at the current price. Eventually, buy back the securities and return them to the owner.
Profit (per share)=Initial Price Ending Price
Short Selling Example
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Suppose you short MSFT at a price of $50. Two months later the price is $40. What is your profit? Profit = $50 - $40 = $10
On a "per-share" basis
Short Selling Example
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Assets=Liabilities + Equity Initially:
Assets = 50 (cash) Liabilities = 50 (value of stock owed) Equity = 0
Two Months Later
Assets = 50 (cash) Liabilities = 40 Equity = 10
Short Selling Steps
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1. Borrow the shares
This is done via your broker. The term of the short-sell is almost always one day, but loans are typically renewed many times. The lender can require you to return the shares whenever he/she wants (short-squeeze)
Short Selling Steps
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2. Shares are sold in the stock market
Proceeds are put in a collateral account Do not have access to these funds until you return the borrowed shares. Money earns interest (risk-free rate) Assets = collateral account (cash) Liability=value of stock owed Equity=Assets-Liability
Step 2: Sell the Borrowed Shares
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You must contribute your own investment equity to the collateral Equity Assets-P(S) account. = margin =
Value of Stock
P(S)
is the "margin" on the short position. Note: the definition of margin is always equity/value of stock, but the computation of equity is different in both cases..
Short Selling Steps
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3. Close out position
Buy the amount of shares you borrowed and return them to broker. Collect all money in collateral account.
MSFT Example
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MSFT: $50/share You short sell 2000 shares Initial Margin is 50% What are Liabilities?
L=100,000=P*S E/100,000=0.50 E=50,000 A=100,000+50,000=150,000
How much equity do you need to provide?
What are assets?
MSFT Example
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Assume you post equity in the form of cash.
You could provide equity in the form of other stocks or bonds.
Assume all cash in collateral account earns 5% interest annually. How much total cash do you have in your collateral account?
Cash in account=Proceeds of short sell + equity 100,000+50,000 = 150,000
MSFT Example
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Suppose price of MSFT is 60 at the end of the year. What are total assets?
A=150000*(1.05)=157,500 L=2000*60=120,000 E=157,500-120,000=37,500
What are liabilities?
What is equity?
What is return? 37,500/50,000-1=-25% What is new margin? 37,500/120,000=0.3125
Margin Example
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Maintenance margin is 35% How far can price increase immediately before getting a margin call?
Ignore interest earned on collateral account.
Margin call [150000-P(2000)]/P(2000) = 0.35 P=150000/ (.35*2000+2000)=55.56
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Short Selling : Concept Check
Suppose you short-sell 2000 shares of MSFT at $50 and the price of MSFT immediately rises to $55.56 and you get a margin call from your broker.
Assume the maintenance margin is 50%.
What are three options available to meet your margin requirements?
Be specific. Hint: look at slides 14&15.
Feel free to work in small groups. You have roughly 5 minutes to work on this
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Short Selling : Concept Check Solution
Close out position
Buy the MSFT stocks for $55.56 Equity left: 150000-2000*55.56=38800
Return
= 38,800/50,000 -1 = -22.4%
Reduce your liability how much?
