16 Oct 2016

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1

Ch. 4 Time Value of Money

The Timeline

A cash flow involves either receiving or paying money

o if money is received (a cash inflow), the sign of the cash flow is positive

o if money is paid (a cash outflow), the sign is negative

A stream of cash flows is a group of several cash flows occurring at different points in time

In general, the length of time between cash flow dates is called a “period”

o A period could be different in terms of context

Depending on the particular context, a period could be a day, or a month, or a year, etc.

The times associated with a cash flow stream are date 0 (today), date 1 (one period from today),

date 2 (two periods from today), etc.

o The cash flows for these dates respectively are C0, C1, C2, etc.

A timeline can be used to represent a stream of cash flows

Each point on the timeline describes the beginning one period and the end of one period

o e.g. Date 1 is the end of Year 1 and beginning of Year 2

In general terms, with n periods:

A specific example: a friend agrees to lend you $5,000 today and in return you agree to pay your

friend $1,700 at the end of each of the next three months

3 Fundamental Rules of TVOM Calculations

Rule 1: Only cash flow values at the same point in time can be compared or combined

A dollar today and a dollar in 1 year are not equivalent

Cash flows must be converted into the same units or moved to the same point in time

Rule 2: To move a cash flow forward in time, compound it

Compounding: moving a value or cash flow forward in time

Let r denote the relevant interest rate per period

To compound a cash flow for n periods, multiply it by (1+r)n

Time value of money: the equivalent value of two cash flows at two different points in time

Simple interest: when an investment only earns interest on principal and no interest on accrued

interest

Compound interest: when an investment earns interest on principal and accrued interest

Geometric or exponential growth

Rule 3: To move a cash flow backward in time, discount it

To discount a cash flow for n periods, divide it by (1 + r)n

FV

(1+r)n

The value of a cash flow cash flow forward in

time

PV

The value today of a cash flow occurring in the

future, discount it to time 0

Discounted Value

The value of a cash flow moved backward in time

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2

Examples

Suppose that r = 4% per year:

i. Consider an initial cash flow of C0 = $500. What is the future value of this cash flow 8 years

from today?

ii. Consider instead a cash flow to be received 5 years from today of $1,200. What is the present

value of this cash flow?

Text figure 4.1 breaks down the future value of an investment over time into the original amount

invested, interest received on this original amount, and interest on previously received interest —

see spreadsheet handout

Valuing a Stream of Cash Flows

Suppose we want to determine the present value of a stream of cash flows C0, C1, C2… Cn:

Example

Suppose r = 3% and C0 = $200, C1 = $150, C2 = $225, and C5 = $500. Calculate PV0 and FV9.

Calculating the Net Present Value (NPV)

NPV = PV(benefits) PV(costs) this is equivalent to defining the NPV of an investment

Opportunity as the present value of the stream of cash flows of that opportunity

o Benefit cash inflow

o Cost cash outflow

Example: an investment that costs $10,000 today will produce cash flows of $4,000 one year from today

and $7,500 two years from today. If the appropriate discount rate is 5%, should you make this

investment?

Spend now!

Borrow $10,612.24 today for 2 years at 5% interest

o Invest $10,000 and spend $612.24

After 2 Years

You owe: $10,612.24 x 1.052 = $11,700

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3

Investment generates $4,000 x 1.05 + $7,500 = $11,700 after 2 years

--------------------------------------------------------------------------------

Spend After 2 years

Lend $10,000 at 5% interest $10,000 x 1.052 = $11,025

Invest in project have $11,700

Have extra $675 $612.24 x 1.052

Perpetuities, Annuities, and Other Special Cases

Recall that we can calculate the present value of a general cash flow stream by adding up the

present values of each of the cash flows, i.e.

In some cases a special pattern of cash flows allows us to use quicker formulas to calculate

present values

o Cash flows that are constant for all periods

o Cash flows that grow at a constant rate each period

Regular Perpetuities

Regular perpetuity: is a stream of equal cash flows received at constant time intervals that goes

on forever

o e.g. Consol: the British government bond that promises the owner a fixed cash flow every

year, forever

Unless otherwise stated, we will assume that the first payment of a regular perpetuity is received

one period from today:

o Referred to as a payment in arrears (first cash flow arrives at the end of the first period)

Present Value of a Regular Perpetuity

For a regular perpetuity with payment C and interest rate r,

The cash flows in the future are discounted for an ever increasing number of periods, so their

contribution to the sum eventually becomes negligible

C no subscript because cash flows are the same

t=1 because first payment is after first period

Infinity because it goes on forever

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