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:

 Interest on Debt = Earnings on Capital x Debt Rate Yield on Capital x Leverage
 Earnings on Equity = Earnings on Capital - Earnings on Capital x Debt Rate Yield on Capital x Leverage

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.

 _____1_____1 - Leverage x [ 1 - Debt RateYield on Capital x Leverage    ]
or
 Yield on Equity = Yield on Capital x 1 - Debt Rate/Yield on Capital1 - Leverage x Leverage

MATHEMATICS

The proceeding can be shown more easily as follows:

 Earnings on Capital = Earnings on Equity + Debt Interest
 Earnings on Capital = Earnings on Equity + Debt Rate x Capital Balance x Leverage
 Earnings on Capital = Earnings on Equity + Debt Rate x Earnings on CapitalYield on Capital x Leverage
 Earnings on Equity = Earnings on Capital x 1 - Debt Rate Yield on Capital x Leverage
 Amount on Equity = Amount of Capital x (1 - Leverage)
 Yield on Equity = Earnings on EquityAmount of Equity = Yield on Capital x 1 - Debt Rate/Yield on Capital x Leverage1   -    Leverage

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.