Introduction to Valuation: The Time Value of Money

Introduction to Valuation: The Time Value of Money
Key Concepts and Skills Be able to compute the future value of an investment made today Be able to compute the present value of cash to be received at some future date Be able to compute the return on an investment

Chapter Outline Future Value and Compounding
Present Value and Discounting More on Present and Future Values

Basic Definitions Present Value – earlier money on a time line
Future Value – The amount an investment is worth after one or more period or later money on a time line Interest rate – “exchange rate” between earlier money and later money It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts.

Future Values Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year? Interest = $1,000(.05) = $50 Value in one year = principal + interest = $1, = $1,050 Future Value (FV) = $1,000( ) = $1,050 Suppose you leave the money in for another year. How much will you have two years from now? FV = $1,000(1.05)(1.05) = $1,000(1.05)2 = $1,102.50 Point out that we are just using algebra when deriving the FV formula. We have 1,000(1) + 1,000(.05) = 1,000(1+.05)

Future Values: General Formula
FV = PV(1 + r)t FV = future value PV = present value r = period interest rate, expressed as a decimal T = number of periods Future value interest factor = (1 + r)t

Effects of Compounding
Simple interest (interest is earned only on the original principal) Compound interest (interest is earned on principal and on interest received) Consider the previous example FV with simple interest = $1, = $1,100 FV with compound interest = $1,102.50 The extra $2.50 comes from the interest of .05($50) = $2.50 earned on the first interest payment

Figure 4.1

Figure 4.2

Future Values – Example 2
Suppose you invest the $1,000 from the previous example for 5 years. How much would you have? FV = $1,000(1.05)5 = $1,276.28 The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1,250, for a difference of $26.28.) It is important at this point to discuss the sign convention in the calculator. The calculator is programmed so that cash outflows are entered as negative and inflows are entered as positive. If you enter the PV as positive, the calculator assumes that you have received a loan that you will have to repay at some point. The negative sign on the future value indicates that you would have to repay 1, in 5 years. Show the students that if they enter the 1,000 as negative, the FV will compute as a positive number. Also, you may want to point out the change sign key (+/-) on the calculator. There seem to be a few students each semester that have never had to use it before. Calculator: N = 5; I/Y = 5; PV = -1,000; CPT FV = 1,276.28

Future Value as a General Growth Formula
Suppose your company expects to increase unit sales of a product by 15% per year for the next 5 years. If you currently sell 3 million units in one year, how many units do you expect to sell during the fifth year? FV = 3,000,000(1.15)5 = 6,034,072 Calculator: N = 5; I/Y = 15; PV = 3,000,000 CPT FV = -6,034,072

TVM on the Calculator Use the highlighted row of keys for solving any of the FV, PV, FVA, PVA, FVAD, and PVAD problems N: Number of periods I/Y: Interest rate per period PV: Present value PMT: Payment per period FV: Future value CLR TVM: Clears all of the inputs into the above TVM keys

Using The Calculator N I/Y PV PMT FV
Inputs N I/Y PV PMT FV Compute Focus on 3rd row of keys (will be displayed in slides as shown above)

Entering the FV Problem
Press: 2nd CLR TVM N I/Y PV 0 PMT CPT FV

Solving the FV Problem N I/Y PV PMT FV Inputs 2 7 -1,000 0 Compute
, N I/Y PV PMT FV Compute 1,144.90 N: 2 periods (enter as 2) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: $1,000 (enter as negative as you have “less”) PMT: Not relevant in this situation (enter as 0) FV: Compute (Resulting answer is positive)

Entering the FV Problem
Press: 2nd CLR TVM N I/Y PV 0 PMT CPT FV

Solving the FV Problem N I/Y PV PMT FV
Inputs , N I/Y PV PMT FV Compute 16,105.10 The result indicates that a $10,000 investment that earns 10% annually for 5 years will result in a future value of $16,

Quick Quiz: Part 1 What is the difference between simple interest and compound interest? Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years. How much would you have at the end of 15 years using compound interest? How much would you have using simple interest? N = 15; I/Y = 8; PV = -500; CPT FV = 1,586.08 Formula: 500(1.08)15 = 500( ) = 1,586.08 (500)(.08) = 1,100

Present Values How much do I have to invest today to have some amount in the future? FV = PV(1 + r)t Rearrange to solve for PV = FV / (1 + r)t When we talk about discounting, we mean finding the present value of some future amount. When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value. Point out that the PV interest factor = 1 / (1 + r)t

PV – One-Period Example
Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today? PV = $10,000 / (1.07)1 = $9,345.79 The remaining examples will just use the calculator keys.

