Equations

Question 1

What is the basic Future Value equation?

Answer 1

The Future Value, FV, of an amount today, PV, after t time periods with interest rate r per time period is given by the following equation:

𝐹𝑉=𝑃𝑉 ∗(1+𝑟)^𝑡

Question 2

What is the basic present value equation?

Answer 2

The Present Value, PV, of a Future Value, FV, taken after t periods of time with interest rate r per time period, is given by the following equation:

𝑃𝑉= 𝐹𝑉 X 1 / (1+𝑟)^𝑡

Question 3

What is the equation for PV of a perpetutity?

Answer 3

PV = C / R

A cash flow sequence that pays a fixed amount, C, at each point in time, forever. PV = PRESENT VALUE & R = interest / discount rate

Question 4

What is the equation for PV of a growing perpetuity?

Answer 4

A cash flow (C) sequence that increase by g% per time period, for ever. R is the interest rate.

PV = C / r - g

This only works when r > g

Question 5

What is the equation for present value of an annuities?

Answer 5

𝑃𝑉 =

C | C 1 |
— - | — X. ——— |
r | r (1+𝑟)^𝑇 |

A series of fixed cash flows, C, for some fixed (or finite) number of periods, T. R is the return of the market.

Question 6

Whats the equation for the future value of an annuity?

Answer 6

𝐹𝑉=

   | 1               1       | C X| —   -   ————  | (1+r)^T
   | r       r X (1+r) ^T|

where c = cash flow
r = return
T = time

Question 7

What is the Effective Annual Rate Of Return (EAR) equation?

Answer 7

If your money is compounded with frequency k, then the EAR is computed as follows:

EAR = ((1 + APR/K)^k) - 1

APR = annual percentage rate

Question 8

What is the equation for real cash flow?

Answer 8

Real cash flow =

nominal cash flow / (1 + i)^t
where i = inflation rate

Question 9

what is the equation for real interest rate?

Answer 9

real interest rate =

(1 + nominal / 1 + inflation ) - 1

Question 10

What is the gordan growth model?

Answer 10

The gordan growth model is

P0= Dividend / r - g

only apples when r > g

Question 11

What is the equation for net present value?

Answer 11

Net Present Value = Present Value of Future Cash Flows – Initial Investment

Question 12

What is the discounted payback period?

Answer 12

Discounted payback period is the minimum number of years, k, such that:

Cf1/ 1+ r + CF2 /(1+r) ^2 > investment

Discounted payback period, in words, is the minimum length of time, k, such that the sum of the discounted cash flows exceeds the initial investment

Question 13

What is the Internal Rate of Return (IRR)?

Answer 13

i=

c / 1 + IRR + ct / (1 + IRR) T

Question 14

What is the profitability index?

Answer 14

profitability index =
Net present value/ Investment

Question 15

What is the retrun for a stock equation?

Answer 15

Return = (𝑃1 − 𝑃0 + 𝐷𝐼𝑉)/𝑃0

Capital Gains: P1 – P0
Dividend: DIV

Question 16

Whats the equation for asset risk premium?

Answer 16

asset risk premium = Asset return – Return of a risk-free asset

Question 17

whats the equation for the average return of a stock?

Answer 17

Average Return:

R̃= 1/ T ∑ rt

Question 18

whats the equation for the variance of a stock?

Answer 18

Varaince
σ² =1/ T ∑ (rt - R̃)²

Question 19

What is the standard deviation/volitlity of a stock equation?

Answer 19

𝜎̂ = √1/ T ∑ (rt - R̃)²

Question 20

What is the equation for beta?

Answer 20

Y = a + b * X

Apply the same framework to compute market risk (beta)

Y is the return of a company’s stock

X is the return of a market index (proxy for market return)

a is the intercept (the expected return of the stock when the market return is zero).

b is the slope of the regression line, which in this context represents the beta.

Question 21

whats the equation for required return?

Answer 21

Required Return = Risk-Free Return + Risk Premium

Question 22

Whats the equation for Capital Asset Pricing Model (CAPM)?

Answer 22

CAPM E(rᵢ) =

rᶠ + 𝛽𝑖 X (E(rᵐ) - rᶠ)

rᶠ = expected return equal to a riskless asset

Beta

Assets with β = 1 should provide an expected return equal to the market (𝑟_𝑚)

Where:

E(ri): The expected return of the asset

rf: The risk-free rate (the return on a riskless asset, like government bonds)

Bi: Beta of the asset, which measures the asset’s sensitivity to market movements

E(rm): The expected return of the market

E(rm) - rf: The market risk premium, which is the extra return investors demand for taking market risk

Question 23

What is the equation for cost of capital?

