# Notes of the Finance course in coursera

This pages uses MathJax extensively. It might not be displayed correctly when JavaScript is disabled.

## Week 1: Time value of money

- \(p\): the original amount of money
- \(n\): number of periods (months, years etc)
- \(r\): interest rate in one period

### Future value

The future value of \(p\) is \(\mathrm{FV}(r, n, 0, p) = p (1+r)^n\).

### Present value

The present value of \(p\) is \(\mathrm{PV}(r, n, 0, p) = \frac{p}{(1+r)^n}\).

## Week 2: Multiple Payments Annuities

- \(C\): Cashflow
- \(\mathrm{PMT}\): Payment

### The Future Value of a Stream of Cash flows

The Future Value of a Stream of Cash flows as of \(n\) Periods from now: \(\mathrm{FV} = \sum_{k=1}^n C_k (1 + r)^{n-k}\).

### The Present Value of a Stream of Cash flows

\(\mathrm{PV} = \sum_{k=1}^n \frac{C_k}{(1+r)^k}\)

### The Future Value of an Annuity

The Future Value of an Annuity paying \(C\) at the *End* of each of \(n\) Periods:
\(\mathrm{FV} = C \, \mathrm{FAF}(r, n)\)
where \(\mathrm{FAF}\) is the \(\mathrm{FV}\) Annuity Factor.

\(\mathrm{FAF}(r, n) =\frac{(1 + n)^n - 1}{r}\)

### The Present Value of an Annuity

The Present Value of an Annuity is: \(\mathrm{FV} = C \, \mathrm{PAF}(r, n)\) where \(\mathrm{PAF}\) is the \(\mathrm{PV}\) Annuity Factor.

\(\mathrm{PAF}(r, n) = \frac{1}{r} \left( 1 - \frac{1}{(1 + r)^n} \right)\)

### The Present Value of a growing Annuity

The Present Value of an Annuity growing at rate \(g\) is: \(\mathrm{PV} = C \, \mathrm{PAF}(r, n, g)\).

\(\mathrm{PAF}(r, n, g) = \frac{1}{r - g} \left( 1 - \frac{(1 + g)^n}{(1 + r)^n} \right)\).

### The Effective Annual Rate

The Effective Annual Rate (EAR) of \(k\) payments in a year is: \(\mathrm{EAR} = \left( 1 + \frac{r}{k} \right)^n - 1\).

### The Present Value a Perpetuity

The Present Value of a Perpetuity is: \(\mathrm{PV} = \frac{C}{r}\).

The Present Value of a Constant Growth Perpetuity is: \(\mathrm{PV} = \frac{C_1}{r - g}\).

## Week 3: Net Present Value

- \(C_k\): Cashflow at time \(k\)
- \(C_0\): Initial investment (likely to be negative)

### The Net Present Value of a Stream of Cash flows

\(\mathrm{NPV} = \sum_{k=0}^n \frac{C_k}{(1+r)^k}\).

### The Internal Rate of Return

The \(\mathrm{IRR}\) is the rate \(r\) that will give a \(\mathrm{NPV}= 0\).

For a perpetuity, the \(\mathrm{IRR}\) can be written as: \(\mathrm{IRR} = \frac{\mathrm{Profit}}{\mathrm{Investment}}\).

## Week 5: Bonds

### Discount Bonds (zero coupon bonds)

In a discount bond, the government borrows money \(P\) at time 0 and returns \(\mathrm{Face \, Value}\) at the end of \(n\) periods.

The price of a discount bond is: \(P = (\mathrm{Face \, Value}) \, \mathrm{PV}(r, n) = \frac{\mathrm{Face \, Value}}{(1+r)^n}\).

The rate \(r\) of a zero coupon bond is called Yield to Maturity.

## Week 5: Stocks

### The Stock Price Formula

The price of a share is: \(P_0 = \sum_{k=1}^n \frac{\mathrm{DIV}_k}{(1+r)^k} + \frac{P_n}{(1+r)^n}\).

### Growth

- \(\mathrm{EPS}\): cash flow per share
- \(\mathrm{PVGO}\): \(\mathrm{PV}\) of Growth Opportunities

The price of a share is: \(P_0 = \frac{\mathrm{EPS}}{r} + \mathrm{PVGO}\).

## Week 8: Diversification

### Diversification

The risk of an \(n\) asset portfolio is: \(\sigma^2 (R_p) = \sigma_p^2 = \sum_i x_i^2 \sigma_i^2 + \sum_{i \ne j} 2x_ix_j\sigma_{ij}\).

### Risk and Return: CAPM

The relationship between risk (beta) and return is linear, with the following form: \(r_i = r_f + (r_m - r_f) \beta\), where:

- \(r_i\): expected rate of return on the equity of the project/idea/firm \(i\)
- \(r_m\): expected rate of return on the "market" portfolio
- \(r_m - r_f\): average market risk premium

## Week 9: Debt and Cost of Capital

### Cost of Capital

- \(E(R_d)\): required rate of return on debt
- \(E(R_e^L)\): required rate of return on the leveraged equity of the firm

Under perfect capital markets, \(E(R_a)\) is just the weighted average of the equity and debt cost of capital, or the weighted average cost of capital (\(\mathrm{WACC}\)):

\(\mathrm{WACC} = E(R_a) = \frac{D}{E_L + D} E(R_d) + \frac{E_L}{E_L + D} E(R_e^L)\).

The expected rate of return on equity of a levered firm increases in proportion to the debut-equity ratio (\(D/E\)), expressed in market values: \(E(R_e^L) = E(R_a) + \frac{D}{E_L} \left( E(R_a) - E(R_d) \right)\).

Similarly, the risk of equity is: \(\beta_e^L = \beta_a + \frac{D}{E_L} \left( \beta_a - \beta_d \right)\).

Or written in a different form: \(\beta_a = \beta_e^L \frac{E_L}{E_L + D} + \beta_d \frac{D}{E_L + D}\).