HellonDr.Ramsey

Can you assist me with my discussion please, Min 150 words due Friday.

Basic Time Value of Money It is a common fact that many lottery winners are ?broke? sooner than later. If you won a $1,000,000 lottery, would you want to collect the lump sum winnings today or receive the monies over time? How does your decision influence the ultimate amount of cash you will collect? Explain the TVM factors you would consider as you make this decision.

Thank you

PRINTED BY: irisgarcia3@student.kaplan.edu. Printing is for personal, private use only. No part of this book may be reproduced or transmitted without publisher's prior permission. Violators will be prosecuted. Chapter 3 Intro TVM LEARNING OBJECTIVES LO1 Introduction to Time Value of Money LO2 Find the Future Value of a Sum LO3 Find the Present Value of a Sum LO1 Introduction to TVM Concepts 1.1 Introduction In this chapter, you'll learn to compute how the value of money changes over time. In addition to corporate finance, this knowledge can come in handy for decisionmaking on personal finance issues like calculating car payments, mortgage payments, and retirement planning. In fact, understanding the Time Value of Money is one of the most important concepts you will learn in this book. 1.2 Simple Interest versus Compound Interest How do you calculate the future value of a sum one period in the future? Suppose money is deposited in a bank account for 1 year. At the end of the year interest is computed and paid. The future value is simply the amount of money in the account at the end of the year. Example 3.1 Simple Interest and OnePeriod Future Value Compute simple interest earned on a 1year $100 deposit that earns 5% per year and compute the future value of the $100 deposit. SOLUTION Write your answers here. The interest is computed as the deposit multiplied by the interest rate: Interest = principal interest rate Interest = $100 0.05 = $5 The future value at the end of the year is the sum of the original deposit and the interest: Future value = $100 + $5 = $105 In the example, the $5 of interest is simple interest because it is interest earned on the original principal. You can use the following formula used in the example to determine future value: FV = PV + (PV i) PV (1 + i) Eq. 3.1 Synonyms and notation The original deposit is also called the principal or the present value and denoted PV. The interest rate, denoted i, is 5%. This rate is called the quoted rate, the nominal rate, or the annual percentage rate (APR). Unless otherwise stated, you should assume that the interest rate is annual. The future value is denoted FV. When there is only one period the interest earned is called simple interest because no interest is earned on previously paid interest. In Section 2, we will calculate a multiperiod future value that will generate compound interest. Simple interest is interest earned on the original principal. Compound interest is interest earned on interest. 1.3 The Timeline What does it mean to say that you received $100 per year for 5 years? Did the money arrive at the beginning, the end, or in the middle of each year? A simple tool to solve this problem is the timeline, a graphic representation of cash flows. Review the following diagram and accompanying video: Figure 3.1 inflows and outflows Inflows, cash you receive, have a positive sign (typically, the + sign is dropped), as implied in the above examples. A cash outflow, cash you spend, has a negative sign, as shown in the following timeline: Figure 3.2 In Figure 3.2, a $100 investment is made at the beginning of the first period and $200 investments are made at the beginning of the second and third periods. Financial calculators and some builtin spreadsheet functions require that you input cash flows with either a positive or negative sign to indicate inflows and outflows. However, when you solve time value of money problems algebraically, you do not make outflows negative. All cash flows are positive. LO2 Future Value of a Sum 2.1 Compound Interest: Future Value over Multiple Periods Simple interest is easy to compute, but most of the time, we want our interest payments to be reinvested. For example, if you put money into a savings account, the bank automatically deposits the monthly or quarterly interest payments back into the account. During subsequent periods, interest is earned on a higher balance. If the interest is reinvested, then, in later periods, interest is earned on earlier interest. This is compound interest. Virtually all of the calculations performed in finance assume that interest is compounded. We'll begin our study of compound interest by finding future balances over multiple periods with annual compounding. Compound interest: Future Value over One Year As we saw in the introduction, if you make a $100 deposit into a bank that pays 5% interest once per year, you'll have $105 at the end of 1 year. The formula for computing