FI 305 Morosan - Chapter 5

Future Value

Refers to the amount of money an investment will grow to over some period of time at some given interest rate. Put another way, future value is the cash value of an investment at some time in the future. The amount an investment is worth after one or more periods.

Finding the number of periods (t)

Start with the basic equation and solve for t (remember your logs). ▪ FV = PV(1 + r)t ▪ t = ln(FV / PV) / ln(1 + r)

Your goal is to have $1 million in your retirement savings on the day you retire. To fund this goal, you will make one lump sum deposit today. If you plan to retire ________ rather than ________ and earn a ________ rate of interest, then you can deposit a smaller lump sum today. a. later; sooner; high b. sooner; later; low c. later; sooner; low d. sooner; later; high e. today; later; high

a. later; sooner; high

PV - Important Relationship 1 - For a given interest rate, the longer the time period, the ___________ the PV.

lower

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 = .03714 = 3.714% Calculator - the sign convention matters! • N = 5 • PV = -1,000 (you pay 1,000 today) • FV = 1,200 (you receive 1,200 in 5 years) • CPT I/Y = 3.714%

Discount Rate - Example 3 Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards 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 = .172688 = 17.27%

PV - Important Relationship 2 - For a given period of time, the higher the interest rate, the __________ the PV.

smaller

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

Future Value Factor =

(1+r)^t

Basic Definitions 1. Present Value 2. Future Value 3. Interest Rate

1. earlier money on a time line 2. later money on a time line 3. "exchange rate" between earlier money and later money - discount rate - cost of capital - opportunity cost of capital - required return

Present Value Factor =

1/(1+r)^t

Future Value =

$1 x (1+r)^t r = rate t = # of periods

Simple Interest

The interest is not reinvested, so interest is earned each period only on the original principal. Interest earned only on the original principal amount invested.

Compounding

The process of leaving your money and any accumulated interest in an investment for more than one period, thereby reinvesting the interest. The process of accumulating interest on an investment over time to earn more interest.

Discount Rate

The rate used to calculate the present value of the future cash flows.

ONLY 18,262.5 DAYS TO RETIREMENT You would like to retire in 50 years as a millionaire. If you have $10,000 today, what rate of return do you need to earn to achieve your goal? The future value is $1,000,000. The present value is $10,000, and there are 50 years until payment. We need to calculate the unknown discount rate in the following:

$10,000 = $1,000,000/(1+r)^50 (1+r)^50 = 100 The future value factor is thus 100. You can verify that the implicit rate is about 9.65 percent.

PV FOR MULTIPLE PERIODS Suppose you need to have $1,000 in two years. If you can earn 7 percent, how much do you have to invest to make sure you have the $1,000 when you need it? In other words, what is the present value of $1,000 in two years if the relevant rate is 7 percent?

$1000 = PV x 1.07 x 1.07 = PV x 1.07^2 = PV x 1.449 PV = $1,000/1.1449 = $873.44

FV interest factor =

(1+r)^t

Discounted Cash Flow (DCF) Valuation

Calculating the present value of a future cash flow to determine its value today.

PV =

FV / (1+r)^t

Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?

FV = 10(1.055)^200 = 447,189.84

FINDING r FOR A SINGLE-PERIOD INVESMENT You are considering a one-year investment. If you put up $1,250, you will get back $1,350. What rate is this investment paying?

First, in this single-period case, the answer is fairly obvious. You are getting a total of $100 in addition to your $1,250. The implicit rate on this investment is thus $100/$1,250 = 8 percent. More formally, from the basic present value equation, the present value (the amount you must put up today) is $1,250. The future value (what the present value grows to) is $1,350. The time involved is one period, so we have: $1,250 = $1,350/(1+r)^1 1 + r = $1,350/1,250 = 1.08 r = .08 or 8%

SAVING FOR COLLEGE You estimate that you will need about $80,000 to send your child to college in eight years. You have about $35,000 now. If you can earn 20 percent per year, will you make it? At what rate will you just reach your goal?

If you can earn 20 percent, the future value of your $35,000 in eight years will be: FV = $35,000 x 1.20^8 = $35,000 x 4.2998 - $150,493.59

FV =

PV (1+r)^t

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 Calculator ▪ 1 N ▪ 7 I/Y ▪ 10,000 FV ▪ CPT PV = -9,345.79

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

Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?