Assume the margin must increase to 50% [150000-55.56*S]/[55.6*S] = 50% S=150000/(.5*55.56+55.56)=1800
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Short Selling : Concept Check Solution
Increase your collateral
How much? [X-2000*55.56]/(2000*55.56)=.50
X=.5*2000*55.56+2000*55.56=166,680 Give broker an additional 16,680 in cash
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Mutual Funds BKM 4.1 & 4.3
Advantages
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Investment Companies (e.g. mutual funds) are financial intermediaries that collect funds from investors and invest those funds in a wide range of securities. Advantages
Diversification Lower transaction costs record keeping and administration Professional Management
Mutual Funds
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Open-end funds
You buy or sell shares directly through the mutual fund Shares are purchased at NAV per share NAV is calculated every day at 4:00pm All orders placed before 4:00pm buy at this NAV Fund companies adjust the number of asset value (NAV) = shares according to Shares outstanding the demand by investors
Market value of assets - Liabilities
Net
Open-End Fund Example
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Fund ABC 4:00pm Monday:
Total Assets = 36 billion Total Liabilities = 1 billion Shares Outstanding = 900 million NAV = 35B/900M = $38.89
Throughout day on Tuesday
Asset Value increases by 1% New flows of $500,000
Open-End Fund Example
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4:00pm on Tuesday
Asset value is now 36B(1.01)=36.36B NAV = 35.36B/900M=39.29
New Flows of $500,000
500,000/39.29 = 12,726 shares
New
investors buy these shares at 39.29 per share
Total shares outstanding = 900M + 12,726 Total Assets = 36.36B + 500,000
Open-End Example
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In this example, there are no fees, and no distributions (dividends or capital gains) Return from owning fund
39.29 ret = - 1 = 1.029% 38.89
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Mutual Fund Investment Policies
Each mutual fund has a specified investment policy which is described in the fund's prospectus. We often classify mutual funds based on the investment policy.
Money Market Funds Equity Funds
Small Value
vs. Large Cap vs. Growth
Sector Funds
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Mutual Fund Costs BKM 4.4
Front-End Loads
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What they are for
Used to pay brokers who sell funds
How they affect price paid
NAV of fund: $13 Front End Load: 4% What price do you pay per share?
P P
= 13 + .04(P) = 13/.96 = 13.54
Front End Loads
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Another way of looking at how they affect price:
Assume front-end = 4% If you invest $1000, only $960 is actually invested The other $40 is gone in fees Your investment must increase to 1000 (4.2%) just to break even, assuming no other fees
Front End Loads
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How they affect returns Fund:
NAV=$10 Front End load of 5% NAV grows by 14% per share each year
Assume you1invest X (1.14) 2 hold investment for two ( - 0.05) $X and return = - 1 = 23.5% years. X
with no fee : (1.14) 2 - 1 = 29.96%
Back End Loads
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What they are for
Used to pay brokers who sell funds 0 to 6% Usually reduced 1% point for each year you hold the fund
How they affect price paid
They don't Assuming no other fees, price is NAV
Back End Loads
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How they affect payoff
NAV of fund: $13 Back End Load: 5% What price do you get when you sell the fund? 13*.95 = $12.35
Back End Loads
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How they affect returns Fund:
NAV=$10 Back End load of 5% Fund earns 14% per share each year
X (1.14) 2$X and hold investment for two Assume you invest (1 - .05) - 1 = 23.5% return = years. X with no fee : return = (1.14) 2 - 1 = 29.96%
Annual Expenses
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Operating expenses
What they are for
Used
to pay for administrative expenses and advisory fees
12b-1 charges
What they are for
Used
to pay for advertising, promotional literature, and brokers
Annual Expenses
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How they affect returns Fund:
NAV=$10 Annual expenses (expense ratio) of 1% Fund earns 14% per share each year
Assume you invest $X and hold investment for two years
X 1.132 return = - 1 = 27.69% Each year gross X return net of fees is 1.14 - 0.01 = 1.13 with no fee : return = (1.14) 2 - 1 = 29.96
Summary
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Assume:
A fund NAV grows by r each year. The fund has front-end load of f. The fund has back-end load of b. The fund has annual expense ratio of a. No capital gains or distributions
The gross return for an investor innthe fund over n periods is Gross Return = ( 1-f)( 1 + r-a) (1 - b)
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Concept Check: Fund or CD's?
Investment Choices: mutual fund,
3% front-end load annual expense ratio of 0.6%.
CD: 5% per year What return must the fund portfolio earn for you to be better off in the fund than in the CD if you plan to hold the investment for 4 years?
Concept Check Solution: Fund or CD's?
