Finance Formulas

Question 1

MM I proposition

Answer 1

The value of the un-leveraged firm is equal to the value of the leveraged firm, perfect capital market assumption

A = U = E + D :
PV (assets) = unlevered equity = equity + debt

Question 2

MM proposition II

Answer 2

The cost of capital (the required rate of return on a portfolio company’s existing securities) of levered equity increases with the firm’s market value debt-equity ratio (D+ / E)

Question 3

Cost of capital of levered equity (return for investors)

Answer 3

based on MM II, r e = r u + D/E (ru - rd)

Question 4

Cost of capital of unlevered equity

Answer 4

rWACC =rU = rA without tax

Question 5

debt to value ratio - risk, cost of equity and debt

Answer 5

D/(E+D)

If the debt-to-equity ratio goes up,the perceived risk goes up - more debt

high considered risky, suggests thatthe company is financing a significant amount of its potential growth through borrowing.

If you don’t make your interest payments, the bank or lender can force you into bankruptcy.

Question 6

cost of equity vs. debt

Answer 6

• cost of equity higher: tax advantages, higher rate of return - wants to take risk since perfect capital market: costly
• return on debt higher if risk higher: debt holders don’t want to loose money, compensation needed

Question 7

Total value of the firm — with and without leverage, since interest expense and taxes included

Answer 7

EBIT ( - interest expense) - taxes = net income

• Might be present value of debt and equity (perpetuity)

Question 8

after tax WACC

Question 9

Interest tax shield — meaning, formula, present value, perpetual

Answer 9

The gain to investors from the tax deduct of interest expense on debt is called the interest tax shield

Interest tax shield = Corporate tax rate x Interest payments (risk-free rate * Debt)

For perpetual debt:
PV(ITS) = interest tax shield / risk-free rate = ITS

Question 10

Total value of levered firm with tax shield

Answer 10

V(L) = V(U) + PV(tax shield) tax shield discounted with risk-free rate

Question 11

The effective borrowing rate - effective return on debt

Answer 11

r(d) = r(d) (1-tc)

Question 12

personal taxes to debt & equity holders: effects

Answer 12

Debt holders (1-tc) - since only pay one type of tax
Equity (1-tc)(1-te) - since also pay dividend taxes on equity

Question 13

Value of equity according to MM

Answer 13

Equity = Value of Cashflows − Value of Debt (project price)

Value of cash flows:
1. cash flow = initial investment outcomes
2. debt repayment = cash flow - initial investment future value
3. total from both scenarios

Initial market value of unlevered equity —> PV of equity, expected cash flow is the value that can be gathered from investors now

Question 14

Trade-off theory - how to calculate distress costs

Answer 14

When securities fairly priced the original shareholders of a firm pay the present value of the costs associated with bankruptcy and financial distress.

total value of a levered firm = firm without leverage + PV(interest tax shield: tax * shield rate) - PV(financial distress cost)

V(L) = V(U) + PV(tax shield) - PV(distress costs)

Question 15

Value of a levered firm with agency costs: meaning and problem defenition

Answer 15

When a firm faces financial distress, shareholders can gain from decisions that increase the risk of the firm sufficiently, even if they have a negative NPV, so they’re gambling with debt holders money (risky investments that might or may not be profitable) — asset substitution problem

V(L) = V(U) + PV(tax shield) - PV(distress costs) - PV(agency costs of debt) - PV(agency benefits of debt)

Question 16

Tax for dividends or capital gains

Answer 16

(1-t(c)) — only corporate tax rate

Question 17

Tax on equity Income

Answer 17

(1-t(c))*(1-t(e)) — corporate tax with tax on equity income = double taxation

Question 18

Effective tax advantage on debt - How high must the marginal corporate tax rate be to offer a tax advantage?

Answer 18

1- (equity income taxes, double)/(debt taxes)

• every dollar debt holders get from interest payments costs equity holders (1-t*) dollars on after-tax basis
• to increase debt by 1, you cut the dividends by (1-t*) dollars
• in case of no personal taxes, t* = t(c)

If asked for a specific tax rate: change the formula.