PV = 19,671.51 / (1.07)^10 = 10,000

Determining the discount rate

PV = FV/(1+r)^t

When we talk about the "value of something, we are talking about the ...

PV unless we specifically indicate that we want the future value

SINGLE-PERIOD PV Suppose you need $400 to buy textbooks next year. You can earn 7 percent on your money. How much do you have to put up today?

We need to know the PV of $400 in one year at 7 percent. Proceeding as in the previous example: Present value×1.07=$400 We can now solve for the present value: PV = $400 x (1/1.07) = $373.83 Thus, $373.83 is the present value. Again, this means that investing this amount for one year at 7 percent will give you a future value of $400.

Kurt won a lottery and will receive $1,000 a year for the next 50 years. The current value of these winnings is called the: a. present value. b. compounded value. c. single amount. d. future value. e. simple amount.

a. present value

Steve just computed the present value of a $10,000 bonus he will receive next year. The interest rate he used in his computation is referred to as the: a. current yield. b. discount rate. c. simple rate. d. compound rate. e. effective rate.

b. discount rate

When we talk about discounting, we mean ...

finding the PV of some future amount

Discount Rate - Example 2 - 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 = .122462 = 12.25%

HOW MUCH FOR THAT ISLAND? To further illustrate the effect of compounding for long horizons, consider the case of Peter Minuit and the American Indians. In 1626, Minuit bought all of Manhattan Island for about $24 in goods and trinkets. This sounds cheap, but the Indians may have gotten the better end of the deal. To see why, suppose the Indians had sold the goods and invested the $24 at 10 percent. How much would it be worth today?

About 391 years have passed since the transaction. At 10 percent, $24 will grow by quite a bit over that time. How much? The future value factor is roughly: (1+r)^t=1.1^391 = apprx. 15,295,000,000,000,000 That is, 15.295 followed by 12 zeroes. The future value is thus on the order of $24 × 15.295 = $367 quadrillion (give or take a few hundreds of trillions). Well, $367 quadrillion is a lot of money. How much? If you had it, you could buy the United States. All of it. Cash. With money left over to buy Canada, Mexico, and the rest of the world, for that matter. This example is something of an exaggeration, of course. In 1626, it would not have been easy to locate an investment that would pay 10 percent every year without fail for the next 391 years.

WAITING FOR GODOT You've been saving up to buy the Godot Company. The total cost will be $10 million. You currently have about $2.3 million. If you can earn 5 percent on your money, how long will you have to wait? At 16 percent, how long must you wait?

At 5 percent, you'll have to wait a long time. From the basic present value equation: $2.3 million = $10,000,000/1.05^t 1.05^t = 4.35 t = 30 years At 16 percent, things are a little better. Verify for yourself that it will take about 10 years.

Interest on Interest Suppose you locate a two-year investment that pays 14 percent per year. If you invest $325, how much will you have at the end of the two years? How much of this is simple interest? How much is compound interest?

At the end of the first year, you will have $325 × (1 + .14) = $370.50. If you reinvest this entire amount and thereby compound the interest, you will have $370.50 × 1.14 = $422.37 at the end of the second year. The total interest you earn is thus $422.37 − 325 = $97.37. Your $325 original principal earns $325 × .14 = $45.50 in interest each year, for a two-year total of $91 in simple interest. The remaining $97.37 − 91 = $6.37 results from compounding. You can check this by noting that the interest earned in the first year is $45.50. The interest on interest earned in the second year thus amounts to $45.50 × .14 = $6.37, as we calculated.

COMPOUND INTEREST You've located an investment that pays 12 percent per year. That rate sounds good to you, so you invest $400. How much will you have in three years? How much will you have in seven years? At the end of seven years, how much interest will you have earned? How much of that interest results from compounding?

Based on our discussion, we can calculate the future value factor for 12 percent and three years as follows: (1+r)^t=1.12^3=1.4049 Your $400 thus grows to: $400×1.4049=$561.97 After seven years, you will have: $400×1.12^7=$400×2.2107=$884.27 Thus, you will more than double your money over seven years. Because you invested $400, the interest in the $884.27 future value is $884.27 − 400 = $484.27. At 12 percent, your $400 investment earns $400 × .12 = $48 in simple interest every year. Over seven years, the simple interest thus totals 7 × $48 = $336. The other $484.27 − 336 = $148.27 is from compounding.