Present Values – Example 2
You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? PV = $150,000 / (1.08)17 = $40,540.34 Key strokes: 1.08 yx 17 +/- = x 150,000 = Calculator: N = 17; I/Y = 8; FV = 150,000; CPT PV = -40,540.34

Present Values – Example 3
Your parents set up a trust fund for you 10 years ago that is now worth $19, If the fund earned 7% per year, how much did your parents invest? PV = $19, / (1.07)10 = $10,000 The actual number computes to –9, This is a good place to remind the students to pay attention to what the question asked and be reasonable in their answers. A little common sense should tell them that the original amount was 10,000, and that the present value calculation doesn’t come out exactly to the penny because the future value was rounded to the nearest penny. Calculator: N = 10; I/Y = 7; FV = 19,671.51; CPT PV = -10,000

PV – Important Relationship I
For a given interest rate – the longer the time period, the lower the present value. What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% 5 years: PV = $500 / (1.1)5 = $310.46 10 years: PV = $500 / (1.1)10 = $192.77 Calculator: 5 years: N = 5; I/Y = 10; FV = 500; CPT PV = N = 10; I/Y = 10; FV = 500; CPT PV =

PV – Important Relationship II
For a given time period – the higher the interest rate, the smaller the present value What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? Rate = 10%: PV = $500 / (1.1)5 = $310.46 Rate = 15%; PV = $500 / (1.15)5 = $248.59 Calculator: 10%: N = 5; I/Y = 10; FV = 500; CPT PV = 15%: N = 5; I/Y = 15; FV = 500; CPT PV =

Figure 4.3

Solving the PV Problem N I/Y PV PMT FV
Inputs ,000 N I/Y PV PMT FV Compute -6,209.21 The result indicates that a $10,000 future value that will earn 10% annually for 5 years requires a $6, deposit today (present value).

Quick Quiz: Part 2 What is the relationship between present value and future value? Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today? If you could invest the money at 8%, would you have to invest more or less than at 6%? How much? Relationship: The mathematical relationship is PV = FV / (1 + r)t. One of the important things for them to take away from this discussion is that the present value is always less than the future value when we have positive rates of interest. N = 3; I/Y = 6; FV = 15,000; CPT PV = -12,594.29 PV = 15,000 / (1.06)3 = 15,000( ) = 12,594.29 N = 3; I/Y = 8; FV = 15,000; CPT PV = -11, (Difference = ) PV = 15,000 / (1.08)3 = 15,000( ) = 11,907.48

The Basic PV Equation - Refresher
PV = FV / (1 + r)t There are four parts to this equation PV, FV, r, and t If we know any three, we can solve for the fourth

Discount Rate Often, we will want to know what the implied interest rate is in an investment Rearrange the basic PV equation and solve for r FV = PV(1 + r)t r = (FV / PV)1/t – 1

Discount Rate – Example 1
You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest? r = ($1,200 / $1,000)1/5 – 1 = = 3.714% It is very important at this point to make sure that the students have more than 2 decimal places visible on their calculator. Efficient key strokes for formula: 1,200 / 1,000 = yx 5 = 1/x - 1 = If they receive an error when they try to use the financial keys, they probably forgot to enter one of the numbers as a negative.

Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? r = ($20,000 / $10,000)1/6 – 1 = = 12.25% Calculator: N = 6; FV = 20,000; PV = -10,000; CPT I/Y = 12.25%

Suppose you have a 1-year old son and you want to provide $75,000 in 17 years toward his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it? r = ($75,000 / $5,000)1/17 – 1 = = 17.27% Calculator: N = 17; FV = 75,000; PV = - 5,000; CPT I/Y = 17.27% This is a great problem to illustrate how TVM can help you set realistic financial goals, and it may lead you to adjust your expectations based on what you can currently afford to save, what rate you can earn, and how long you have to reach your goals.

Quick Quiz: Part 3 What are some situations in which you might want to compute the implied interest rate? Suppose you are offered the following investment choices: You can invest $500 today and receive $600 in 5 years. The investment is considered low risk. You can invest the $500 in a bank account paying 4% annually. What is the implied interest rate for the first choice and which investment should you choose? Implied rate: N = 5; PV = -500; FV = 600; CPT I/Y = 3.714% r = (600 / 500)1/5 – 1 = 3.714% Choose the bank account because it pays a higher rate of interest. How would the decision be different if you were looking at borrowing $500 today and either repaying at 4% or repaying $600? In this case, you would choose to repay $600 because you would be paying a lower rate.