Question 19

Debt Overhang Problem

Answer 19

Positive NPV investments are not made since all the benefits are appropriated by debt holders, equity holders do not gain from the investments since for them it’s a negative NPV investment

Question 20

Signaling theory of debt

Answer 20

• firm can use leverage to convince investors that it has information that the firm will grow, even if it cannot provide verifiable details about the sources of growth.
• investors know that firm would be at risk of defaulting without growth opportunities, so they will interpret the additional leverage as a credible signal of the firm’s confidence

Question 21

the dividend-capture theory

Answer 21

states that, absent transaction costs, investors can trade shares at the time of the dividend so that non-taxed investors receive the dividend.

Question 22

Unlevered firm — what is the price of each share?

Answer 22

P = PV(value of firm or cash flows) / # of shares

• might be perpetuity

Question 23

Unlevered firm — has cash reserves, what is the price of each equity share?

Answer 23

P = PV(value of firm) / # shares + cash / # shares

Question 24

Unlevered firm — dividend prices (cum/ex)?

Answer 24

P cum = div + PV(future div. - perpetuity)
P ex = PV(future div. - perpetuity)

Question 25

Company wants to raise money by selling shares to pay dividends for investors - How many shares has to sell?

Answer 25

raising amount in $ / share price = needed amount of shares to sell

Note: the overall dividend is higher, but the number of shares is also higher, but does not affect the shareholders.

the new div. per share
PV(firm v) / new # of shares

Question 26

Instead of dividends, repurchases on the open market: how many shares will repurchase & effect on shares outstanding and dividends?

Answer 26

1. cash for repurchasing ÷ price per share = amount of shares
2. original share amount - repurchase amount + share left outstanding

The net effect is that the share price remains unchanged, however future dividend changes

Question 27

How to calculate pre-tax and post-tax profits on dividends?

Answer 27

𝜋 𝑃𝑟𝑒−𝑡𝑎𝑥 = (𝑃 𝑒𝑥−𝑃 𝑐𝑢𝑚) + 𝐷 — pre-tax profit

𝜋 𝑎𝑓𝑡𝑒𝑟−𝑡𝑎𝑥 = (𝑃 𝑒𝑥−𝑃 𝑐𝑢𝑚) (1−𝜏 𝐶𝐺) + 𝐷 (1− 𝜏 𝐷) — after tax profit with equity capital gain and dividend taxes

Question 28

What is the optimal payout policy?

Answer 28

Almost same as a bond payment

(1 - tax on dividends) * all dividend payments, discounted with return on equity + price ex-div, discounted with return on equity

div and Pex - n+1, while discount with n

Question 29

equity in the firm has two effects — how do we quantify the two aspects separately? pre, post, ownership change

Answer 29

• Pre-Money Valuation: valuation of outstanding shares at the time
• Post-Money Valuation: valuation of shares at the price which new equity is sold

What is the pre-money valuation of your firm?

pre-money valuation = # shares (1.75mill) × price of funding ($1)

What is the post-money valuation of your firm?

new # of shares (all combined 1.75mill) × price of funding ($1)

How does your ownership change if you accept the round of financing?

******precentage of ownership calculated = (the money YOU contributed) / (the whole amount invested)******

Question 30

2. typical methods of setting the IPO price

Answer 30

1. Compute the present value of the estimated future cash flows — as usual
2. Estimate the value by examining comparable IPOs.price (post share amount x price of shares) / earnings = value of comparable firm

Question 31

the 4 IPO puzzles

Answer 31

1. Under-pricing : The Winner’s curse
2. Issuance is cyclical
3. Costs of IPO are high, not usual for companies to make
4. Long-run performance of newly public company usually bad

Question 32

B (risk) for unlevered firm

Answer 32

WCC w/o tax with beta on both

Question 33

B for equity

Answer 33

Bu + D/E (Bu - Bd)
same as return on equity

Question 34

In case of permanent debt - Firm value with leverage

Answer 34

V(L) = V(U) + t*D