# Notes of the Finance course in coursera

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## 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: $$\beta_a = \beta_e^L \frac{E_L}{E_L + D} + \beta_d \frac{D}{E_L + D}$$.