Finding the Number of Periods
Start with basic equation and solve for t (remember your logs) FV = PV(1 + r)t t = ln(FV / PV) / ln(1 + r) Remind the students that ln is the natural logarithm and can be found on the calculator. The rule of 72 is a quick way to estimate how long it will take to double your money. # years to double = 72 / r where r is a percent, not a decimal.

Number of Periods – Example 1
You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? t = ln($20,000 / $15,000) / ln(1.1) = 3.02 years Calculator: I/Y = 10; FV = 20,000; PV = - 15,000; CPT N = 3.02 years

Number of Periods – Example 2
Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

Example 2 Continued How much do you need to have in the future?
Down payment = .1($150,000) = $15,000 Closing costs = .05($150,000 – 15,000) = $6,750 Total needed = $15, ,750 = $21,750 Using the formula t = ln($21,750 / $15,000) / ln(1.075) = 5.14 years Point out that the closing costs are only paid on the loan amount, not on the total amount paid for the house.

We will use the “Rule-of-72”.
Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? We will use the “Rule-of-72”.

Approx. Years to Double = 72 / i%
The “Rule-of-72” Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? Approx. Years to Double = 72 / i% 72 / 12% = 6 Years [Actual Time is 6.12 Years]

Solving the Period Problem
Inputs , ,000 N I/Y PV PMT FV Compute 6.12 years The result indicates that a $1,000 investment that earns 12% annually will double to $2,000 in 6.12 years. Note: 72/12% = approx. 6 years

Table 4.4

Quick Quiz: Part 4 When might you want to compute the number of periods? Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money? Calculator: PV = -500; FV = 600; I/Y = 6; CPT N = 3.13 years Formula: t = ln(600/500) / ln(1.06) = 3.13 years

Comprehensive Problem
You have $10,000 to invest for five years. How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4.5% annual return? How long will it take your $10,000 to double in value if it earns 5% annually? What annual rate has been earned if $1,000 grows into $4,000 in 20 years? N = 5 PV = -10,000 At I/Y = 5, the FV = 12,762.82 At I/Y = 4.5, the FV = 12,461.82 The difference is attributable to interest. That difference is 12, – 12, = 301 To double the 10,000: I/Y = 5 FV = 20,000 CPT N = 14.2 years Note, the rule of 72 indicates 72/5 = 14 years, approximately. N = 20 PV = -1,000 FV = 4,000 CPT I/Y = 7.18%

Types of Annuities An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equal time periods. Ordinary Annuity: Payments or receipts occur at the end of each period. Annuity Due: Payments or receipts occur at the beginning of each period.

Parts of an Annuity Today 0 1 2 3 $100 $100 $100 Equal Cash Flows
(Ordinary Annuity) End of Period 1 End of Period 2 End of Period 3 $ $ $100 Today Equal Cash Flows Each 1 Period

Parts of an Annuity Today 0 1 2 3 $100 $100 $100 Equal Cash Flows
(Annuity Due) Beginning of Period 1 Beginning of Period 2 Beginning of Period 3 $ $ $100 Equal Cash Flows Each 1 Period Today

Overview of an Ordinary Annuity -- FVA
Cash flows occur at the end of the period n n+1 i% R R R R = Periodic Cash Flow FVAn FVAn = R(1+i)n-1 + R(1+i)n R(1+i)1 + R(1+i)0

Example of an Ordinary Annuity -- FVA
Cash flows occur at the end of the period 7% $1, $1, $1,000 $1,070 $1,145 FVA3 = $1,000(1.07) $1,000(1.07)1 + $1,000(1.07)0 = $1,145 + $1,070 + $1, = $3,215 $3,215 = FVA3

Hint on Annuity Valuation
The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period.

