The Time Value of Money

The Time Value of Money Lecture 4

Learning Objectives • Understand what is meant by “the time value of money.” • Understand the relationship between present and future value. • Describe how the interest rate can be used to adjust the value of cash flows – both forward and backward – to a single point in time. • Calculate both the future and present value of: (a) an amount invested today; (b) a stream of equal cash flows (an annuity); and (c) a stream of mixed cash flows.

The Time Value of Money Would you prefer to have $1 million now or $1 million 10 years from now? Of course, we would all prefer the money now! This illustrates that there is an inherent monetary value attached to time.

What is The Time Value of Money? • A dollar received today is worth more than a dollar received tomorrow • This is because a dollar received today can be invested to earn interest • The amount of interest earned depends on the rate of return that can be earned on the investment • Time value of money quantifies the value of a dollar through time

Uses of Time Value of Money • Time Value of Money, or TVM, is a concept that is used in all aspects of finance including: • Bond valuation • Stock valuation • Accept/reject decisions for project management • Financial analysis of firms • And many others!

Types of TVM Calculations • There are many types of TVM calculations • The basic types include: • Present value of a lump sum (single amount) • Future value of a lump sum (single amount) • Present and future value of cash flow streams • Present and future value of annuities • Keep in mind that these forms can, should, and will be used in combination to solve more complex TVM problems

Basic Rules • The following are simple rules that you should always use no matter what type of TVM problem you are trying to solve: • Stop and think: Make sure you understand what the problem is asking. You will get the wrong answer if you are answering the wrong question. • Draw a representative timeline and label the cash flows and time periods appropriately. • Write out the complete formula using symbols • Substitute the actual numbers to solve. • While these may seem like trivial and time consuming tasks, they will significantly increase your understanding of the material and your accuracy rate.

Present Value of a Lump Sum (single amount) • Present value calculations determine what the value of a cash flow received in the future would be worth today (time 0) • The process of finding a present value is called “discounting” (hint: it gets smaller) • The interest rate used to discount cash flows is generally called the discount rate

Formulas • Common formulas that are used in TVM calculationsfor lump sum: • Present value (PV) of a lump sum (single amount): PV = CFt / (1+i)tOR PV = FVt/ (1+i)t • Future value (FV) of a lump sum (single amount): FVt = CF0 * (1+i)tORFVt = PV * (1+i)t • where • i = rate of interest per compounding period • t = time period • CF = Cash flow (the subscripts t and 0 mean at time t and at time zero, respectively)

Example of PV of a Lump Sum (single amount) • How much would $100 received five years from now be worth today if the current interest rate is 10%? • Draw a timeline The arrow represents the flow of money and the numbers under the timeline represent the time period. Note that time period zero is today. i = 10% ? $100 0 1 2 3 4 5

Example of PV of a Lump Sum (single amount) • Write out the formula using symbols: PV = CFt / (1+i)t • Insert the appropriate numbers: PV = 100 / (1 + 0.1)5 • Solve the formula: PV = $62.09

Future Value of a Lump Sum (single amount) • You can think of future value as the opposite of present value • Future value determines the amount that a sum of money invested today will grow to in a given period of time • The process of finding a future value is called “compounding” (hint: it gets larger)

Example of FV of a Lump Sum (single amount) • How much money will you have in 5 years if you invest $100 today at a 10% rate of return? • Draw a timeline i = 10% $100 ? 0 1 2 3 4 5

Example of FV of a Lump Sum (single amount) • Write out the formula using symbols: FVt = CF0 * (1+i)t ORFVt = PV * (1+i)t • Substitute the numbers into the formula: FV = $100 * (1+0.1)5 • Solve for the future value: FV = $161.05

Some Things to Note • In both of the examples, note that if you were to perform the opposite operation on the answers (i.e., find the future value of $62.09 or the present value of $161.05) you will end up with your original investment of $100. • This illustrates how present value and future value concepts are intertwined.

Present Value of a Cash Flow Stream • A cash flow stream is a finite set of payments that an investor will receive or invest over time. • The PV of the cash flow stream is equal to the sum of the present value of each of the individual cash flows in the stream. • The PV of a cash flow stream can also be found by taking the FV of the cash flow stream and discounting the lump sum at the appropriate discount rate for the appropriate number of periods.

Formulas • Present value of a cash flow stream: n PV = S [CFt / (1+i)t] t=0 • Future value of a cash flow stream: n FV = S [CFt * (1+i)n-t] t=0 n = number of time periods

Example of PV of a Cash Flow Stream • Joe made an investment that will pay $100 the first year, $300 the second year, $500 the third year and $1000 the fourth year. If the interest rate is ten percent, what is the present value of this cash flow stream? • Draw a timeline: $100 $300 $500 $1000 0 1 2 3 4 ? ? i = 10% ? ?