In 4 years:
CD: gross return = (1.05)4 = 121.55% gross Fund: return = .97 (1+r-0.006)4
.97 (1+r-0.006)4 = 1.2155 (1+r-0.006)4 = 1.2155/.97=1.253 1+r =0.006+1.2531/4
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Exchange traded funds & other BKM 4.6 investment
CEFs and ETFs: Similarities
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Closed-end funds and Exchange Traded Funds Buy a portfolio of stocks, issue fixed number of shares Shares trade like stocks
Buy Can
or sell at real time prices on an exchange short-sell
CEFs and ETFs: Differences
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ETF holdings are pegged to an index. CEFs are actively managed.
ETF holdings are transparent. Investors rarely know exactly what is in the portfolio backed by a CEF share. ETF fees are cheap. CEF fees are higher.
ETFs CEFs
generally trade very near NAV. generally trade at a significant discount to
NAV
CEFs and ETFs: Differences
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Arbitraging discounts (ETF or CEF prices are below NAV)
ETF investors have the right to exchange their shares for the underlying securities of the fund. Not so for CEFs.
Investors
can buy the ETF shares, redeem for underlying securities, sell securities for market value and make an instant arbitrage profit.
Arbitrage
using CEF: buy CEF, short all stock in CEF
CEFs and ETFs: Differences
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Arbitraging premiums (ETF or CEF prices are above NAV)
ETF investors have the right to exchange securities for new "creation units" of the ETF. Not so for CEFs.
Investors
can buy the underlying securities, redeem for ETF shares, sell the ETF at market value and make an instant arbitrage profit.
Arbitrage
using CEF: short CEF, buy all shares in CEF
ETFs vs. Open Ended Funds
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ETF advantages over open-ended mutual funds:
buy at real time prices, can short.
ETF disadvantages to open-ended funds
prices can move away slightly from NAV pay the bid-ask spread when trading an ETF
Hedge Funds
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A private investment pool, open to wealthy or institutional investors. Not registered as mutual funds and not subject to SEC regulation. Pursues more speculative policies. Name comes from the fact that hedge funds want to create market-neutral strategies by going long in some assets and going short in related assets.
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Hedge Funds vs. Mutual Funds Funds Mutual Hedge Funds
Buy publicly traded securities. Little use of leverage or short-sales. Hold broad mix of assets. Relatively low fees that do not depend on performance Investors can sell back shares at any time Heavy Regulation Relatively low Buy also non-public securities, currencies and commodities. Wide use of leverage and short-sales. Holdings are often concentrated. Relatively high fees that depend on performance. Investor money is "locked up" for long periods Light Regulation Very high investments necessary.
Investment methods Diversification Fees Withdrawls Regulation Initial investments
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Investing and Expectation
CFA pp. 129-131, 144,
Investing
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You're considering whether you should buy a stock for $10 What would you like to know?
What
Not
will be the return on my investment?
possible to predict with certainty
How
can I make a reasonable prediction about the return? Can I measure the uncertainty of that prediction? Pieces of a puzzle.
What
is the probabilityunderstand each95%?before we put Let's first I lose 10%? piece
them together.
Investing
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Random Variable:
A quantity whose value is uncertain. Example: the return on stock ABC over the next year
Event:
A specified set of outcomes Example: all outcomes less than -10%
PDF
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Example
0.75 for r = 10% P(r ) = 0.25 for r = -15%
Discrete PDF
Only
a finite number of outcomes The value of the function, P(r), tells us the exact probability of getting a given outcome. The sum of the probabilities across all possible outcomes must equal 1.
Not very realistic. Why do we use them?
Continuous PDF
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Example: Normal PDF
Continuous PDF
Infinite number of outcomes
Expected Value
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Expectation or Expected Value
Notation:
E[r] or
What Expectation is:
The
average value of an infinite number of outcomes of a random variable. we never can observe an infinite number of outcomes. we know the PDF, we can calculate the
But
If
Expectation
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For a discrete probability function with n n outcomes
E[r ] = i =1 ri P(ri )
0.75 for r = 10% P(r ) = Example 0.25 for r = -15%
Expectation
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For a continuous probability function
E[r ] = rP (r )dr
-
The idea is the same as for a discrete PDF. We just integrate across all possible values rather than sum over the discrete values.