DIVIDEND GROWTH The TICO Corporation currently pays a cash dividend of $5 per share. You believe the dividend will be increased by 4 percent each year indefinitely. How big will the dividend be in eight years?

Here we have a cash dividend growing because it is being increased by management; but once again the calculation is the same: FV = $5 x 1.04^8 = $5 x 1.3686 = $6.84 The dividend will grow by $1.84 over that period.

Compound Interest

Interest earned on both the initial principal and the interest reinvested from prior periods.

Interest on interest

Interest earned on the interest on an investment over time to earn more interest.

Present Value

The current value of future cash flows discounted at the appropriate discount rate. PV = $1 x (1/(1/+r)^t) = $1/(1+r)^t

EVALUATING INVESTMENTS To give you an idea of how we will be using present and future values, consider the following simple investment. Your company proposes to buy an asset for $335. This investment is very safe. You would sell off the asset in three years for $400. You know you could invest the $335 elsewhere at 10 percent with very little risk. What do you think of the proposed investment?

This is not a good investment. Why not? Because you can invest the $335 elsewhere at 10 percent. If you do, after three years it will grow to: $335 x (1+r)^t = $335 x 1.1^3 = $335 x 1.331 = $445.89 Because the proposed investment pays out only $400, it is not as good as other alternatives we have. Another way of seeing the same thing is to notice that the present value of $400 in three years at 10 percent is: $400 x (1/(1+r)^t) = $400/1.1^3 = $400/1.331 = $300.53 This tells us that we have to invest only about $300 to get $400 in three years, not $335. We will return to this type of analysis later on.

SAVING UP You would like to buy a new automobile. You have $50,000 or so, but the car costs $68,500. If you can earn 9 percent, how much do you have to invest today to buy the car in two years? Do you have enough? Assume the price will stay the same.

What we need to know is the present value of $68,500 to be paid in two years, assuming a 9 percent rate. Based on our discussion, this is: PV - $68,500/1.09^2 = $68,500/1.1881 = $57,655.08 You're still about $7,655 short, even if you're willing to wait two years.

DECEPTIVE ADVERTISING? Businesses sometimes advertise that you should "Come try our product. If you do, we'll give you $100 just for coming by!" If you read the fine print, what you find out is that they will give you a savings certificate that will pay you $100 in 25 years or so. If the going interest rate on such certificates is 10 percent per year, how much are they really giving you today?

What you're actually getting is the present value of $100 to be paid in 25 years. If the discount rate is 10 percent per year, then the discount factor is: 1/1.1^25 = 1/10.8347 = .0923 This tells you that a dollar in 25 years is worth a little more than nine cents today, assuming a 10 percent discount rate. Given this, the promotion is actually paying you about .0923 × $100 = $9.23. Maybe this is enough to draw customers, but it's not $100.

Andy deposited $3,000 this morning into an account that pays 5 percent interest, compounded annually. Barb also deposited $3,000 this morning at 5 percent interest, compounded annually. Andy will withdraw his interest earnings and spend it as soon as possible. Barb will reinvest her interest earnings into her account. Given this, which one of the following statements is true? a. After five years, Andy and Barb will both have earned the same amount of interest. b. Barb will earn more interest in Year 2 than Andy. c. Barb will earn more interest in Year 1 than Andy will. d. Andy will earn more interest in Year 3 than Barb will. e. Andy will earn compound interest.

b. Barb will earn more interest in Year 2 than Andy.

