RETURN ON EQUITY CALCULATION

SUMMARY

When a company makes a profitable investment using leveraged funds, the yield on its equity is larger than the yield on its capital. Assuming that the cash returned by the investment is used to retire the equity and the debt in proportion to the leverage, the yield on the equity is

Yield on Capital - (Debt Rate x Leverage)
1 - Leverage

THE PROBLEM

This article considers the analysis of a single investment. The capital for the investment comes from equity and debt, and the amount of the debt divided by the total capital is the leverage factor. The leverage factor varies from zero to one, and is higher for higher leverage.

We assume that the cash returned by the investment is used to pay back the equity and debt principal in proportion to the leverage. For 90% leverage, 10% of each principal payment goes to pay off the equity and 90% goes to pay off the debt. Therefore, the leverage is constant, and the equity and debt balances are reduced to zero at the same time.

The cash returned by the investment less the principal payment equals the interest on the debt plus the earnings on the equity.


AN EXAMPLE

Consider the following investment:

Quarter		Cash From Investment		Earnings on Capital at 8% A.T.		Principal Payment		Capital Balance
0		0		0				-1,000,000
1		220,000		20,000		200,000		-800,000
2		316,000		16,000		300,000		-500,000
3		410,000		10,000		400,000		-100,000
4		102,000		2,000		100,000		0

The yield on capital is 8% after tax. However, the yield on equity is higher. For example, if 90% debt at 4% after tax were used to make the investment, we would have:

Quarter		Earnings on Equity		Equity Payment		Equity Balance		4% A.T. Interest on Debt		Debt Payment		Debt Balance
0		0				$-100,000		0		0		$900,000
1		$11,000		$20,000		$-80,000		$9,000		$180,000		720,000
2		8,800		30,000		$-50,000		7,200		270,000		450,000
3		5,500		40,000		-10,000		4,500		360,000		90,000
4		1,100		10,000		0		900		90,000		0

The principal payments are split between the equity and debt in proportion to the leverage. The interest on the debt is calculated from the debt rate and the previous debt balance. The earnings on the equity is the cash from the investment less the equity payment less the debt payment less the debt interest. A consistent yield of 44% is shown.


THE EFFECT OF LEVERAGE ON EARNINGS

The earnings on equity is less than the earnings on capital because of debt interest payments. The debt interest payments are equal to the earnings on capital times the leverage factor times the ratio of debt rate to yield on capital, or:

THE EFFECT OF LEVERAGE ON YIELD

The amount of the equity investment is equal to the total capital times one minus the leverage and therefore the yield will be higher by one over this difference. However, earnings are lower, and so the yield is different by a factor.

MATHEMATICS

The proceeding can be shown more easily as follows:

The same result can be obtained by solving the two equations:

Yc x C x dt = Ye x e x dt + Yd x D x dt     (Earnings Balance)
C = E + D                                                  (Capital Balance)

Where E and Y are the equity balance and yield rate, D and Yd are the loan balance and interest rate, and C and Yc are the capital balance and yield rate.