the balance after one period is given in Eq 3.1: FV1 = PV0 (1 + i) FV1 = $100 (1 + 0.05) = $105 This calculation is shown on the timeline in Figure 3.3. At time period 0, we see a present value of $100. At time period 1, the future value is FV1 = $105. Notice the use of the subscripts in the above formula. The subscript denotes the point on the timeline when the cash flow occurs. Figure 3.3 Compound interest: Future Value over Two Years Now suppose that you leave the deposit in the bank to compound for another year, without withdrawing any money. Using the above formula, the balance grows to $110.25: FV2 = FV1 (1 + i) FV2 = $105 (1 + 0.05) = $110.25 This calculation is shown under period 2 on the timeline in Figure 3.3. During the first year, the account earned $5, but during the second year, it earned $5.25. The extra $0.25 is interest earned on the first period's interest ($5.00 0.05 = $0.25)it is compound interest. We can simplify these calculations by noting that FV1 is equal to PV0 (1 + i) and then by substituting PV0 (1 + i) for FV1 into the equation: FV2 = PV0 (1 + i) (1 + i) FV2 = PV0 (1 + i)2 FV2 = $100 (1 + 0.05)2 FV2 = $100 1.1025 = $110.25 The process of computing a future balance over multiple periods is called compounding because the investor is earning compound interest. Compound interest: Future Value over Multiple Years Similarly, each subsequent period of compounding increases the exponent by one. The generalized equation for finding the future value of a deposit is FVn = PV0 (1 + i)n Eq. 3.2 where Let's illustrate with an example. Example 3.2 Future Value over Multiple Years Suppose your grandmother gave you $1,000 when you graduated from high school. Instead of using it to buy clothes, you decided to invest it and to not touch the balance for 40 years, until you retire. If you managed to earn 10% per year, what is the future value of your investment? To work out the future value of $1,000 invested for 40 years at 10%, let's first lay out the data that we have and draw a timeline to describe the problem. You should make a habit of doing this for each time value problem that you attempt. The timeline for the problem is SOLUTION Write your answers here. To find the future value of $1,000 over 40 years compounded at 10: FVn = PV0 (1 + i)n FV40 = $1,000 (1.10)40 FV40 = $1,000 45.25926 FV40 = $45,259.26 Calculating interest Earned To find the interest earned, we subtract the principal (the original amount) from the ending balance: Interest earned = FV PV Interest earned = $45,259.26 $1,000 = $44,259.26 Write your answers here. It's Time to Do a SelfTest 1. Practice computing the future values. Answer Simple and Compound Interest Compared If simple interest, rather than compound interest, had been earned in Example 3.2, then $100 per year would have been earned ($1,000 0.10 = $100). Over a period of 40 years, the total simple interest earned would have been 40 times $100 or $4,000. Since the total amount of interest earned was $44,259.26, the difference ($44,259.26 $4,000) of $40,259.26 was earned because of compounding. Put another way, $40,259.26 in interest was earned on interest. In this example, more interest was earned on the interest than was earned on the original principal! This is the magic of compound interest. Compounding Rules of Thumb It's important that the student of finance develop an understanding of how the compounding process works beyond simply applying the equations. Here are some rules that will help you think of the theory of compound interest in more meaningful ways. 1. Future balance increases if periods and/or interest rates increase. First, as the number of compounding periods increases, the future balance increases. Second, as the interest rate increases, the future balance increases. The videos show how the future balance changes given different interest rates and numbers of compounding periods. 2. Compound interest theory applies to any growth. One of the more important features of TVM calculations is that the methods can be applied to anything that grows. We can use the same equations to find future sales, if sales grow at a constant rate. The method can be applied to any constant growth situation, whether it be money, sales, profits, or dividends. 