Expectation
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At times we are interested in the expectation of a function of a random variable.
0.65 for r = 8% Exampler ) = P( 0.35 for r = -10%
E[r]=.65*(.08)+(.35)*(-.10)=0.017 What is E[r2]?
Expectation
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Given a discrete PDF we now know how to calculate the expected value of a random variable, and the expected value of a function of a random variable. PDFs are like number generating machines Sometimes, we don't know exactly everything about the machine, but we
Expectation
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Example
0.65 for r = 8% P (r ) = 0.35 for r = -10%
E[r]=.65*(.08)+(.35)*(-.10)=0.017 Suppose the above PDF generates 100 numbers, and we take the sample average.
100 is much less than infinity: the average of these 100 numbers is not likely to be
Estimation
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Suppose we observe the following numbers generated by a PDF: (.10, .10, . 10, .10, .10,.10, -.08,-.08,-.08,-.08). Estimate E[r]
[r]=(.10+.10+ . 10+.10+.10+.10-.08-.08-.08-.08)/10=0.028
0.65 for r = 10% P(r ) = 0.35 for r = -8%
Note that it looks like these numbers may have been generated from a PDF with two
Estimation
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Suppose we observe the following numbers generated by a PDF: (.10, .10, .10, .10, . 10,.10, -.08,-.08,-.08,-.08). Estimate
E[r2] [r2]=[.102+.102+ .102+.102+.102+.102+(-.08)2+(-.08)2+
(-.08)2+(-.08)2]/10=0.009
Also note that .65*(.102)+.35*(-.08)2=0.00874
Expectation Summary
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Given a discrete PDF we can find the n true expectation by
E[r ] = i =1 ri P(ri )
n We can also find the)true expectation E[ g (r )] = i =1 g (r P (ri ) of a function of a random variable, g(r) as
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Excel Random Number Generation
Data Data Analysis Random Number Generation "Number of Variables" = number of samples to create "Number of Random Numbers" = sample size Choose "Discrete" for the distribution "Value and Probability Input Range"
Here is where you specify the outcomes, and the probability of each outcome.
Example
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Statistics Rule #1
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Rule 1: If x and y are random variables and a and b are constants, then E[ax + by] = aE[x] + bE[y]
See equation 4-13 in CFA reading, p.153
Statistics Rule #1: Example
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0.80 for rA = 10% and rB = 5% (expansion) P ( rA , rB ) = ) 0.20 for rA = 0% and rB = 3% (recession
rz = .3rA + .7rB Portfolio z: 30% A, 70% B
return = (constant random variable)+ (constant random variable)
Note the return for portfolio z is of the form [r ] = .3 E[r ] + .7 E[r ] E
z A B
= .3 .08 + .7 .046 = 5.62%
we can therefore use stat rule #1
Statistics Rule #1: Example
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Let's see why stat rule #1 works. Note that the PDF (10%)the(5%) = 6.5% on for + .7 return .80 for r = .3 P( r ) = is portfolio z.20 for r = .3 (0) + .7 (3%) = 2.1
z z z
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Variance
CFA pp. 103-105.
Measuring Uncertainty
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For some stocks we can be more confident about our forecast than others.
Discrete Examples
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Investment #1
0.75 for r = 10% P(r ) = 0.25 for r = -15%
E[r]=3.75%
1 for r = 3.75% P(r ) = Investment #2 0 otherwise
Variance
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When using E[r] as a forecast, we know the bias will always be zero E[r-E[r]] Investment #1:
E[r-
E[r]]=.75*(.10-.0375)+.25*(-.15-.0375)=0
Investment #2:
E[r-E[r]]=1*(.0375-.0375)
=0
Variance
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How about finding the expected squared bias? Investment #1
E[(rE[r])2]=.75*(.10-.0375)2+.25*(-.15-.0375)2 =0.0117
Investment #2
Variance
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For a discrete PDF with n outcomes the true variance is:
Var (r ) = P (ri )( ri - E[r ]) 2
i =1
n
We never know the true PDF for stock returns But we do observe outcomes for stock returns
^2 = i Estimate the variance as theaverage n squared deviation from mean. (ri - r ) 2
Variance: Concept Check
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Suppose you are given four months of data on Microsoft. This data reveals that Microsoft earned the following returns: 0.10, 0.05, 0, -.03 over the past 4 months.
