Set out how leverage shapes a firm's returns, including the leverage effect, and show how different transaction types alter leverage and the balance sheet.

FRM® Part II · Liquidity and Treasury Risk Measurement and Management · Liquidity and Leverage · FRM2.LTR.2.d

The idea

Balance-sheet leverage is assets divided by equity. Debt-to-equity is that figure minus one. Leverage magnifies equity returns in both directions. With ROA the return on assets, Kd the cost of debt and D/E debt-to-equity, the return on equity is ROE = ROA + (ROA − Kd) × D/E. So equity earns more than assets only when the asset return exceeds the borrowing cost. ROA is before interest, and ROA and Kd are both after tax or both ignore tax. Equity volatility is roughly (assets ÷ equity) × asset volatility. This is the leverage effect. Transactions change leverage differently. Financing a purchase by repo or margin loan grows assets and liabilities together, so assets ÷ equity rises. Selling assets to repay debt shrinks both and lowers it. Derivatives and total return swaps carry embedded leverage, meaning notional relative to margin, that is largely off the balance sheet. So balance-sheet leverage understates economic leverage. With a haircut h, the maximum assets ÷ equity is 1 ÷ h. Typical traps are using D/E where assets ÷ equity is needed, and forgetting that ROE falls below ROA when ROA is below Kd.

Formulas

Return on equity with leverage

ROE = ROA + (ROA − Kd) × (D ÷ E)

ROA: the return on assets, before interest
Kd: the cost of debt
D: debt
E: equity

ROA and Kd must both be after tax or both ignore it. The multiplier is debt-to-equity, not assets ÷ equity; when ROA is below Kd, leverage pulls ROE below ROA.

Equity volatility under leverage

σE = (A ÷ E) × σA

A: total assets
E: equity
σA: the volatility of asset returns

The leverage effect, approximately. A fall in asset value equal to E ÷ A wipes out the equity, which is why a price move close to the haircut is so damaging.

A practice question

From the question bank, with the reason each wrong option is wrong.

A firm has total assets of USD 600 million and equity of USD 48 million. What is its balance-sheet leverage, measured as assets divided by equity?

11.5Why not: This is debt-to-equity, assets/equity minus one.
0.08Why not: Inverts the ratio: this is equity divided by assets.
1.09Why not: Divides assets by debt rather than by equity.
12.5 - correct

Assets/equity = 600/48 = 12.5. Debt is 552, so debt-to-equity is 11.5; the two measures differ by exactly one.

A harder one

A firm earns a return on assets of 2.5% and pays 4.0% on its debt. Its debt-to-equity ratio is 7. What is its return on equity?

13.0%Why not: Reverses the sign of the spread, treating the borrowing cost in excess of ROA as a gain.
-9.5%Why not: Uses assets/equity (8) rather than debt/equity (7) as the multiplier.
-8.0% - correct
2.5%Why not: Assumes leverage has no effect on ROE, ignoring that the cost of debt exceeds the asset return.

ROE = 2.5% + (2.5% - 4.0%) x 7 = 2.5% - 10.5% = -8.0%. Leverage magnifies losses when ROA is below the cost of debt.

The hardest tier

A fund has assets of USD 400 million and equity of USD 50 million. Asset prices fall 5%. To restore assets/equity of 8 it sells assets at the new prices and uses all proceeds to repay debt. How much must it sell, closest to?

USD 160 millionWhy not: Applies the target to the post-loss equity but starts from the pre-loss asset value of 400.
USD 110 millionWhy not: Treats the target of 8 as debt-to-equity, so the required assets/equity is 9 and the sale is too small.
USD 20 millionWhy not: Sells only an amount equal to the loss rather than the amount needed to restore the ratio.
USD 140 million - correct

After the fall: assets = 380, equity = 30 (leverage 12.7). Target assets = 8 x 30 = 240. Sale = 380 - 240 = 140. Losses cut equity, so leverage jumps and forces deleveraging much larger than the loss.

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