Valuation Using Table III
FVAn = R (FVIFAi%,n) FVA3 = $1,000 (FVIFA7%,3) = $1,000 (3.215) = $3,215

Solving the FVA Problem
Inputs ,000 N I/Y PV PMT FV Compute 3,214.90 N: 3 periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Not relevant in this situation (no beg value) PMT: $1,000 (negative as you deposit annually) FV: Compute (Resulting answer is positive)

Future Value of Ordinary Annuity
Suppose you begin saving for your retirement by depositing $2,000 per year. If the interest rate is 7.5%, how much will you have in 40 years? FV(Ordinary) = $2,000( – 1)/.075 = $454,513.04 FV = 2,000( – 1)/.075 = 454,513.04 Remember the sign convention!!! 40 N 7.5 I/Y -2,000 PMT CPT FV = 454,513.04

Overview View of an Annuity Due -- FVAD
Cash flows occur at the beginning of the period n n i% R R R R R FVADn = R(1+i)n + R(1+i)n R(1+i)2 + R(1+i) = FVAn (1+i) FVADn

Example of an Annuity Due -- FVAD
Cash flows occur at the beginning of the period 7% $1, $1, $1, $1,070 $1,145 $1,225 FVAD3 = $1,000(1.07) $1,000(1.07)2 + $1,000(1.07)1 = $1,225 + $1,145 + $1, = $3,440 $3,440 = FVAD3

Valuation Using Table III
FVADn = R (FVIFAi%,n)(1+i) FVAD3 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,440

Solving the FVAD Problem
Inputs ,000 N I/Y PV PMT FV Compute 3,439.94 Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting to “BGN” first. Don’t forget to change back! Step 1: Press 2nd BGN keys Step 2: Press 2nd SET keys Step 3: Press 2nd QUIT keys

Future Value of Annuity Due
You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? FV = $10,000[(1.083 – 1) / .08](1.08) = $35,061.12 Note that the procedure for changing the calculator to an annuity due is similar on other calculators. Calculator 2nd BGN 2nd Set (you should see BGN in the display) 3 N -10,000 PMT 8 I/Y CPT FV = 35,061.12 2nd BGN 2nd Set (be sure to change it back to an ordinary annuity) What if it were an ordinary annuity? FV = 32,464 (so receive an additional 2, by starting to save today.)

Overview of an Ordinary Annuity -- PVA
Cash flows occur at the end of the period n n+1 i% R R R R = Periodic Cash Flow PVAn PVAn = R/(1+i)1 + R/(1+i)2 R/(1+i)n

Example of an Ordinary Annuity -- PVA
Cash flows occur at the end of the period 7% $1, $1, $1,000 $ $ $ $2, = PVA3 PVA3 = $1,000/(1.07) $1,000/(1.07) $1,000/(1.07)3 = $ $ $ = $2,624.32

Hint on Annuity Valuation
The present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the end of the first cash flow period.

Valuation Using Table IV
PVAn = R (PVIFAi%,n) PVA3 = $1,000 (PVIFA7%,3) = $1,000 (2.624) = $2,624

Solving the PVA Problem
Inputs , N I/Y PV PMT FV Compute 2,624.32 N: 3 periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Compute (Resulting answer is positive) PMT: $1,000 (negative as you deposit annually) FV: Not relevant in this situation (no ending value)

Overview of an Annuity Due -- PVAD
Cash flows occur at the beginning of the period n n i% R R R R R: Periodic Cash Flow PVADn PVADn = R/(1+i)0 + R/(1+i) R/(1+i)n = PVAn (1+i)

Example of an Annuity Due -- PVAD
Cash flows occur at the beginning of the period 7% $1, $1, $1,000 $ $ $2, = PVADn PVADn = $1,000/(1.07)0 + $1,000/(1.07) $1,000/(1.07)2 = $2,808.02

Valuation Using Table IV
PVADn = R (PVIFAi%,n)(1+i) PVAD3 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808

Solving the PVAD Problem
Inputs , N I/Y PV PMT FV Compute 2,808.02 Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting to “BGN” first. Don’t forget to change back! Step 1: Press 2nd BGN keys Step 2: Press 2nd SET keys Step 3: Press 2nd QUIT keys

Annuity – Sweepstakes Example
Suppose you win the Publishers Clearinghouse $10 million. The money is paid in equal annual installments of $333, over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? PV = $333,333.33[1 – 1/1.0530] / .05 = $5,124,150.29 Calculator: 30 N; 5 I/Y; 333, PMT; CPT PV = 5,124,150.29

Finding the Payment Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = % per month). If you take a 4-year loan, what is your monthly payment? $20,000 = C[1 – 1 / ] / C = $488.26 Calculator: 4(12) = 48 N; 20,000 PV; 8/12 = I/Y; CPT PMT =

