Chapter 6 Time Value of Money. Introduction Why money has a time value –The opportunity cost of capital concept Time value of money and risk –Typically

Chapter 6Chapter 6Chapter 6Chapter 6

Time Value of MoneyTime Value of Money

Introduction• Why money has a time value

– The opportunity cost of capital concept

• Time value of money and risk– Typically risk and expected returns are related

to each other

• Investment decisions and the Time Value of Money– Allows us to compare present and future cash

flows on a comparable basis (apples to apples).

Basic Time DiagramBasic Time Diagram

• Principal (P): The amount borrowed or invested

• Interest rate (i): A percentage of the outstanding principle.

• Number of periods (n): (years or fractional of) that principal is outstanding.

• Present value (PV): present value of a single amount of cash, or series of cash flows.

• Future value (FV): future value of a single amount of cash, or series of cash flows.

Variables Used in Time Value of Variables Used in Time Value of Money ComputationsMoney Computations

Time Value of Money – From Present Value to Future Value

and Back• A simple case is investing $1.00• Use of a time line is helpful

Present and future values are calculated for us in the text:

• Future value of $1, Table 6.2• Present value of $1, Table 6.3• Future value of an annuity of $1, Table 6.5• Present value of an annuity of $1, Table 6.7

Compound Interest TablesCompound Interest Tables

Present Value of a Single (Lump) Sum

Rate

Periods 10% 12% 14%

1 0.90909 0.89286 0.87719

2 0.82645 0.79719 0.76947

3 0.75131 0.71178 0.67497

4 0.68301 0.63552 0.59208

5 0.62092 0.56743 0.51937

Rate

Periods 10% 12% 14%

1 0.90909 0.89286 0.87719

2 0.82645 0.79719 0.76947

3 0.75131 0.71178 0.67497

4 0.68301 0.63552 0.59208

5 0.62092 0.56743 0.51937

Excerpt from Present Value of $1, Table 6.3

2 0.82645 0.79719 0.76947 3 0.75131 0.71178 0.67497 4 0.68301 0.63552 0.59208 5 0.62092 0.56743 0.51937

2 0.82645 0.79719 0.76947 3 0.75131 0.71178 0.67497 4 0.68301 0.63552 0.59208 5 0.62092 0.56743 0.51937

Present Value of a Single (Lump) Sum

Present value factor of $1 for 2 periods at 12%.Present value factor of $1 for 2 periods at 12%.

$100 $100 ×× 0.797 = $79.70 present value 0.797 = $79.70 present value

Future Value Exercise• $100,000 is put into a mutual fund yielding a

12% annual return. How much would the fund be worth in 23 years?

• $100,000 * 13.552 = $1,355,200

Present Value of a Series of Cash Flows

• Annuity – a series of receipts (or payments) of the same amount spaced over regular time intervals (periods)– Example– Beauty of the PV factors for annuities

• Uneven series of receipts (or payments)– Example– No single PV factor may be used, the present

value of each cash flow must be calculated, then summed

Annuity

11 22 33 44 55 66

$100$100 $100$100 $100$100 $100$100 $100$100 $100$100

An investment that involves a series of IDENTICAL cash flows at the end of each year is called an annuityannuity.

PV of An Annuity Example

Lacey Company purchased a tract of land on which a $60,000 payment will be due each year for the next five years. What is the

present value of this stream of cash payments when the discount rate is 12%?


We could solve the problem like this . . .

Present Value of an Annuity of $1Excerpted from Table 6.6

Periods 10% 12% 14%1 0.9091 0.8929 0.8772 2 1.7355 1.6901 1.6467 3 2.4868 2.4018 2.3216 4 3.1699 3.0373 2.9137 5 3.7908 3.6048 3.4331


We could solve the problem like this . . .

Periods 10% 12% 14%1 0.9091 0.8929 0.8772 2 1.7355 1.6901 1.6467 3 2.4868 2.4018 2.3216 4 3.1699 3.0373 2.9137 5 3.7908 3.6048 3.4331

$60,000 × 3.605 = $216,300$60,000 × 3.605 = $216,300

Future Value of a Series of Cash Flows

The same techniques applied to calculating the PV of an annuity or uneven stream of cash flows may be used to calculate the FV of cash flows– Example

The Rule of 72• A quick way to estimate the approximate

number of periods it takes to double the value of an investment.

72 % return

Text ProblemsNote: some problems require the use of formulas )or

tables outside the text)• E 33(a) FV single sum• E 33(b) “ “• E 34(a) PV single sum• E 34(b) PV series of cash receipts• E 35(a) FV annuity + lump sum• E 35(b) “ ”• E 36(a) Lottery winnings – PV vs. annuity• E 36(b) Lottery winnings – PV vs. annuity

The Net Present Value Method

To determine net present value we . . .Calculate the present value of cash inflows,Calculate the present value of cash outflows,Subtract the present value of the outflows from

the present value of the inflows.

