# Definitions and Equations

## Future Value

### Compounding Factor for T Periods (Non continuous)

$\displaystyle CF_T = (1+\frac{r}{m})^{mT}$ where r is the annual interest rate, m is the number of time per year the rate is compounded, and T is the number of years that the rate is compounded.

### Compounding Factor for T Periods (Continuous)

$\displaystyle CF_T = e^{rt}$ where r is the annual interest rate and T is the number of years that the rate is compounded.

### Effective Annual Rate (Non continuous)

$\displaystyle \hat{r} = (1+\frac{r}{m})^m - 1$

### Effective Annual Rate (Continuous)

$\displaystyle \hat{r} = e^r - 1$

### Effective Monthly Rate (Non continuous)

$\displaystyle \hat{r}_M = (1+\frac{r}{m})^{\frac{m}{12}} - 1$

### Sum of Regular Deposits

If C amount of money is saved every time period for T time periods with an interest rate of r, the value of the savings after T years is $\displaystyle PV = \frac{C}{r}*(1-\frac{1}{(1+r)^T})*(1+r)^T$

## Discounted Value

### Discount Factor over T Periods

$\displaystyle DF_T = \frac{1}{(1+\frac{r}{m})^{T*m}}$ where r is the interest rate and m is the number of times in the period T that the interest is compounded. When discounting a cash flow, make sure that if the cash flow is in nominal terms, then r is also in nominal terms. Similarly, if the cash flow is in real terms, then make sure that the r is also in real terms.

### Present Value with constant R

$\displaystyle PV = \sum_{t=1}^T \frac{C_t}{(1+r)^t}$

## Perpetuities

### Regular Perpetuities

The amount of money that has to be invested at an interest rate of r, to yield a constant yearly return of C is $\displaystyle PV = \frac{C}{r}$ . Note that the first time C is paid out one year after the investment.

### Deferred Perpetuities

The amount of money that has to be invested at an interest rate of r, to yield a constant yearly rate of return of C in t years from now is $\displaystyle PV = \frac{C}{r}*\frac{1}{(1+r)^t}$ . Note that the first time C is paid out is in year t+1.

### Growing Perpetuities

The amount of money that has to be invested at an interest rate of r, to yield a yearly rate return of C that grows by g every year is $\displaystyle PV = \frac{C}{r-g}$ . Note that the first time a payment is made is one year after the investment.

## Annuities

### Regular Annuities

The amount of money that has to be invested at an interest rate of r, to yield a constant yearly return of C for T years is given by $\displaystyle PV = \frac{C}{r}*(1-\frac{1}{(1+r)^T})$ . Note that the first annuity is paid out one year from the date of the investment. Note that the first deposit in the bank is being made in time period 1.

### Growing Annuities

The amount of money that has to be invested at an interest rate of r, to yield a yearly return of C which grows by g each year for T years is given by $\displaystyle PV = \frac{C}{r-g}*(1-(\frac{1+g}{1+r})^T)$ . Note that the first annuity is paid out one year from the date of the investment.

## Interest Rate

### Real Interest Rate

The real interest rate given the nominal interest rate r and the inflation rate i is given as $\displaystyle R=\frac{1+r}{1+i}-1$

## Bond Pricing

### Forward Rates

The interest rate to borrow a sum of money between years t-1 and t is $\displaystyle f_t = \frac{(1+r_t)^t}{(1+r_{t-1})^{t-1}}-1$

### Spot Rate

The spot rate is defined as the interest rate per year for T years starting today.

### Price of Bond given Spot Rates

The price of a bond maturing in T years with a semi annual coupon rate of i% and face value of C is given as:

$\displaystyle P = \frac{\frac{i%}{2}*C}{(1+r_{0.5})^{0.5}} + \frac{\frac{i%}{2}*C}{(1+r_{1.0})} + ... + \frac{(1+\frac{i%}{2})*C}{(1+r_T)^T}$

### Price of Bond given Discount Factors

The price of a bond maturing in T years with a semi annual coupon rate of i% and a face value of C is given as: $\displaystyle P = \frac{i%}{2}*C*DF_{0.5} + \frac{i%}{2}*C*DF_{1} + ... + (\frac{i%}{2} + 1)*c*DF_{T}$

### Quoted Treasure Bill

Treasury bills are quoted in terms of a discount rate and this can be converted to price according the the following formula:

$\displaystyle Price = F*(1-d*\frac{N}{360})$ where F is the face value of the bond, d is the quotes discount rate and N is the number of days in which the bond will reach maturity.

### Quoted Treasury Coupon Securities and Treasury Strips

Treasury Coupon Securities and Treasury Strips are quoted in 32nds so quote of 89:15 means that the price of the security is $\displaystyle 89 \frac{15}{32}$ .

### Bond Equivalent Yield

The bond equivalent yield is the y that solves $\displaystyle PV = \sum_{t=1}^T \frac{C_t}{(1+y)^t} + \frac{F}{(1+y)^T}$ where C is the coupon payment of the bond, T is the number of years till the bond's maturity and F is the face value of the bond.

### Yield of a Bond

In general, people in the industry define the yield of a bond as the effective annual rate under semi-annual compounding. Therefore, it is essential to convert the bond equivalent yield to this number.

## Cash from Operations

There are three ways to calculate cash from operations:

• $\displaystyle CFO = (R+X-E-D) * (1-t_c) + D$
• $\displaystyle CFO = R+X-E - t_c * (R+X-E-D)$
• $\displaystyle CFO = (R+X-E)*(1-t_c) + t_c*D$

where:

1. R is revenue
2. E is expenses
3. D is depreciation
4. X is extraordinary gains
5. $\displaystyle t_c$ is the corportate tax rate

## Equivalent Annual Cash Flow

If there are two investments that cost different amount for a different number of years, the cost of each investment can be compared by finding the equivalent annula cash flow for each of the investments (EACF). To find the EACF, solve for C where: $\displaystyle PV = \frac{C}{r} * (1-\frac{1}{(1+r)^T})$

Where:

1. PV is the present value of the cost of the investment
2. r is the interest rate
3. T is the number of years that the investment will continue to be useful.

## Profitability Index

The profitability index (PI) measures how profitable an investment is relative to the money invested. $\displaystyle PI = \frac{NPV}{C}$

Where:

1. NPV is the net present value of the profits generated from the investment
2. C is the capital invested