PV of a Single Cash Flow

Watch Video 1.1, which explains the PV of a single cash flow.

Now that we have a working understanding of single cash flow calculations, let's look at an example.

Single Cash Flow Example

If you pay $1,100 for an investment project that will return $1,611 in 5 years, is it a good investment? Assume i = 10%.

 

PV of Multiple Uneven Cash Flows

A multiple uneven cash flow is a case where different amounts of cash flows are spread over several periods.

Multiple Uneven Cash Flows Example

You want to zero out your bank account over the next 3 years using the following cash flows (CF): $1,000 in Year 1, $600 in Year 2, and $400 in Year 3. Find the PV of your account assuming a discount rate of 10%.

PV of Annuities

An annuity is a fixed stream of payments (cash flows) for a fixed number of periods. Annuities are often used for a retired person to secure a steady cash flow for a fixed number of years or for as long as the retired person remains living. A lottery jackpot may also be awarded as an annuity, a fixed amount every year for a fixed number of years.

Annuity Example

Find the PV of a stream of constant cash flows (CFs): a payment of $1,000 per year for 5 years. Assume a discount rate of 10%. Payments are made at the end of each period.

Ordinary Annuity and Annuity Due

In an ordinary annuity, payments are made at the end of each period. In an annuity due, payments are made at the beginning of each period. Consider Figure 1.1 to help you understand this concept.

It will be important to keep in mind that if you are using Excel to calculate PV of annuities, the value for type will change from 1 to 0 depending on when the payment is made. As indicated in the image, the type will be 0 for an ordinary annuity and 1 for annuity due.

Since the cash flows of an annuity due are the same cash flows from an ordinary annuity shifted one period earlier, we get the following formulas:

$PV(\text{annuity due for} t \text{periods})=\left(1+r\right)\times \text{PV}\left(\text{ordinary annuity for} t \text{periods}\right)$

 

$PV\left(\text{annuity due for }t \text{periods}\right)=\left(1+r\right)\times PMT\times \left(\frac{1}{r}\right)\times \left(1-\frac{1}{{\left(1+r\right)}^t}\right)$

Equivalently, 

$PV\left(\text{annuity due for} t \text{periods}\right)=PMT+\left(1+r\right)\times PV\left(\text{ordinary annuity for }\left(t-1\right)\text{ periods}\right)$

 

$=PMT+PMT\times \left(\frac{1}{r}\right)\times \left(1-\frac{1}{{\left(1+r\right)}^{\left(t-1\right)}}\right)$

 

Perpetuity: Annuity With No Maturity

A perpetuity is an annuity with no maturity, or a stream of fixed-amount cash flows that continues forever. The United Kingdom government has issued perpetuities in the past, called consols. Preferred stocks promise a fixed amount of dividend every period forever.

This is the formula for PV of perpetuity:

${\mathrm{PV}}_{\text{Perpetuity}}=\frac{\text{Payment}}{r}$

 

Perpetuity: Annuity With No Maturity Example

Find the PV of a stream of $1,000 per year forever. Assume an interest of 10%.

Growing Perpetuity

A growing perpetuity is a stream of payments that start in Year 1, growing at a rate, g, each year successively forever.

$C{F}_1=CF \text{at} t =1; C{F}_2=CF\times \left(1+g\right)\text{ at} t =2; \dots , C{F}_t=CF\times {\left(1+g\right)}^{\left(t-1\right)}\text{ at} t= t, C{F}_{t+1}= CF\times {\left(1+g\right)}^t \text{at} t=n+1, \dots$

Then PV of (CF₁, CF₂, ..., CF_n+1, ..., CF_forever) at t = 0 is

$P{V}_0 =\frac{C{F}_1}{\left(r-g\right)}$

This is often called the Gordon model and is used in stock valuation. Take note that the left side of the equation, PV, is a value at time 0, while CF on the right side is at t = 1, one period later.

Growing Perpetuity Example

In 2011, ABC Company paid $2 dividend per share. The company announced that all future dividends will grow at 3% annually.

1. What will be the dividend amount in 2015?
2. What is the present value in 2011 of all future dividends starting in 2012? Assume an interest rate of 13%.

Growing Annuity

A growing annuity is a stream of payments that start in Year 1, growing at a rate, g, successively until year n, then stop.

$C{F}_1=CF \text{at} t =1; C{F}_2=CF\times \left(1+g\right) \text{at} t =2; \dots , C{F}_n=CF\times {\left(1+g\right)}^{\left(n-1\right)} \text{at} t= n$

 

$$\begin{gathered} PV\text{ of}\left(C{F}_1, C{F}_2, ... , C{F}_n\right) \text{at} t=0 \text{is:} \\ PV(\text{Growing Annuity at} t=0): \\ P{V}_0 =\frac{C{F}_1}{r-g}\left(1-\frac{{\left(1+g\right)}^n}{{\left(1+r\right)}^n}\right) \end{gathered}$$

Growing Annuity Example

John will retire 1 year from today, and he expects a Social Security payment of $30,000 next year. It is expected to grow annually at 3%, and he expects to receive Social Security payments for the next 20 years. What is the present value of his Social Security payments? Use a 10% discount rate.