Typical Cash Outflows

InitialInitialinvestmentinvestment

Repairs andRepairs andmaintenancemaintenance

IncrementalIncrementaloperatingoperating

costscosts

WorkingWorkingcapitalcapital

Typical Cash Inflows

ReductionReductionof costsof costs

Salvage valueSalvage value

IncrementalIncrementalrevenuesrevenues

Release ofRelease ofworkingworkingcapitalcapital

The Net Present Value Method (NPV)

General decision rule . . .


Lester Company has been offered a five year contract to provide component parts for a large manufacturer.


• At the end of five years the working capital will be released and may be used elsewhere by Lester.

• Lester Company uses a discount rate of 10%.

Should the contract be accepted?Should the contract be accepted?


Annual net cash inflows from operations



Present value of an annuity of $1 factor for 5 years at 10%.


Present value of $1 factor for 3 years at 10%.


Present value of $1 factor for 5 years at 10%.


Accept the contract because the project has a positivepositive net present value.

Expanding the Net Present Value Method

To compare two competing investment projects we can use the following net present value approaches:– Total-cost – Calculate NPV for each project.

Accept the project with the higher NPV– Incremental cost – Determine the cash flow

differences between alternatives, and calculate the NPV of these cash flows. Accept one alternative over the other if the differential NPV is positive.

Least Cost Decisions

In decisions where revenues are not directly involved, managers should choose the

alternative that has the least total cost from a present value perspective.

Let’s look at the Home Furniture CompanyLet’s look at the Home Furniture Company..


• Home Furniture Company is trying to decide whether to overhaul an old delivery truck now or purchase a new one.

• The company uses a discount rate of 10%.

HomeFurniture


Old TruckOverhaul cost now 4,500$ Annual operating costs 10,000 Salvage value in 5 years 250 Salvage value now 9,000

Information Information about the about the trucks . . .trucks . . .


Buy the New Truck YearCash Flows

10% Factor

Present Value

Purchase price Now $(21,000) 1.0000 $ (21,000)Annual operating costs 1-5 (6,000) 3.7908 (22,745)Salvage value of old truck Now 9,000 1.0000 9,000 Salvage value of new truck 5 3,000 0.62092 1,863 Net present value (32,882)

Keep the Old Truck YearCash Flows

10% Factor

Present Value

Overhaul cost Now $ (4,500) 1.0000 $ (4,500)Annual operating costs 1-5 (10,000) 3.7908 (37,908)Salvage value of old truck 5 250 0.62092 155 Net present value (42,253)


Home Furniture should purchase the new truck.

Net present value of costs associated with purchase of new truck (32,883)$ Net present value of costs associated with remodeling existing truck (42,255) Net present value in favor of purchasing the new truck 9,372$

The Internal Rate of Return Method

• The internal rate of return is the interest yield promised by an investment project over its useful life.

• The internal rate of return is computed by finding the discount rate that will cause the net present value of a project to be zero.


• Decker Company can purchase a new machine at a cost of $104,320 that will save $20,000 per year in cash operating costs.

• The machine has a 10-year life.


Future cash flows are the same every year in this example, so we can calculate the

internal rate of return as follows:

Investment required Net annual cash flows

PV factor for theinternal rate of return

=

$104, 320 $20,000

= 5.216


Find the 10-period row, move across until you find the factor 5.216. Look at the top of the

column and you find a rate of 14%14%.

Periods 10% 12% 14%1 0.909 0.893 0.877 2 1.736 1.690 1.647

. . . . . . . . . . . .9 5.759 5.328 4.946 10 6.145 5.650 5.216

Using the present value of an annuity of $1 table . . .


• Decker Company can purchase a new machine at a cost of $104,320 that will save $20,000 per year in cash operating costs.

• The machine has a 10-year life.

The internal rate of return internal rate of return on this project is 14%.

If the internal rate of return is equal to or If the internal rate of return is equal to or greater than the company’s required rate of greater than the company’s required rate of

return, the project is acceptable.return, the project is acceptable.

Net Present Value vs. Internal Rate of Return

Net Present ValueNet Present Value Easier to use.Easier to use.

Assumes cash inflows will Assumes cash inflows will be reinvested at the be reinvested at the discount rate. This is a discount rate. This is a realistic assumption.realistic assumption.

Documents

Chapter 6 Time Value of Money. Introduction Why money has a time value –The opportunity cost of capital concept Time value of money and risk –Typically