Time Value of Money (TVM)

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Contents

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• What is TVM

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• Why TVM

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• The Role of TVM

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• Calculations

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• Annuities

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One of the most important principle in finance

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Decision Dilemma—Take a Lump Sum or Annual Installments

• A couple won a Lucky Draw.

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• They had to choose between a single lump sum GHS104 million, or GHS198 million paid out over 25 years (or GHS7.92 million per year).

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• The winning couple opted for the lump sum.

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• Did they make the right choice? What basis do we make such an economic comparison?

What is Time Value of Money

• The time value of money refers to the observation that it is better to receive money sooner than later.

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• It involves the idea that money that is available at the present time is worth more than the same amount in the future.

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Why Time Value of Money?

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• Opportunity Cost - Money has a time value because it can earn more money over time (earning power).

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• Inflation - Money has a time value because its purchasing power changes over time (purchasing power).

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The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that are spread out over time.
• Time value of money allows comparison of cash flows from different periods.
• Question: Your father has offered to give you some money and asks that you choose one of the following two alternatives:
• GHS1,000 today, or
• GHS1,100 one year from now.
• What do you do?

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The Role of Time Value in Finance (cont.)

• The answer depends on what rate of interest you could earn on any money you receive today.
• For example, if you could deposit the GHS1,000 today at 12% per year, you would prefer to be paid today.
• Alternatively, if you could only earn 5% on deposited funds, you would be better off if you chose the GHS1,100 in one year.

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The Time Value of Money

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GHS1 today

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Relationship

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GHS1 future

Present Value

Future Value

Interest is the factor contributing to Time Value of Money

Simple Interest= Principal x Interest rate x time period

= P_o(i)(n)

• Simple Interest

Interest paid (earned) on only the original amount, or principal, borrowed (lent).

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• Compound Interest

Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

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Types of Interest

Opportunity Cost

Opportunity cost = Alternative use

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• The opportunity cost of money is the interest rate that would be earned by investing it.
• It is the underlying reason for the time value of money
• Any person with money today knows they can invest those funds to be some greater amount in the future.
• Conversely, if you are promised a cash flow in the future, it’s present value today is less than what is promised!

Key Components

• Present value (PV) - This is your current starting amount. It is the money you have in your hand at the present time. 

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• Future value (FV) - This is your ending amount at a point in time in the future.

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• The number of periods (n) - This is the timeline for your investment. It is usually measured in years, but it could be any scale of time such as quarterly, monthly, or even daily.

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• Interest rate (i) - This is the growth rate of your money over the lifetime of the investment. It is stated in a percentage value, such as 8% or .08.

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Future Value versus Present Value

• Suppose a firm has an opportunity to spend GHS15,000 today on some investment that will produce GHS17,000 spread out over the next five years as follows:

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• Is this a wise investment?
• To make the right investment decision, managers need to compare the cash flows at a single point in time.

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Figure 5.1 �Time Line

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Figure 5.2 �Compounding and Discounting

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Future Value of a Single Amount: The Equation for Future Value

• We use the following notation for the various inputs:
• FV_n = future value at the end of period n
• PV = initial principal, or present value
• r = annual rate of interest paid. (Note: On financial calculators, I is typically used to represent this rate.)
• n = number of periods (typically years) that the money is left on deposit
• The general equation for the future value at the end of period n is

FV_n = PV × (1 + r)ⁿ

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Future Value of a Single Amount: The Equation for Future Value

Jane Farber places GHS800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of five years.

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This analysis can be depicted on a time line as follows:

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FV₅ = GHS800 × (1 + 0.06)⁵ = GHS800 × (1.33823) = GHS1,070.58

Calculations

There are really only four different things you can be asked to find using this basic equation:

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FV_n=PV₀ (1+r)ⁿ

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• Find the initial amount of money to invest (PV₀)
• Find the Future value (FV_n)
• Find the rate (r)
• Find the time (n)

Calculations cont..�Solving for the Rate (r)

You have asked your father for a loan of GHS10,000 to get you started in a business. You promise to repay him GHS20,000 in five years time.

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What compound rate of return are you offering to pay?

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FV_t=PV₀ (1+r)ⁿ

GHS20,000= GHS10,000 (1+r)⁵

2=(1+r)⁵

2^1/5=1+r

1.14869=1+r

r = 14.869%

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Calculations cont..�Solving for Time (n)

You have GHS150,000 in your RRSP (Registered Retirement Savings Plan). Assuming a rate of 8%, how long will it take to have the plan grow to a value of GHS300,000?

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FV_t=PV₀ (1+r)ⁿ

GHS300,000= GHS150,000 (1+.08)ⁿ

2=(1.08)ⁿ

log 2 =log 1.08 × n

0.301029995 = 0.033423755 × n

t = 9.00 years

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Calculations cont..�Solving for the Future Value (FV_n)

You have GHS650,000 in your pension plan today. Because you have retired, you and your employer will not make any further contributions to the plan. However, you don’t plan to take any pension payments for five more years so the principal will continue to grow.