3. Any length of period can be used. Time does not have to be measured in years. The formula can be used with any length period: days, weeks, months, quarters, or years. This period is called the compounding period or the conversion period. This period is the basic unit of time in all time value of money problems. However, whatever compounding period is used, the interest rate must be defined over the same period. For example, if time is measured in months, then the interest rate must be expressed as a \"per month\" rate. If time is measured in weeks, then the interest rate must be expressed as a \"per week\" rate. When no compounding period is stated, then you should assume that interest is compounded annually. Write your answers here. It's Time to Do a SelfTest 2. You are about to graduate from college at the age of 22. You just learned that your grandfather invested $10,000 in your name when you were born and it's available for you to withdraw today. If the stock market earned an average 12% over the past 22 years, how much should be in the account today? Algebraic Answer Excel Answer Calculator Answer 3. Suppose that, instead of investing the $10,000 in the stock market, your grandfather invested it in Treasury bonds, which averaged 5.5%. How much would the investment be worth today? Algebraic Answer Excel Answer Calculator Answer 2.2 Future Value of Mixed Streams of Cash Flows Suppose you expect to receive a number of separate cash flows over time, which will be different from one another. Now, suppose you want to know what their combined future value will be. This might occur if you're interested in computing your retirement balance, but know you'll save a different amount each year as your salary increases. The solution is to compute the future value of each cash flow separately and add them together. Example 3.3 Future Value of Mixed Streams You expect to receive $100 at the end of the first period, $200 at the end of the second period, and $300 at the end of the third period. If you will earn 10% by investing these cash flows, what will the future balance be at the end of the sixth period? The timeline for the problem is SOLUTION Write your answers here. The first cash flow ($100) goes forward five years (count the number of intervals in the timeline) so n = 5. The future value at date 6 is $161.05. The date 2 cash flow ($200) compounds for 4 years (n = 4) and the future value is $292.82. The date 3 cash flow ($300) compounds for 3 years (n = 3) and the future value is $399.30. The sum of the three future values is $853.17 2.3 Solving for the Interest Rate In all of the examples shown, we've used the basic future value equation to solve for how much a deposit today can grow to in the future. There are times, however, when we already know both the future and present values and want to know either the interest rate or the length of time it took to earn the future balance. Next, we will learn how to find the interest rate and then, we will learn how to solve for the length of time. We can use the future value of a lump sum (Eq. 3.2) to solve for the interest rate (when it is the unknown): Eq. 3.3 Example 3.4 Future Value, Solving for the Interest Rate Just as you were about to enter college, you learned that a great aunt had established a college trust fund for you when you were born, 20 years ago. She deposited $5,000 initially. If the balance is now $19,348.42, what average compounded rate of return has been earned? SOLUTION Write your answers here. We can use Eq. 3.3 to solve for the unknown interest rate: i = [19,348.42/5,000]1/20 1 = [3.8697]1/20 1 i = 1.0700 1 = 0.0700 or 7% Write your answers here. It's Time to Do a SelfTest 4. Calculate the average annual growth rate if an investment of $500 grows to $1200 in 4 periods. Algebraic Answer Excel Answer Calculator Answer 5. If a company's dividend was $2.00 5 years ago and it is $3.00 now, what was the average annual dividend growth rate? Algebraic Answer Excel Answer Calculator Answer 2.4 Solving for the Number of Periods If you know the PV, FV, and the return you can compute the number of periods. Example 3.5 Future Value, Solving for n How long does it take to double your money at 3%? For example, if inflation averages 3% per annum, then how many years will it take for the price of a pint of beer to double from $1.50 to $3.00? PV0 = $1.50 FVn = $3 n = ? i = 3% SOLUTION Write your answers here. First, we set this up as a future value problem where the unknown is the number of compounding periods, n: FVn = PV0 (1 + i)n FVn/PV0 = (1 + i)n The videos show how to solve for n with a financial calculator or spreadsheet. The Explain It shows how to solve for n using algebra and a property of logarithms. 2.5 Future Value with NonAnnual Compounding Annual compounding refers to a situation in which interest is paid once per year. It is common for interest to be paid more frequently than once per year. Most bank accounts, for example, pay interest every month. We must adjust our formula to allow for any interest payment schedule that may arise. To calculate a future value with nonannual compounding: 1. Define m as the number of compounding periods in the year. For example, with semiannual compounding, m = 2. 2. Always match the interest rate to the length of the compounding period. If there are m compounding periods, then the periodic rate for each period is i/m. For example, if the interest rate is 12% per year, then the semi annual periodic rate is 6%, the quarterly rate is 3%, and the monthly rate is 1%. 