Finding the Number of Payments – Example 6.6
Start with the equation and remember your logs. $1,000 = $20(1 – 1/1.015t) / .015 .75 = 1 – 1 / 1.015t 1 / 1.015t = .25 1 / .25 = 1.015t t = ln(1/.25) / ln(1.015) = months = 7.75 years And this is only true if you don’t charge anything more on the card! You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000. This is an excellent opportunity to talk about credit card debt and the problems that can develop if it is not handled properly. Many students don’t understand how it works and it is rarely discussed. This is something that students can take away from the class, even if they aren’t finance majors. Calculator: 1.5 I/Y 1,000 PV -20 PMT CPT N = MONTHS = 7.75 years

Finding the Number of Payments – Another Example
Suppose you borrow $2,000 at 5% and you are going to make annual payments of $ How long is it before you pay off the loan? $2,000 = $734.42(1 – 1/1.05t) / .05 = 1 – 1/1.05t 1/1.05t = = 1.05t t = ln( ) / ln(1.05) = 3 years Sign convention matters!!! 5 I/Y 2,000 PV PMT CPT N = 3 years

Finding the Rate Suppose an insurance company offers to pay you $1000 per year for 10 years, if you will pay $6710 up front. What rate is implicit in this 10 years annuity. The next slide talks about how to do this without a financial calculator.

Annuity – Finding the Rate
Trial and Error Process Choose an interest rate and compute the PV of the payments based on this rate Compare the computed PV with the actual loan amount If the computed PV > loan amount, then the interest rate is too low If the computed PV < loan amount, then the interest rate is too high Adjust the rate and repeat the process until the computed PV and the loan amount are equal

Annuities– Basic Formulas

Annuities– Interest Factors

Table 5.2

Mixed Flows Example Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%? 10% $ $600 $400 $400 $100 PV0

How to Solve? 1. Solve a “piece-at-a-time” by discounting each piece back to t=0. 2. Solve a “group-at-a-time” by first breaking problem into groups of annuity streams and any single cash flow group. Then discount each group back to t=0.

“Piece-At-A-Time” 0 1 2 3 4 5 10% $600 $600 $400 $400 $100
10% $ $600 $400 $400 $100 $545.45 $495.87 $300.53 $273.21 $ $ = PV0 of the Mixed Flow

“Group-At-A-Time” (#1)
10% $ $600 $400 $400 $100 $1,041.60 $ $ $1, = PV0 of Mixed Flow [Using Tables] $600(PVIFA10%,2) = $600(1.736) = $1,041.60 $400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57 $100 (PVIF10%,5) = $100 (0.621) = $62.10

Example Time Line 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93

Multiple Cash Flows – PV Example
If you are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of 4th year and interest rate is 12% on every investment what you should you pay for this investment? Find the PV of each cash flow and add them Year 1 CF: $200 / (1.12)1 = $178.57 Year 2 CF: $400 / (1.12)2 = $318.88 Year 3 CF: $600 / (1.12)3 = $427.07 Year 4 CF: $800 / (1.12)4 = $508.41 Total PV = $ = $1,432.93 The students can read the example in the book. You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one? Point out that the question could also be phrased as “How much is this investment worth?” Calculator: Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = Total PV = = 1,432.93 Remember the sign convention. The negative numbers imply that we would have to pay 1, today to receive the cash flows in the future.

Multiple Cash Flows – PV Another Example
You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? PV = $1,000 / (1.1)1 = $909.09 PV = $2,000 / (1.1)2 = $1,652.89 PV = $3,000 / (1.1)3 = $2,253.94 PV = $ , , = $4,815.92 Calculator: N = 1; I/Y = 10; FV = 1,000; CPT PV = N = 2; I/Y = 10; FV = 2,000; CPT PV = -1,652.89 N = 3; I/Y = 10; FV = 3,000; CPT PV = -2,253.94

Multiple Cash Flows – FV Example
Suppose you deposit $4000 at the end of each year for next three years at 8% interest rate, you currently have $7000 in the account. Find the value at year 3 of each cash flow and add them together. How much you have in three years? In four years? Today (year 0): FV = $7,000(1.08)3 = $8,817.98 Year 1: FV = $4,000(1.08)2 = $4,665.60 Year 2: FV = $4,000(1.08) = $4,320 Year 3: value = $4,000 Total value in 3 years = $8, , , ,000 = $21,803.58 Value at year 4 = $21,803.58(1.08) = $23,547.87 Point out that you can find the value of a set of cash flows at any point in time; all you have to do is get the value of each cash flow at that point in time and then add them together. The students can read the example in the book. It is also provided here. You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in three years? In four years? Point out that there are several ways that this can be worked. The book works this example by rolling the value forward each year. The presentation will show the second way to work the problem. Calculator: Today (year 0 CF): 3 N; 8 I/Y; -7,000 PV; CPT FV = 8,817.98 Year 1 CF: 2 N; 8 I/Y; -4,000 PV; CPT FV = 4,665.60 Year 2 CF: 1 N; 8 I/Y; -4,000 PV; CPT FV = 4,320 Year 3 CF: value = 4,000 Total value in 3 years = 8, , , ,000 = 21,803.58 Value at year 4: 1 N; 8 I/Y; -21, PV; CPT FV = 23,547.87 I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.