Answer 23

Cost Of Capital =

total required income / value of investment

= D/V X rdebt + E/V X requity

= weighted average of cost of debt and cost of equity or WACC

Question 24

What is the after-tax cost of debt?

Answer 24

The after-tax (e.g., effective) cost of debt is:

(1-Tc) X r debt

Tc = tax

Question 25

What is the cost of equity equation?

Answer 25

R(equity) = Rf + B x (Rm - Rf)

Where:

1. R(equity):

The cost of equity is the return required by equity investors to compensate them for the risk of investing in the company.

It represents the minimum return that shareholders expect for providing capital to the business.

2. Rf (Risk-free rate):

The return on a riskless asset, such as government bonds.

It sets the baseline return for any investment, representing the compensation for the time value of money with no risk.

3. B (Beta):

Measures the volatility or sensitivity of the company’s stock returns relative to the market.

It reflects the company’s systematic risk, which cannot be diversified away.

Beta > 1: The stock is more volatile than the market.

Beta < 1: The stock is less volatile than the market.

4. Rm (Expected market return):

The return investors expect from the overall market, such as a broad market index like the S&P 500.

5. Rm - Rf (Market risk premium):

The extra return investors demand for taking on the risk of investing in the market instead of a risk-free asset.

Explanation

The cost of equity formula comes from the Capital Asset Pricing Model (CAPM). It calculates the return required by shareholders, considering the time value of money (Rf) and the risk associated with the company’s stock relative to the market (B x (Rm - Rf)).

Question 26

what is the equation for rprefered?

Answer 26

R(preferred) = Div / Price of Preferred Stock

Where:

1. R(preferred):

The cost of preferred equity is the return required by investors who hold the company’s preferred stock.

It represents the rate the company must pay to compensate preferred shareholders for their investment.

2. Div:

The annual dividend paid per share of preferred stock.

Since preferred stock dividends are typically fixed, this value does not change.

3. Price of Preferred Stock:

The current market price of a single share of preferred stock.

Question 27

What is the weighted average cost of capital equation?

Answer 27

WACC =

( D/V X (1-Tc)R debt )
+
(P/V X Rprefered)
+
(E/V X Requity)

1. D/V (Debt-to-value ratio):

The proportion of the company’s total capital that comes from debt financing.

: Total debt (e.g., bonds, loans).

: Total value of the firm’s financing, which is the sum of debt (D), preferred equity (P), and common equity (E).

D/V = Debt ÷ Total Capital.

2. (1 - Tc) (Tax adjustment):

This accounts for the tax shield provided by debt. Since interest payments on debt are tax-deductible, the company effectively pays less for its debt.

: The corporate tax rate.

3. R(debt):

The cost of debt, which is the effective interest rate the company pays on its borrowed funds.

4. P/V (Preferred equity-to-value ratio):

The proportion of the company’s capital structure financed by preferred stock.

: Total value of preferred stock.

P/V = Preferred Stock ÷ Total Capital.

5. R(preferred):

The cost of preferred equity, calculated as:
R(preferred) = Annual Preferred Dividend ÷ Price of Preferred Stock.

6. E/V (Equity-to-value ratio):

The proportion of the company’s capital structure financed by common equity.

: Total value of common equity (market capitalization).

E/V = Equity ÷ Total Capital.

7. R(equity):

The cost of equity, which is the return required by equity investors, calculated using the CAPM formula:
R(equity) = Rf + Beta x (Rm - Rf).

Question 28

What is Rdebt?

Answer 28

Rdebt: The bonds yeild or bank interest rate

Question 29

Perpetual NPV equation?

Answer 29

Perpetual NPV =
PV of investment - cost

McDonalds Inc. wants to introduce a new type of burger which would cost $30 million, but it will generate a perpetual cash flow of $5 million per year. Assuming that McDonalds’ WACC is 11%, is the investment worth doing?

5 mil / 11% - 30 million = 15.45 million

Question 30

what is the value of levered and unlveered firm under the Modigliani and Miller Proposition 1 (without taxes) ?

Answer 30

VL = VU = 𝐸𝐵𝐼𝑇/𝑊𝐴𝐶𝐶
​
Vl: Value of a levered firm (a firm with debt).

Vu: Value of an unlevered firm (a firm without debt).

EBIT: Earnings Before Interest and Taxes. This is the firm’s operating profit.

WACC: Weighted Average Cost of Capital, which is the firm’s overall cost of capital, accounting for both equity and debt financing.