3. Always adjust the number of periods by multiplying the number of years by the number of compounding periods per year. For example, 5 years (n = 5) with semiannual compounding (m = 2) is 10 periods (n m). The following formula adjusts the generalized equation for future value by inserting the periodic interest rate and the number of compounding periods. Eq. 3.4 Example 3.6 Multiple Compounding Periods per Year What is the future value of a $1,500 deposit after 20 years with an annual interest rate of 8% compounded quarterly? In this example, we know that i = 8% m = 4 PV = $1,500 n = 20 First, solve for the periodic rate: SOLUTION Write your answers here. i/m = 8%/4 = 2%. Second, substitute these numbers into Eq. 3.4: The future value of a $1,500 deposit after 20 years at a rate of 8% compounded quarterly is $7,313.16 In Example 3.6, we found that $1,500 deposited for 20 years at 8% compounded quarterly is $7,313.16. However, the future value with annual compounding is only $6,991.44. It turns out that the more frequently interest is paid, the greater is the future value. Now, look at Figure 3.4. This figure shows the future value of $100 at 12%, compounded at different frequencies. As the compounding periods get smaller the increase in future value from additional compounding periods increases at a decreasing rate. Figure 3.4 Effect of Different Compounding Frequencies on Future Value (i = 12%) Write your answers here. It's Time to Do a SelfTest 6. Practice computing future values with multiple compounding periods per year. Answer 2.6 Effective Interest Rate If you make a $100 deposit into an account that pays 12% compounded once per year, at the end of the year you will have $112. Would you be any better off if the bank compounded (made interest payments) any more frequently? To answer this let's compute what you would have if the deposit was compounded semiannually. After 6 months, you would be paid half of the quoted rate, or $6. This $6 would then earn additional interest for you during the second halfyear. The interest earned on this interest would be $0.36, or 6% of $6. Your total earnings would then be $6 each half year plus $0.36 or $12.36. So you are better off with compounding. The 12% in this example is the quoted interest rate and the 12.36% is the effective interest rate (EIR). The EIR is the rate you actually earn on your investment after taking into account the compounding frequency. To determine the exact effective interest rate, given multiple compounding periods per year, we find the FV of $1 after 1 year and then subtract the initial dollar. What's left is the interest for the year, which is the effective interest rate. The FV of $1 is found by applying Eq. 3.2 where PV = 1 and n = 1. The effective interest rate is computed by subtracting 1. Eq. 3.5 computes the effective interest rate: Eq. 3.5 where i = the quoted annual rate m = the number of compounding intervals Example 3.7 Effective Interest Rate Calculation What is the effective interest rate for a nominal rate of 12% which is compounded monthly? In this example, we know that i = 12% m = 12 SOLUTION Write your answers here. Substitute these numbers into Eq. 3.5: An annually compounded interest rate of 12.68% is equivalent to earning a 12% annual rate compounded 12 times per year. Table 3.1 Effective Interest Rates with 12% Annual Rate Compounding interval Eq. 3.5 Effective Rate Annual FV = (1.12)1 1 12.00% Semiannual FV = (1.06)2 1 12.36 Quarterly FV = (1.03)4 1 12.55 Monthly FV = (1.01)12 1 12.68 Weekly FV = (1.0023)52 1 12.73 Daily FV = (1.0003288)365 1 12.7475 Table 3.1 shows the effective rate at different compounding intervals, when the annual rate is 12%. Suppose you are attempting to choose between two bank savings accounts. The first pays 5% compounded annually, and the second pays 4.9% compounded monthly. Which would you prefer? To answer this, you must compute the effective rate of each alternative and pick the largest one. The effective rate of the first is unchanged, 5%. The effective rate of the second is 5.01%. So, you would choose the bank offering 4.9% compounded monthly. Write your answers here. It's Time to Do a SelfTest 7. Practice computing effective interest rates. Answer Ready to do LO2 topic homework 1? LO3 Present Value of a Sum Often, we need to determine what the value is today of sums that will be received in the future. For example, if a firm is evaluating an investment that will generate future income, it must compare today's expenditures with expected future revenues. To compare sums across time, future values must be adjusted to what they're worth today. Luckily, if the concepts discussed under future value make sense, it won't be difficult to understand present value. 