What is the expected return of Microsoft? What is the variance of Microsoft?
Variance: Concept Check
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Example:
Sample of returns: 0.10, 0.05, 0, -.03
(0 1: Step.10 + 0.05 + 0 - .03) = .03 r= 4
[(0.10 - .03) 2 + (0.05 - .03) 2 + (0 - .03) 2 + (-.03 - .03) 2 ] ^ = = .00245 Step 2: 4
2
Standard Deviation
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Standard Deviation = Square Root(variance)
Both for discrete and continuous PDF's
Notation Variance: 2 Standard Deviation:
Examples
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Stdev= 0.018
Stdev= 0.034
Statistics Rule #2
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If z = ax + c, where a and c are constants, and x is a random variable, then 2 [ z ] = a 2 2 [ x] [ z ] = a [ x]
Example
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Investment in a risky stock
0.75 for rS = 10% (State 1) P(rS ) = 0.25 for rS = -15% (State 2)
E[rS]=3.75% Var[rS]=0.0117
Investment 2in a risk-free bond 1 for r = %
B P (rB ) = 0 otherwise
E[rB]=2%
Example
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Your portfolio: 80% in the stock, 20% in the risk-free bond The realized return for this portfolio is
rp = (0.80 rS ) + (0.20 rB )
Note the realized return takes on the portfolio return = (constant random variable)+ (constant) form
2 p = 0.80 2 0.0117 = 0.0075
p = 0.80 0.0117 = 0.087
This means we can use stat rule #2
Example
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Now let's get a better understanding of stat rule 2 Note that the PDF for portfolio returns is
0.75 for rp = ( 0.80 10% ) + ( 0.20 2% ) = 8.4% P (rp ) = 0.25 for rp = (0.80 -15%) + (0.20 2%) = -11.6%
Let's find the variance as expected squareddeviation from meanrp ] = (0.80 .0375) + (0.20 0.02) = 3.4% E[ We know from stat rule #1 that the expected 2 p = .75 (.084 - .034) 2 + .25 (-.116 - .034) 2 = 0.0075 return is
p = 0.0075 = 0.087
Hence, the variance and standard deviation are
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The normal distribution
BKM 5.6
The Normal Distribution
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We often assume that returns are normally distributed.
Much analysis is far more tractable if we assume returns are normal.
The
normal distribution is symmetric absent this symmetry using variance as a risk measure is flawed. normal distribution is stable - if the return on stocks are normal, the return on any portfolio must also be normal normal distribution only requires two parameters mean (i.e. expected return) and
The
The
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Are returns normally distributed?
Probability of extreme losses greatly exceeds normal distribution probability.
Peak tends to be higher than predicted by normal distribution. (i.e. Leptokurtic)
Normal Distribution
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Assume the PDF for your investment return is a normal distribution. If we know E[r] and [r] we can integrate under the normal curve over any region using Excel! That is we can find the probability the return will fall within any given range.
Normal Distribution
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Assume the PDF for your investment return is a normal distribution with E[r]=10% and [r]=0.15
P(r<-20%) = normdist(0.20,0.10,0.15,1)=2.275%
P(r>30%)=1-normdist(0.30,0.10,0.15,1)=9.12% P(-20%<r<30)=
normdist(0.30,0.10,0.15,1)-normdist(-
Normal Distribution
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5% Value-at-Risk (VAR)
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Find the losses you expect to incur with 5% probability. Example:
Over the next year: E[r]=0.10, =0.20 5% VAR = 0.10 1.64*0.20=-0.23 During any given year, you have a 5% chance of losing 23% or more of your portfolio value.