TVM: Multiple Cash Flows

Valuing Contracts with Multiple Cash Flows

Introduction

In this presentation we’ll cover the valuation of contracts that are comprised of multiple cash flows. These contracts will cover:

Annuities
Perpetuities
Growing Annuities and Perpetuities
Irregular Cash Flows

Annuities

An annuity is a contract which pays a fixed amount at the end of each period for a fixed number of periods. Many common financial contracts are annuities, such as fixed rate mortgages and auto loans. If the payments are in the beginning of a period (such as renting an apartment) the contract is known as an annuity due.

The annuity is defined by its:

`r` rate per period
`n` number of periods
`C` cash flow per period

It is important that each quantity you use is over the same period, i.e. don’t use a yearly rate with monthly payments.

The Present Value of an Annuity

To calculate the present value of an annuity we can simply discount each payment individually, to the same period, and sum them. In other words we can:

`PV_0 = \frac{C}{(1+r)^1} + \frac{C}{(1+r)^2} + ... + \frac{C}{(1+r)^n}`

Note the PV is at time 0 (one period before the first payment in time 1). This calculation is very easy to do in a spreadsheet. However annuities are old, so historically we have used a simplified version of the above equation. Specifically the above is equal to:

`PV_0 = C(\frac{1 - \frac{1}{(1+r)^n}}{r})`

Applications of the PV of an Annuity

You can use the above formula to find your monthly mortgage payment. Say you are going to borrow $150,000 to buy a house, and your 30-year mortgage rate is 5%. We can plug these values into the formula:

`\$150,000 = C(\frac{1 - \frac{1}{(1 + \frac{5\%}{12})^{12(30)}}}{\frac{5\%}{12}})`

rearranging for `C` (the monthly payment) gives:

`C = \$805.23`

PV of an Annuity Calculator

The Future Value (FV) of an Annuity

We can instead push each cash flow into the last period, and find the total value of the payments then. This is the FV of the annuity.

The FV at the last period of the annuity (time `n`) is simply:

`FV_n = C(1+r)^{n-1} + C(1+r)^{n-2} + ... + C`

This is equivalent to:

`FV_n = \frac{C(1 + r)^n - 1}{r}`

Applications of the FV of an Annuity

Calculating the FV of an annuity is most often used in retirement calculations. For example, if you put $300 per month into an account earning 4% annual interest, how much money would you have in the account in 30 years?

You will have `\frac{\$300(1 + \frac{4\%}{12})^{30(12)}}{\frac{4\%}{12}} = \$298,214.80`

Future Value of an Annuity Calculator

A Growing Annuity

A growing annuity is a contract which pays a constantly increasing amount at the end of each period for a set number of periods.

For example, the following is a growing annuity: a contract which pays `\$100` in the next period, and `\$100(1 + r)^i` in period `i`, where `i` ranges from 1 to the final period `n`, and `r` is the growth rate per period. The value of a growing annuity at time `0` is:

PV of a Growing Annuity

The PV of a growing annuity is:

`PV_0 = \frac{C}{(1+r)} + \frac{C(1+g)}{(1+r)^2} + ... + \frac{C(1+g)^{n-1}}{(1+r)^n}`

which can simplified to:

`PV = C(\frac{1-(\frac{1+g}{1+r})^n}{r - g})`

Feel free to try the calculation, and check your answer on the calculator below.

FV of a Growing Annuity

The FV of a growing annuity (in the last period `n`) is:

`FV_n = C(1 + r)^{n-1} + C(1+g)(1+r)^{n-2} + ... + C(1+g)^{n-1}`

which can be simplified to:

`FV_n = C(\frac{(1+r)^n - (1+g)^n}{r - g})`

where `r \ne g`.

Again, you can try the calculation, and check your answer on the calculator below.

Perpetuity

A perpetuity is a contract which pays a fixed amount at the end of each period for an infinite number of periods. Despite having an infinite number of payments, the PV is a finite amount (assuming a positive interest rate). This is because later payments become negligible.

The present value of a perpetuity is:

`PV = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \ldots =\frac{C}{r}`

Applications of the Perpetuity

Likely the best known perpetuity is preferred stock. Preferred stock pays a fixed dividend, unlike common stock whose dividend changes over time.

Say ABC company’s preferred stock pays a fixed dividend of $7 per year every year. The discount rate is 10%. Then the preferred stock is worth:

`PV = \frac{\$7}{0.10} = \$70`

A Growing Perpetuity

A growing perpetuity is a contract which pays a constantly increasing amount at the end of each period for an infinite number of periods. That is, if the first payment is `\$C`, then the following payments are `\$C(1+g)`, `\$C(1+g)^2` off to infinity.

Assuming `g < r` the PV of the growing perpetuity is:

$PV = + + + … =

If `g \geq r` then the value is infinite.

Applications of the Growing Perpetuity

Perhaps the most famous application is the Gordon Growth Model of stock valuation. The model assumes a stock’s dividends grow at a constant rate. Since stock is infinitely lived, we can find the PV of the dividends (the stock’s value), as the PV of a growing perpetuity.

Say XYZ Corp pays yearly dividends, and next year’s dividend will be $5. XYZ’s dividends are expected to grow at 3% in perpetuity, and its discount rate is 7%. The value of XYZ’s stock is:

`PV = \frac{\$5}{7% - 3%} = \$125`

Irregular Cash Flows

A contract with Irregular cash flows is any contract that has a cash flow structure other than those listed above.

As mentioned earlier, annuity formulas allowed computationally easy valuation, and therefore use, of the above contracts. However present computing power makes valuation of any stream of cash flows very easy.
So the valuation of irregular cash flows can simply be done by applying the formulas for single cash flows to each individual cash flow in the contract.