3.1 The Present Value Equation Take another look at the future value equation (Eq. 3.2): FVn = PV0 (1 + i)n The future value is defined in terms of present value. To find the equation for computing present value, we need to rearrange Eq. 3.2. If both sides are divided by (1 + i)n, we get the equation for present value: Eq. 3.6 The term 1/(1 + i)n (or (1 + i)n) is called the present value interest factor. Example 3.8 Solving for the Present Value What is the present value of $100, which will be received in 10 years, if the interest rate is 5%? SOLUTION Write your answers here. FV = $100 n = 10 i = 0.05 Present Value with NonAnnual Compounding Just as we discussed with future value, the compounding period in Eq. 3.6 does not have to be a full year. Frequently, there are shorter compounding periods such as semiannual, monthly, weekly, or daily. When the compounding period is not annual we must be sure to use a periodic interest rate that reflects the length of the period. The periodic rate is simply i/m, where m is the number of compounding periods in a year. For example, if the nominal rate is 10% (i = 10%) and the compounding period is semiannual (m = 2), then the periodic rate is 10%/2 or 5%. The formula for present value with nonannual compounding is analogous to the formula for the future value with nonannual compounding (Eq. 3.4). Eq. 3.7 Example 3.9 Solving for the Present Value with NonAnnual Compounding What is the present value of $100, which will be received in 10 years, if the annual interest rate is 6% and there is monthly compounding of invested funds? SOLUTION Write your answers here. FV = $100 n = 10 years m = 12 n m = 10 12 = 120 months i/m = 6%/12 = 0.5% per month Some insights about Present Values 1. The present value interest factor, 1/(1 + i)n, is just the inverse of the future value factor, (1 + i)n. 2. If i = 0, then 1/(1 + i)n = 1. If interest rates are zero, then future values are equivalent to present values. 3. Interest rates and the present value are inversely related. When the interest rate rises, the present value falls. 4. The process of computing the present value of a future sum is called discounting. Remember that a future sum isn't worth as much as a sum you have today. To convert future amounts to their present values, the future amount must be reduced, or discounted. 5. You can think of discounting as moving dollars to the left on the timeline. Whenever dollars are moved to the left, their value gets smaller. The further to the left and the higher the interest rate, the less the dollars are worth. 6. You don't always want to move cash flows all the way to time period 0. Present value calculations include moving future sums to any earlier period. Write your answers here. It's Time to Do a SelfTest 8. Practice computing present values. Answer 3.2 Present Value of a Mixed Stream of Cash Flows Thus far, we've focused on computing the present value of a single, lumpsum future cash flow. However, there are many occasions when you must find the present value of a series of unequal cash flows. For example, if you're evaluating an investment in a business, it's unlikely each year's cash flows will be the same. To find the present value of mixed streams, simply find the present value of each cash flow individually, then add them together. Example 3.10 Present Value of Mixed Streams You're contemplating introducing a new product line expected to generate cash flows of $100 at the end of the first year, $200 at the end of the second year, $300 at the end of the third year, and $400 at the end of the fourth year. What is the present value of the cash flows if the appropriate interest rate is 10%? SOLUTION Write your answers here. The timeline for the problem is The first cash flow ($100) is discounted 1 year (n = 1) and the present value at date 0 is $90.91. The date 2 cash flow ($200) is discounted 2 years (n = 2) and the present value at date 0 is $165.29. The date 3 cash flow ($300) is discounted 3 years (n = 3) and the present value at date 0 is $225.39. The date 4 cash flow ($400) is discounted 4 years (n = 4) and the present value at date 0 is $273.21. The sum of the four present values is $754.80. Write your answers here. It's Time to Do a SelfTest 9. Suppose you estimate the cost of your room, board, and tuition for the next 4 years of your education (you've decided to get your MBA) as follows: $5,000 due at the end of the first year, $5,500 due at the end of the second year, $6,000 due at the end of the third year, and $6,500 due at the end of the fourth year. Assume a discount rate of 7%. How much must you invest today to pay for your future education? Algebraic Answer Excel Answer Calculator Answer 3.3 The Intuition behind Present Values Because the concept of present value is central to the study of finance, you need to understand the intuition behind the numbers. What does it mean that the present value of $100 to be received in 1 year at 10% is $90.91? 