Multiple Cash Flows – FV Example
Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? FV = $500(1.09)2 + $600(1.09) = $1,248.05 Calculator: Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV = Year 1 CF: 1 N; -600 PV; 9 I/Y; CPT FV = Total FV = = 1,248.05

Example 2 Continued How much will you have in 5 years if you make no further deposits? First way: FV = $500(1.09)5 + $600(1.09)4 = $1,616.26 Second way – use value at year 2: FV = $1,248.05(1.09)3 = $1,616.26 Calculator: First way: Year 0 CF: 5 N; -500 PV; 9 I/Y; CPT FV = Year 1 CF: 4 N; -600 PV; 9 I/Y; CPT FV = Total FV = = 1,616.26 Second way – use value at year 2: 3 N; -1, PV; 9 I/Y; CPT FV = 1,616.26

Multiple Cash Flows – FV Example 3
Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%? FV = $100(1.08)4 + $300(1.08)2 = $ $ = $485.97 FV = 100(1.08) (1.08)2 = = Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV = Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV = Total FV = =

Example 3 Time Line 1 2 3 4 5 $100 $300 $136.05 $349.92 $485.97

Frequency of Compounding
General Formula: FVn = PV0(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today

Impact of Frequency Julie Miller has $1,000 to invest for 2 years at an annual interest rate of 12%. Annual FV2 = 1,000(1+ [.12/1])(1)(2) = 1,254.40 Semi FV2 = 1,000(1+ [.12/2])(2)(2) = 1,262.48

Impact of Frequency Quarterly FV2 = 1,000(1+ [.12/4])(4)(2) = 1,266.77
Monthly FV2 = 1,000(1+ [.12/12])(12)(2) = 1,269.73 Daily FV2 = 1,000(1+[.12/365])(365)(2) = 1,271.20

Solving the Frequency Problem (Quarterly)
Inputs 2(4) / , N I/Y PV PMT FV Compute The result indicates that a $1,000 investment that earns a 12% annual rate compounded quarterly for 2 years will earn a future value of $1,

Solving the Frequency Problem (Daily)
Inputs 2(365) 12/ , N I/Y PV PMT FV Compute The result indicates that a $1,000 investment that earns a 12% annual rate compounded daily for 2 years will earn a future value of $1,

Finding the Payment Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = % per month). If you take a 4-year loan, what is your monthly payment? $20,000 = C[1 – 1 / ] / C = $488.26 Calculator: 4(12) = 48 N; 20,000 PV; 8/12 = I/Y; CPT PMT =

Effective Annual Interest Rate
The actual rate of interest earned (paid) after adjusting the nominal rate (which does not take inflation into account) for factors such as the number of compounding periods per year. (1 + [ i / m ] )m - 1

BW’s Effective Annual Interest Rate
Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)? EAR = ( 1 + 6% / 4 ) = = or 6.14%!

Steps to Amortizing a Loan
1. Calculate the payment per period. 2. Determine the interest in Period t (Loan balance at t-1) x (i% / m) 3. Compute principal payment in Period t. (Payment - interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal payment from Step 3) 5. Start again at Step 2 and repeat.

Amortizing a Loan Example
Julie Miller is borrowing $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1: Payment PV0 = R (PVIFA i%,n) $10,000 = R (PVIFA 12%,5) $10,000 = R (3.605) R = $10,000 / = $2,774

Amortizing a Loan Example
[Last Payment Slightly Higher Due to Rounding]

Solving for the Payment
Inputs , N I/Y PV PMT FV Compute The result indicates that a $10,000 loan that costs 12% annually for 5 years and will be completely paid off at that time will require $2, in annual payments.

Steps to Solve Time Value of Money Problems
1. Read problem thoroughly 2. Determine if it is a PV or FV problem 3. Create a time line 4. Put cash flows and arrows on time line 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional)

Introduction to Valuation: The Time Value of Money

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