Question 31

Modigliani & Miller Proposition 2 (without taxes) equation?

Answer 31

𝑟_𝑒^𝐿 = 𝑟_𝑒^𝑈 + (𝑟_𝑒^𝑈 – rd) ∗𝐷/𝐸

reL: Cost of equity for a levered firm (a firm with debt).

reU : Cost of equity for an unlevered firm (a firm with no debt).

rd: Cost of debt (the interest rate paid on the firm’s borrowings).

D: Market value of the firm’s debt.

E: Market value of the firm’s equity.

(r eU −r d ): The risk premium that reflects the additional return required by equity holders due to the firm taking on debt.

Question 32

What is the Interest Tax Shield equation?

Answer 32

Interest Tax Shield = 𝑇_𝑐×(𝑟_𝑑×𝐷)

Question 33

What is the PV of the interest tax shields, assuming perpetual cash flows?

Answer 33

PV (tax shield) =
(𝑇𝑐×(𝑟𝑑×𝐷))/𝑟𝑑 (same as 𝑇𝑐 ×𝐷)

Components:

PV(tax shield): Present value of the tax shield. This is the benefit a firm receives by deducting interest payments on debt from its taxable income.

T c : Corporate tax rate. It represents the percentage of taxable income that the firm pays in taxes.

r d: Cost of debt. This is the interest rate the firm pays on its debt.

D: Total amount of debt in the firm’s capital structure.

Question 34

what si teh equation for a RLe: Cost of equity for a levered firm (a firm with debt)?

Answer 34

R𝐿e =
Re𝑈 + (Reu𝑈 – rd)∗𝐷/𝐸∗ (1− TC)

RLe: Cost of equity for a levered firm (a firm with debt).

ReU: Cost of equity for an unlevered firm (a firm with no debt).

r d: Cost of debt (interest rate paid on the firm’s borrowings).

D: Market value of the firm’s debt.

E: Market value of the firm’s equity.

T C: Corporate tax rate. It reflects the tax shield on interest payments, which reduces the overall cost of debt.

Question 35

what is the Firm Value with Taxes and Bankruptcy Costs?

Answer 35

Firm Value with Taxes and Bankruptcy Costs

VL =
VU + PV (interest tax shields) – PV (bankruptcy costs)

Question 36

what is the beta of a market portfolio?

Answer 36

The beta of a market portfolio is always 1

Question 37

What is the z score equation?

Answer 37

The z score equation is:

Z = 1.2 X (𝑊𝑂𝑅𝐾𝐼𝑁𝐺 𝐶𝐴𝑃𝐼𝑇𝐴𝐿) / (𝑇𝑂𝑇𝐴𝐿 𝐴𝑆𝑆𝐸𝑇𝑆)

+ 1.4* (𝑅𝐸𝑇𝐴𝐼𝑁𝐸𝐷 𝐸𝐴𝑅𝑁𝐼𝑁𝐺𝑆) / (𝑇𝑂𝑇𝐴𝐿 𝐴𝑆𝑆𝐸𝑇𝑆)

+ 3.3* 𝐸𝐵𝐼𝑇 / (𝑇𝑂𝑇𝐴𝐿 𝐴𝑆𝑆𝐸𝑇𝑆)

+ 0.6* (𝑀𝐴𝑅𝐾𝐸𝑇 𝑉𝐴𝐿𝑈𝐸 𝑂𝐹 𝐸𝑄𝑈𝐼𝑇𝑌) / (𝑇𝑂𝑇𝐴𝐿 𝐿𝐼𝐴𝐵𝐼𝐿𝐼𝑇𝐼𝐸𝑆)

+ 1* 𝑆𝐴𝐿𝐸𝑆 / (𝑇𝑂𝑇𝐴𝐿 𝐴𝑆𝑆𝐸𝑇𝑆)

Question 38

How do you interpret the results of the z score analsyis?

Answer 38

If the the Z-score is less than 1.80 then it is a High Risk Area.

if the the z score is between 1.80 and 2.99 its a Grey Area

If the Z-score is above 2.99 then its a Low Risk Area

Question 39

what is the distance to default equation?

Answer 39

Distance to Default (DD) =

(𝑉 X(1+𝜇^Τ − 𝐹) /𝜎

=

(𝐸_𝑉 − 𝐹)/𝜎

𝐸_𝑉 − 𝐹 shows how much money firm value (𝐸_𝑉 ) is away from the default point (F)

The market value of the firm or the market value of its assets (V)

The face value of its liabilities (F)

The growth of the market value of assets (μ)

The volatility of the market value of assets (σ)