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Chapter 4

• Draw a timeline illustrating a given set of cash flows.

• List and describe the three rules of time travel.

• Calculate the future value of:• A single sum.• An annuity, starting either now or sometime in the future.• An uneven stream of cash flows, starting either now or sometime in the future.

• Calculate the present value of:• A single sum.• An annuity, starting either now or sometime in the future.• An infinite stream of identical cash flows.• An infinite stream of cash flows growing at a constant rate• An uneven stream of cash flows, starting either now or sometime in the future.

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• Given three out of the following four inputs for a single sum, compute the fourth: • (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate.

• Given four out of the following five inputs for an annuity, compute the fifth: • (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate, (e)

periodic payment.

• Given cash flows and present or future value, compute the internal rate of return for a series of cash flows.

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A timeline is a linear representation of the timing of potential cash flows.

Drawing a timeline of the cash flows will help you visualize the financial problem.

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Assume that you made a loan to a friend. You will be repaid in two payments, one at the end of each year over the next two years.

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Differentiate between two types of cash flows

Inflows are positive cash flows.

Outflows are negative cash flows, which are indicated with a – (minus) sign.

Timelines can represent cash flows that take place at the end of any time period – a month, a week, a day, etc.

You will need to match the interest rate and time period

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Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments.

The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow.

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Financial decisions often require combining cash flows or comparing values. Three rules govern these processes.

Table 4.1 The Three Rules of Time Travel

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A dollar today and a dollar in one year are not equivalent.

It is only possible to compare or combine values at the same point in time. Which would you prefer: A gift of $1,000 today or $1,210 at a later date?

To answer this, you will have to compare the alternatives to decide which is worth more. One factor to consider: How long is “later?”

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To move a cash flow forward in time, you must compound it.

Suppose you have a choice between receiving $1,000 today or $1,210 in two years.

You believe you can earn 10% on the $1,000 today, but want to know what the $1,000 will be worth in two years. The time line looks like this:

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Future Value of a Cash Flow

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Problem Suppose you have a choice between receiving $5,000 today or $10,000 in five years. You believe you can earn 10% on the $5,000 today, but want to know what the $5,000 will

be worth in five years.

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Solution The time line looks like this:

In five years, the $5,000 will grow to:

$5,000 × (1.10)5 = $8,053

The future value of $5,000 at 10% for five years is $8,053.

You would be better off forgoing the gift of $5,000 today and taking the $10,000 in five years.

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To move a cash flow backward in time, we must discount it.

Present Value of a Cash Flow

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Problem Suppose you are offered an investment that pays $10,000 in five years. If you expect to earn a 10% return, what is the value of this investment today?

Solution

The $10,000 is worth:

$10,000 / (1.10)5 = $6,209

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Recall the 1st rule: It is only possible to compare or combine values at the same point in time. So far we’ve only looked at comparing single cash flows.

Suppose we plan to save $1000 today, and $1000 at the end of each of the next two years.

If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today?

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Annuity: equal payments at equal time intervals

Ordinary Annuity: Payments occur at the end of each period

Annuity Due: Payments occur at the beginning of each period Because you earn interest for an additional period an annuity due is worth (1+r) times as

much as an ordinary annuity PVAD = (1+r)PVA FVAD = (1+r)FVA

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The time line would look like this:

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THE THREE RULES OF TIME TRAVEL

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Based on the first rule of time travel we can derive a general formula for valuing a stream of cash flows

If we want to find the present value of a stream of cash flows, we simply add up the present values of each.

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Present Value of a Cash Flow Stream

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Problem What is the future value in three years of the following cash flows if

the compounding rate is 5%?

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Solution

Or

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Calculating the NPV of future cash flows allows us to evaluate an investment decision.

Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs).

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Problem Would you be willing to pay $5,000 for the following stream of cash flows

if the discount rate is 7%?

Solution The present value of the benefits is:

3000 / (1.05) + 2000 / (1.05)2 + 1000 / (1.05)3 = 5366.91

The present value of the cost is $5,000, because it occurs now.

The NPV = PV(benefits) – PV(cost)

= 5366.91 – 5000 = 366.91

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Perpetuities

When a constant cash flow will occur at regular intervals forever it is called a perpetuity.

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The value of a perpetuity is simply the cash flow divided by the interest rate.

Present Value of a Perpetuity

inperpetuity

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To find a simpler formula, suppose you invest $100 in a bank account paying 5% interest. As with the perpetuity, suppose you withdraw the interest each year. Instead of leaving the $100 in forever, you close the account and withdraw the principal in 20 years.

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For the general formula for the PV of an (ordinary) annuity:

For an annuity due:

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1 1 11

1

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NOTE: Because the first payment occurs today, this is called an annuity due

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Future Value of an (ordinary) annuity

For an annuity due

1 1

D 1 1 1 1

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Growing Perpetuity

Assume you expect the amount of your perpetual payment to increase at a constant rate, g.

NOTE: We must have g < r

growingperpetuity

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Growing Annuity The present value of a growing annuity with the initial cash flow c, growth rate g, and

interest rate r is defined as:

Present Value of a Growing Annuity

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1 1 1

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Problem You want to begin saving for your retirement. You plan to contribute $12,000 to the

account at the end of this year. You anticipate you will be able to increase your annual contributions by 3% each year for

the next 45 years. If your expected annual return is 8%, how much do you expect to have in your retirement

account when you retire in 45 years?

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Solution:The PV of the series of deposits (PVA) is

This tells us what the deposits are worth today. We want to know what the are worth in 45 years

FV = PV(1+r)n = $211,567(1+ .08)45 = $6,753,314

$12,000 1

.08 .03 1

1 .031 .08

$211,567

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The same time value of money concepts apply if the cash flows occur at intervals other than annually.

The interest and number of periods must be adjusted to reflect the new time period.

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Sometimes we know the present value or future value, but do not know one of the variables we have previously been given as an input.

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For example, when you take out a loan you may know the amount you would like to borrow, but may not know the loan payments that will be required to repay it.

Recall that

We need to find C:

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In some situations, you know the present value and cash flows of an investment opportunity but you do not know the internal rate of return (IRR), the interest rate that sets the net present value of the cash flows equal to zero.

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In addition to solving for cash flows or the interest rate, we can solve for the amount of time it will take a sum of money to grow to a known value.

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