1. Buying on Margin and Short Selling
BKM: 3.6 – 3.7
Today we want to talk about what everyone does when they buy their house – buy on margin.
If you don’t own it, you can buy it

2. Buying on Margin Buying on Margin Borrow cash to buy more of the asset
Incur liability to pay back cash
Increase expected return
Increase standard deviation and probability of large losses.

3. Buying on Margin You put $10,000 in Microsoft (200 shares)
Price of MSFT=50
Suppose you have $10,000 and you expect Microsoft to perform well in the near future.
You put $10,000 in Microsoft (200 shares)
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
What are the returns of this trading strategy if Microsoft stock increases or falls by 25% during the next year?

4. Return of Buying on Margin
MSFT increases 25%
MSFT decreases 25%
Assets
Pay Back Loan
(Liability)
Equity
Return
32.5%
-42.5%
15,000*1.25=18,750
15,000*.75=11,250
-5,500
-5,500
5750
13,250

5. “Get” from Buying on Margin
“Get” = IE(1+rs) + L(1+rs) – L(1+rf)
IE = Investment Equity
L = Loan amount
rs= return on asset
rf = risk-free rate
Use algebra to rearrange:
P = IE(1+rs) + L(rs – rf)

6. Return from Buying on Margin
return = get/pay – 1
In this case, price = IE

7. Returns and Buying on Margin
Again, the return is a weighted sum of the returns of the assets in your portfolio.
Weights are a fraction of you 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.
The weights still add to one.

8. 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%
If Miscrosoft returns -25%
Your return: 1.5*(-.25)+(-.5)*.1=-42.5%

9. Returns and Buying on Margin
What is the return on the portfolio If MSFT price increases immediately by 25%?
Can ignore interest owed on margin account
Return = 1.5(.25) = 37.5%

10. Margin and Stat Rule #1 Example: E[rs]=16%, rf=10% w=1.5
E[rp]=1.5(.16)-0.5(.10)=19%

11. Margin and Stat Rule #2 Example: What is 5% VAR if I just buy MSFT?
ss=.18
w=1.5
sp=(1.5).18=.27
What is 5% VAR if I just buy MSFT?
*.18=-13.52%
What is 5% VAR on margin position?
*.27=-25.28%

12. Margin and Losses Lower bound for any stock: 0
Assets = Liabilities + Equity
Equity = Assets - Liabilities
Lower bound for assets = 0
Lower bound for equity = - Liabilities

13. Margin Limits SEC mandates limitations on borrowing.
Limit is defined in terms of “the margin”.
P=Price of security
S=Shares owned
L=Value of Loan

14. Margin Limits Initial margin must exceed 50%
Maintenance margin set by broker
MSFT example:
Initial margin =
(300* )/15000= 67%

15. Example of Margin Calls
Assume MSFT drops within a year by 40%.
Price of MSFT=
Your margin equals:
[300* (1.10)]/[300(30)]=39%
Assume Broker has set maintenance margin at 40%.
Your current margin is lower than the maintenance margin and you will receive a margin call from your broker.
Broker mandates you close position, or increase margin back to 50%

16. 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* =1000

17. Three Possible Options to Satisfy Margin Call
Increase your position in MSFT
How much?
[S* ]/S*30=.50
S=5000/(30-.5*30)=334
Buy 34 more shares

18. Example
How far can price fall immediately before getting a margin call? (Ignore interest on loan.)
(P* )/(P*300)=.40
P* =.40*P*300
P*(300-.4*300)=5000
P=5000/(300-.4*300)=27.78

19. Short Selling Securities are sold by someone who does not own them.
Borrow the securities from somebody.
Sell the securities at the current price.
Eventually, buy back the securities and return them to the owner.
Profit=Initial Price – Ending Price

20. Short Selling Example
Suppose you short MSFT. Two months later the price is $40. What is your profit?
Profit = $ = 10
On a “per-share” basis

21. Short Selling Example 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

22. Short Selling Steps 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)

23. Short Selling Steps 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

24. Step 2: Sell the Borrowed Shares
You must contribute your own investment equity to the collateral account.
is the “margin” on the short position.
Your total assets include
The proceeds from the short-sell
Any extra equity you have provided
Can be in the form of other security.

25. Short Selling Steps 3. Close out position
Buy the amount of shares you borrowed and return them to broker.
Collect all money in collateral account.

26. MSFT Example MSFT: $50/share You short sell 2000 shares
Initial Margin is 50%
What are Liabilities?
L=100,000=P*S
How much equity do you need to provide?
E/100,000=0.50
E=50,000
What are assets?
A=100,000+50,000=150,000

27. MSFT Example 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

28. MSFT Example Suppose price of MSFT is 60 at the end of the year.
What are total assets?
A=150000*(1.05)=157,500
What are liabilities?
L=2000*60=120,000
What is equity?
E=157, ,000=37500
What is return? 37500/ =-25%
What is new margin? 37500/120000=0.3125

29. Margin Example Margin call [150000-P(2000)]/P(2000) = 0.35
Maintenance margin is 35%
How far can price increase immediately before getting a margin call?
Ignore interest earned on collateral account.
Margin call
[ P(2000)]/P(2000) = 0.35
P=150000/ (.35* )=55.56

30. Three Possible Options to Satisfy Margin Call
Close out position
Buy the MSFT stocks for $55.56
Equity left: *55.56=38800
Reduce your liability – how much?
Assume the margin must increase to 50%
[ *S]/[55.6*S] = 50%
S=150000/(.5* )=1800
Buy back 2000 shares with new equity and return to broker

31. Three Possible Options to Satisfy Margin Call
Increase your collateral
How much?
[X-2000*55.56]/(2000*55.56)=.50
X=.5*2000* *55.56=166,680
Give broker an additional 16,680 in cash

32. Short Selling and Portfolio Weights
Again, we can use portfolio weights:
Equity:50,000
Position earning risk-free: 150,000
Weight=150/50=3.0
Position in MSFT: liability of 100,000
Weight=-100/50=-2
MSFT return: 60/50-1=20%
Portfolio return:
3*.05+(-2)*(.20)=-25%

33. Short Selling and Portfolio Weights
Suppose when you took the short MSFT position you expected MSFT to return -10% with a standard deviation of 0.25
What is expected return and standard deviation of your portfolio?

34. Short Selling and Stat Rules
E[rp]=3*.05+(-2)*(-.10)=35%
Stat Rule #2
= (-2)2*.252=0.25
= 2*(0.25)=0.50

35. Short Selling and Portfolio Weights
At the beginning of the year, what is your 5% VAR?
5% VAR= *.50=-47%
If you get a -47% return, what happened to MSFT?
-.47=-2*rH+3*.05
rH=(-.47-3*.05)/(-2)=0.31
Note: given that MSFT does have en expected return of -10% and a stdev of .25, the probability rH is 0.31 or greater is 5%

36. Borrowing as Short-Selling
You can think of borrowing as short-selling a risk-free bond.
Borrow $ at 10% Short-sell 1 bond
FV = 1000, Price =
Get $ now
Get $ now
Pay $1000 in future
Pay $1000 in future