Example of PV of a Cash Flow Stream • Write out the formula using symbols: n PV = S [CFt / (1+i)t] t=0 OR PV = [CF1/(1+i)1]+[CF2/(1+i)2]+[CF3/(1+i)3]+[CF4/(1+i)4] • Substitute the appropriate numbers: PV = [100/(1+0.1)1]+[300/(1+0.1)2]+[500/(1+0.1)3]+[1000/(1+0.1)4] • Solve for the present value: PV = $90.91 + $247.93 + $375.66 + $683.01 PV = $1397.51

Future Value of a Cash Flow Stream • The future value of a cash flow stream is equal to the sum of the future values of the individual cash flows. • The FV of a cash flow stream can also be found by taking the PV of that same stream and finding the FV of that lump sum using the appropriate rate of return for the appropriate number of periods.

Example of FV of a Cash Flow Stream • Assume Joe has the same cash flow stream from his investment but wants to know what it will be worth at the end of the fourth year • Draw a timeline: $100 $300 $500 $1000 0 1 2 3 4 $1000 i = 10% ? ? ?

Example of FV of a Cash Flow Stream • Write out the formula using symbols n FV = S [CFt * (1+i)n-t] t=0 OR FV = [CF1*(1+i)n-1]+[CF2*(1+i)n-2]+[CF3*(1+i)n-3]+[CF4*(1+i)n-4] • Substitute the appropriate numbers: FV = [$100*(1+.1)4-1]+[$300*(1+.1)4-2]+[$500*(1+.1)4-3] +[$1000*(1+.1)4-4] • Solve for the Future Value: FV = $133.10 + $363.00 + $550.00 + $1000 FV = $2046.10

Annuities • An annuity is a cash flow stream in which the cash flows are all equal and occur at regular intervals. • Note that annuities can be a fixed amount, an amount that grows at a constant rate over time, or an amount that grows at various rates of growth over time. We will focus on fixed amounts.

Formulas • Present value of an annuity: PVA = PMT * {[1-(1+i)-t]/i} • Future value of an annuity: FVAt = PMT * {[(1+i)t –1]/i} • PMT = payment • PVA = present value of an annuity • FVA = future value of an annuity

Example of PV of an Annuity • Assume that Sally owns an investment that will pay her $100 each year for 20 years. The current interest rate is 15%. What is the PV of this annuity? • Draw a timeline $100 $100 $100 $100 $100 0 1 2 3 …………………………. 19 20 ? i = 15%

Example of PV of an Annuity • Write out the formula using symbols: PVA = PMT * {[1-(1+i)-t]/i} • Substitute appropriate numbers: PVA = $100 * {[1-(1+0.15)-20]/0.15} • Solve for the PV PVA = $100 * 6.2593 PVA = $625.93

Example of FV of an Annuity • Assume that Sally owns an investment that will pay her $100 each year for 20 years. The current interest rate is 15%. What is the FV of this annuity? • Draw a timeline $100 $100 $100 $100 $100 0 1 2 3 …………………………. 19 20 ? i = 15%

Example of FV of an Annuity • Write out the formula using symbols: FVAt = PMT * {[(1+i)t –1]/i} • Substitute the appropriate numbers: FVA20 = $100 * {[(1+0.15)20 –1]/0.15} • Solve for the FV: FVA20 = $100 * 102.4436 FVA20 = $10,244.36

Problem #1 You must decide between $25,000 in cash today or $30,000 in cash to be received two years from now. If you can earn 8% interest on your investments, which is the better deal?

Problem #1 Solution 1. Write out the formula using symbols: FVt = CF0 * (1+i)t ORFVt = PV * (1+i)t 2. Substitute the numbers into the formula: FV = 25000* (1+0.08)2 3. Solve for the future value: FV = $29160

Possible Answers - Problem 1 • $25,000 in cash today • $30,000 in cash to be received two years from now • Either option O.K.

Problem #2 • What is the value of $100 per year for four years, with the first cash flow one year from today, if one is earning 5% interest, compounded annually? Find the value of these cash flows four years from today.

Problem #2 Solution 1. Write out the formula using symbols: FVAt = PMT * {[(1+i)t –1]/i} 2. Substitute the appropriate numbers: FVA4 = 100 * {[(1+0.05)4 –1]/0.05} 3. Solve for the FV: FVA4 = 100 * 4.3101 FVA4 = $431.01

The Time Value of Money