Leverage and Margin
Kevin Crotty
BUSI 448: Investments
Where are we?
Last time:
- Adverse Selection
- Market structure
- Liquidity
Today:
- Leverage
- Margin
- Repurchase agreements
Leverage
Leverage
Leverage is investing borrowed money.
- The return, good or bad, on every $1 of your own money is amplified.
Example
Initial capital to invest of $100,000 + borrow $50,000
Buy $150,000 of stocks
| Assets | Liab/Eq | ||
|---|---|---|---|
| Stocks | 150,000 | Debt Equity | 50,000 100,000 |
| Total | 150,000 | Total | 150,000 |
- Leverage ratio \(=\frac{\text{Assets}}{\text{Equity}}\)
- Example is levered 1.5 to 1
- More jargon: 50% leverage
One possible future
Suppose the stocks go up 10% and you’re charged 2% interest on the loan (rolled into the debt balance)
| Assets | Liab/Eq | ||
|---|---|---|---|
| Stocks | 165,000 | Debt Equity | 51,000 114,000 |
| Total | 165,000 | Total | 165,000 |
The return is 14% (114,000/100,000-1).
You made 10% plus one half of (10% minus 2%) \(= 0.10 + 0.5(0.10-0.02) = 0.14\)
“one-half” because you borrowed 50%.
Levered return
Let \(w = \frac{\text{Debt}}{\text{Initial Equity}}\).
Levered portfolio return is:
\[ -w \cdot r_{\text{borrow}} + (1 + w) \cdot r_{\text{stock}} \]
We can rewrite this as:
\[ r_{\text{stock}} + w \cdot (r_{\text{stock}} - r_{\text{borrow}})\,.\]
The return in the example is:
\[ 0.10 + 0.5(0.10-0.02) = 0.14\]
Another possible future
- Suppose the stocks fell by 10%.
- You lose 10% plus one half of (\(-\) 10% \(-\) 2% ).
- So, your loss is 16% on your $100,000 investment.
| Assets | Liab/Eq | ||
|---|---|---|---|
| Stocks | 135,000 | Debt Equity | 51,000 84,000 |
| Total | 135,000 | Total | 135,000 |
- Check: 84,000/100,000 -1 = -16%.
The good and the bad
- You always make the stock return plus the fraction borrowed times (stock return minus borrowing rate).
- With 50% leverage and a 2% interest charge,
\[+10\text{%} \rightarrow +14\text{%}\]
\[-10\text{%} \rightarrow -16\text{%}\]
Levered S&P Returns
- SPY with leverage in today’s notebook
Margin
Margin
Margin: borrowing from your broker to purchase securities
Percent margin = \(\frac{\text{Equity}}{\text{Total Asset Value}}\)
Initial margin requirement set by the Fed’s Reg T: 50%
- Broker may set a higher initial margin requirement
Maintenance margin requirement set by broker
- Protects broker agains default by borrower if asset values drop.
Example with margin
Initial balance sheet
| Assets | Liab/Eq | ||
|---|---|---|---|
| Stocks | 150,000 | Margin loan Equity | 50,000 100,000 |
| Total | 150,000 | Total | 150,000 |
\[ \begin{align*} \text{Percent Margin} &= \frac{\text{Equity}}{\text{Total Asset Value}} \\ &= \frac{100,000}{150,000} \\ &= 66.67\% \end{align*} \]
Example with price drop of 10%
Balance sheet after stocks drop by 10% (and margin interest of 2% rolled into loan)
| Assets | Liab/Eq | ||
|---|---|---|---|
| Stocks | 135,000 | Margin loan Equity | 51,000 84,000 |
| Total | 135,000 | Total | 135,000 |
\[ \begin{align*} \text{Percent Margin} &= \frac{\text{Equity}}{\text{Total Asset Value}} \\ &= \frac{84,000}{135,000} \\ &= 62.22\% \end{align*} \]
Margin Calls
A margin call occurs when the percent margin falls below the maintenance margin set by the broker.
- Suppose the maintenance margin on the account in our example is 35%.
- How much could the stock value drop before a margin call occurs? (Ignore the interest expense on the margin loan.)
A margin call occurs when:
\[ \frac{\text{Equity}}{\text{Total Asset Value}} < \text{Maintenance Margin}\,.\]
Margin Calls
- \(S_0\) = initial stock value
- \(L\) = margin loan amount
- \(MM\) = maintenance margin percentage
- \(r\) = stock return
A margin call occurs when:
\[ \frac{S_0(1+r) - L}{S_0(1+r)} < MM\,.\]
Solving for \(r\):
\[ r < \frac{L}{S_0(1-MM)} - 1.\]
Example
Margin call occurs if stock return is less than:
\[r < \frac{50,000}{150,000(1-0.35)} - 1 = -48.7\%\]
Balance sheet with -50% return
| Assets | Liab/Eq | ||
|---|---|---|---|
| Stocks | 75,000 | Margin loan Equity | 50,000 25,000 |
| Total | 75,000 | Total | 75,000 |
\[\text{Percent Margin} = \frac{25,000}{75,000} = 33.3\% \]
Margin Loan Rates
- It pays to shop around.
- Interactive Brokers charges
- Fed Funds rate plus 1.5% on the first $100,000.
- and falling further after that.
- Fidelity rate schedule
Repurchase agreements
Repurchase agreements (repos)
- Simultaneously sell a security and agree to repurchase the same, or similar, asset at a later date at an agreed price.
- A repo can be thought of as a collateralized loan
- cash borrower pays the lender interest at the repo rate.
- Initial collateral is usually greater than the notional loan amount.
- difference is a haircut or repo margin.
Repo transaction
Repo rates
\[ \text{Repo rate} = \text{short-term rate} - \text{collateral-specific fee} \]
- General collateral: repo rates slightly below federal funds rate
- Special collateral: repo rates lower because cash lender (security borrower) wants a particular security
- Repo rates are lower:
- higher credit quality bonds
- more liquid bonds
- harder to find bonds
Term of repos
- Repos are short-term
- Majority are overnight
Source: Krishnamurthy, Nagel, Orlov
Numerical example
A dealer needs to finance $20 million par value of 10-year Treasury notes for 1 day. The current market value of the securities is $19,576,026.65. A corporation is willing to take the other side of the repo at a repo rate of 6% with a 1% haircut.
At initiation, the dealer surrenders the notes and receives $19,380,266.39 ($19,576,026.65*99%) in cash.
In 1 day, the corporation returns the notes and is paid $19,383,496.43 in cash. The interest on the cash loan is calculated as 3,230.04 (19,380,266.39 \(\cdot\) 6% \(\cdot\) (1/360).
Credit risk and repos
- Both parties are exposed to credit risk.
- The cash lender is exposed to the possibility of default on the cash borrower’s part.
- If the market value of the collateral declines, the lender may have a loss.
- The cash borrower is exposed to the possibility that the cash lender cannot return the collateral (if the market value of the collateral increases)
Mitigating credit risks
The haircut is designed to protect the cash lender. If the collateral market value declines, the lender may still be made whole if the drop is less than the haircut.
Higher haircuts for riskier borrowers and/or less liquid collateral.
Marking-to-market
- if collateral MV declines, cash borrower can send cash or additional securities to the cash lender.
- if collateral MV increases, cash lender can send cash or the collateral securities to the cash borrower
Empirical evidence on haircuts
Source: Krishnamurthy, Nagel, Orlov
For next time: Short-selling + Limits to arbitrage
BUSI 448