RETURN ON EQUITY CALCULATION

SUMMARYWhen 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 PROBLEMThis 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 EXAMPLEConsider the following investment:

QuarterCash From

InvestmentEarnings

on Capital

at 8% A.T. Principal

PaymentCapital

Balance0 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:

QuarterEarnings

on EquityEquity

PaymentEquity

Balance4% A.T.

Interest

on Debt Debt

PaymentDebt

Balance0 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 EARNINGSThe 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 | = | x | Debt Rate | x | Leverage |

= | - | x | Debt Rate | x | Leverage |

THE EFFECT OF LEVERAGE ON YIELDThe 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 Rate | x | Leverage ] |

= | x | 1 - | Debt Rate/Yield on Capital1 - Leverage |
x | Leverage |

MATHEMATICSThe proceeding can be shown more easily as follows:

Earnings on Capital | = | + | Debt Interest |

= | + | x | x |

= | + | x | Earnings on Capital | x |

= | x | 1 - | Debt Rate | x |

= | x |

= | Earnings on Equity | = | x | 1 - | Debt Rate/Yield on Capital x 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.