Which one of these will increase the present value of a set amount to be received sometime in the future? a. Decrease in both the future value and the number of time periods b. Decrease in the interest rate c. Increase in the discount rate d. Increase in the time until the amount is received e. Decrease in the future value

b. Decrease in the interest rate

Phillippe invested $1,000 ten years ago and expected to have $1,800 today He has neither added nor withdrawn any money since his initial investment. All interest was reinvested and compounded annually. As it turns out, he only has $1,680 in his account today. Which one of the following must be true? a. He earned simple interest rather than compound interest. b. He earned a lower interest rate than he expected. c. He ignored the Rule of 72 which caused his account to decrease in value. d. He did not earn any interest on interest as he expected. e. The future value interest factor turned out to be higher than he expected.

b. He earned a lower interesr rate than he expected

Nan and Neal are twins. Nan invests $5,000 at 7 percent at age 25. Neal invests $5,000 at 7 percent at age 30. Both investments compound interest annually. Both twins retire at age 60 and neither adds nor withdraws funds prior to retirement. Which statement is correct? a. If both Nan and Neal wait to age 70 to retire they will have equal amounts of savings. b. Nan will have more money than Neal at any age. c. Neal will earn more interest on interest than Nan. d. Nan will have less money when she retires than Neal. e. Neal will earn more compound interest than Nan.

b. Non will have more money than Neal at any age.

Terry is calculating the present value of a bonus he will receive next year. The process he is using is called: a. growth analysis. b. discounting. c. reducing. d. compounding. e. accumulating.

b. discounting

You are investing $100 today in a savings account. Which one of the following terms refers to the total value of this investment one year from now? a. Present value b. Discounted value c. Future value d. Principal amount e. Invested principal

c. Future Value

Chang Lee is going to receive $20,000 six years from now. Soo Lee is going to receive $20,000 nine years from now. Which one of the following statements is correct if both individuals apply a discount rate of 7 percent? a. In future dollars, Soo Lee's money is worth more than Chang Lee's money. b. Twenty years from now, the value of Chang Lee's money will equal the value of Soo Lee's money. c. In today's dollars, Chang Lee's money is worth more than Soo Lee's. d. Soo Lee's money is worth more than Chang Lee's money given the 7 percent discount rate. e. The present values of Chang Lee's and Soo Lee's money are equal.

c. In today's dollars, Change Lee's money is worth more than Soo Lee's.

Christina invested $3,000 five years ago and earns 2 percent annual interest. By leaving her interest earnings in her account, she increases the amount of interest she earns each year. The way she is handling her interest income is referred to as: a. discounting. b. accumulating. c. compounding. d. simplifying. e. aggregating.

c. compounding

Your grandmother has promised to give you $10,000 when you graduate from college. If you speed up your graduation by one year and graduate two years from now rather than the expected three years, the present value of this gift will: a. remains constant b. equal $10,000 c. increase d. decrease e. be less than $10,000

c. increase

Discount

calculate the present value of some future amount

This afternoon, you deposited $1,000 into a retirement savings account. The account will compound interest at 6 percent annually. You will not withdraw any principal or interest until you retire in 40 years. Which one of the following statements is correct? a. The total amount of interest you will earn will equal $1,000 × .06 × 40. b. The future value of this amount is equal to $1,000 × (1 + 40).06. c. The interest you earn in Year 6 will equal the interest you earn in Year 10. d. The present value of this investment is equal to $1,000. e. The interest amount you earn will double in value every year.

d. The present value of this investment is equal to $1,000.

The process of determining the present value of future cash flows in order to know their value today is referred to as: a. future value interest factoring. b. compound interest valuation. c. interest on interest valuation. d. discounted cash flow valuation. e. complex factoring.

d. discounted cash flow valuation

Which one of these will increase the present value of a set amount to be received sometime in the future? a. Increase in the time until the amount is received b. Decrease in both the future value and the number of time periods c. Decrease in the future value d. Increase in the discount rate e. Decrease in the interest rate

e. Decrease in the interest rate.

Renee invested $2,000 six years ago at 4.5 percent interest. She spends all of her interest earnings immediately so she only receives interest on her initial $2,000 investment. Which type of interest is she earning? a. Free interest b. Compound interest c. Complex interest d. Interest on interest e. Simple interest

e. Simple Interest

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% of the loan amount for closing costs. • Assume the type of house you want will cost 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?

• 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 = 15,000 + 6,750 = 21,750 • Compute the number of periods • Using the formula: ▪ t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years • Using a financial calculator: ▪ PV = -15,000, FV = 21,750, I/Y = 7.5, CPT N = 5.14 years

FV AS A GENERAL GROWTH FORMULA Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?

▪ FV = 3,000,000(1.15)^5 = 6,034,072