Assuming a rate of 8%, forecast the value of your pension plan in 5 years.

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FV_t=PV₀ (1+r)ⁿ

FV₅= GHS650,000 (1+.08)⁵

FV₅ = GHS650,000 × 1.469328077

FV₅ = GHS955,063.25

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Calculations cont..�Finding the amount of money to invest (PV₀)

You hope to save for a down payment on a home. You hope to have GHS40,000 in four years time; determine the amount you need to invest now at 10%

• This is a process known as discounting

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FV_n=PV₀ (1+r)ⁿ

GHS40,000= PV₀ (1.1)⁴

PV₀ = GHS40,000/1.4641=GHS27,320.53

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Personal Finance Example

Paul Amidu has an opportunity to receive GHS300 one year from now. If he can earn 6% on his investments, what is the most he should pay now for this opportunity?

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PV × (1 + 0.06) = GHS300

PV = GHS300/(1 + 0.06) = GHS283.02

Present Value of a Single Amount: The Equation for Present Value

The present value, PV, of some future amount, FV_n, to be received n periods from now, assuming an interest rate (or opportunity cost) of r, is calculated as follows:

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Present Value of a Single Amount: The Equation for Future Value

Pam Valenti wishes to find the present value of GHS1,700 that will be received 8 years from now. Pam’s opportunity cost is 8%.

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This analysis can be depicted on a time line as follows:

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PV = GHS1,700/(1 + 0.08)⁸ = GHS1,700/1.85093 = GHS918.46

Annuities

An annuity is a stream of equal periodic cash flows, over a specified time period. These cash flows can be inflows of returns earned on investments or outflows of funds invested to earn future returns.

• An ordinary (deferred) annuity is an annuity for which the cash flow occurs at the end of each period
• An annuity due is an annuity for which the cash flow occurs at the beginning of each period.
• An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

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Finding the Future Value of an Ordinary Annuity

• You can calculate the future value of an ordinary annuity that pays an annual cash flow equal to CF by using the following equation:

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• As before, in this equation r represents the interest rate and n represents the number of payments in the annuity (or equivalently, the number of years over which the annuity is spread).

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Finding the Present Value of an Ordinary Annuity

• You can calculate the present value of an ordinary annuity that pays an annual cash flow equal to CF by using the following equation:

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• As before, in this equation r represents the interest rate and n represents the number of payments in the annuity (or equivalently, the number of years over which the annuity is spread).

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Finding the Present Value of an Ordinary Annuity (cont.)

Tomtom Company, a small producer of plastic toys, wants to determine the most it should pay to purchase a particular annuity. The annuity consists of cash flows of GHS700 at the end of each year for 5 years. The required return is 8%.

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This analysis can be depicted on a time line as follows:

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Table 5.2 Long Method for Finding the Present Value of an Ordinary Annuity

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Finding the Future Value of an Annuity Due

• You can calculate the future value of an annuity due that pays an annual cash flow equal to CF by using the following equation:

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• As before, in this equation r represents the interest rate and n represents the number of payments in the annuity (or equivalently, the number of years over which the annuity is spread).

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Finding the Present Value of an Annuity Due

• You can calculate the present value of an ordinary annuity that pays an annual cash flow equal to CF by using the following equation:

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• As before, in this equation r represents the interest rate and n represents the number of payments in the annuity (or equivalently, the number of years over which the annuity is spread).

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Finding the Present Value of a Perpetuity

• A perpetuity is an annuity with an infinite life, providing continual annual cash flow.
• If a perpetuity pays an annual cash flow of CF, starting one year from now, the present value of the cash flow stream is

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PV = CF ÷ r

Personal Finance Example

Prof Bawa wishes to donate an amount to his alma mater (SSD UBIDS). The university indicated that it requires GHS200,000 per year to support the endowment fund, and the endowment would earn 10% per year. To determine the amount Prof Bawa must give the university to fund the endowment, we must determine the present value of the GHS200,000 perpetuity discounted at 10%.

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PV = GHS200,000 ÷ 0.10 = GHS2,000,000

Future Value of a Mixed Stream

Sam Industries, a cabinet manufacturer, expects to receive the following mixed stream of cash flows over the next 5 years from one of its small customers.

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Future Value of a Mixed Stream

If the firm expects to earn at least 8% on its investments, how much will it accumulate by the end of year 5 if it immediately invests these cash flows when they are received?

This situation is depicted on the following time line.

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Present Value of a Mixed Stream

Frey Company, a shoe manufacturer, has been offered an opportunity to receive the following mixed stream of cash flows over the next 5 years.

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Present Value of a Mixed Stream

If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity?

This situation is depicted on the following time line.

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