1. It means you're indifferent to which you getthe $90.91 today or the $100 in 1 year. What if you don't really need the money today, but you expect to need it in 1 year? In that case, you can invest the $90.91 today and it will grow to be exactly $100 by the time you need it. The interest rate used to discount the $100 back to the present is selected so that you'll be properly compensated for the risk you may not be paid and for the delay in receiving the cash flow. 2. If you have the present value of a future sum, you can exactly match that future sum by investing what you have today at the discount rate. 3. It is the current value of a promised payment in the future. 3.4 Using Present Values to Value Assets There are a number of different methodologies used to find the fair value of assets. By fair value we mean the price that you would expect to pay (receive) if you buy (sell) the asset in a competitive market. The primary valuation methodology presented in this book is the discounted cash flow (DCF) method. With the DCF methodology, the current value of an asset is the present value of all future cash flows. The current price must be such that the seller is indifferent between continuing to receive the cash flow stream provided by the asset and receiving the offer price. That is exactly what present values do. Suppose you're interested in investing in a Van Gogh painting called Sunflowers. If you think you can hold the painting for 2 years and then resell it for $1,200,000, how much would you pay for the painting today? The only cash flow from this investment is the final sales price. No periodic payments will be received. To compute the current price of the painting, you will first need to determine an appropriate discount rate to use for computing the present value of the cash flows. Investments in art are high risk and therefore require a high discount rate. Let's assume you require a 16% return to compensate you for the risk of the investment. To find the current price of the painting, discount the future cash flow back to the present at 16%. This provides a present value of $891,795.48 [$1,200,000/11.1622]. In fact, this painting actually sold for $883,123. The buyer must have expected a slightly lower future price or required a slightly higher return than assumed by our example. A commercial building is valued using the same technique. The preferred way to value commercial real estate is the income approach, which involves computing the present value of the building's cash flows and using this to estimate the property's value. Why did the Van Gogh painting not sell for the present value of its expected cash flows and why might a building not sell for its PV of projected income? It is because competition for valuable investments in the market causes prices to adjust so that they represent the best estimate of value by market participants. Not everyone agrees about what the future cash flows are going to be. Assets like these will sell to the investor who either expects the largest cash flows or sees the least risk in the investment. Let's summarize how to find the DCF value of a business asset. 1. Identify the cash flows that result from owning the asset. 2. Determine what discount rate is required to compensate the investor for \"the risk\" of holding the asset. 3. Find the present value of the cash flows estimated in step 1 using the discount rate determined in step 2. This discounted cash flow methodology is the process used to value all business assets and securities. We will employ it repeatedly in the remainder of this book to value bonds, stocks, projects, and firms. The next video takes a brief look at an alternative valuation method called the Bigger Fool theory. Write your answers here. It's Time to Do a SelfTest 10. You are offered a tree (a giant Sitka Spruce) that will grow to its optimum size in 10 years, at which time its lumber will be worth $10,000. What will you pay today for the tree if your alternative investments can generate a return of 5%? Algebraic Answer Excel Answer Calculator Answer Ready to do LO3 topic homework 1? Chapter Summary Concepts You Key Equations and Terms Should Know Solution Tools Extra Practice MyFinanceLab LO1 Introduction to Time Value of Money Study Plan 3.LO1 principal, present value, quoted rate, nominal rate, annual percentage rate (APR), future value, simple interest, compound interest, timeline Future Value in One Period FV =PV + (PV i) Eq. 3.1 LO2 Find the annual compounding, compounding, Future Value compounding period, conversion period, of a Sum periodic rate, effective interest rate (EIR) Future Value of Multiple Years Study Plan 3.LO2 FVn = PV0 (1 + i)n Eq. 3.2 Solving for the Interest Rate Eq. 3.3 Solving for n Finding the Future Value of Mixed Streams of Payments Eq. 3.4 Multiple Compounding Periods per Year Finding Effective Interest Rate Compounded Monthly Eq. 3.5 LO3 Find the discounting, discounted cash flows (DCF), Present Value Bigger Fool theory of a Sum Study Plan 3.LO3 Solving for the Present Value Eq. 3.6 Present Value of NonAnnual Compounding Eq. 3.7 Finding the Present Value of Mixed Streams of Payments It's time to do your chapter