Principles of Managerial Finance Chapter 4 PDF

Principles of Managerial Finance Sixteenth Edition Chapter 4 Time Value of Money Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Learning Objectives Understand the Valu...

Principles of Managerial Finance Sixteenth Edition Chapter 4 Time Value of Money Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Learning Objectives Understand the Valuation Principle, and how it can be used to identify decisions that increase the value of the firm Assess the effect of interest rates on today’s value of future cash flows Be able to determine the time value of money: ▪ Future Value. ▪ Present Value. ▪ Present Value of an Annuity. ▪ Future Value of an Annuity. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Chapter Outline 3.1 Time Value of Money 3.2. Uses of Time Value of Money 4. Interest Rates 5. Time Value of Money (TVM) Calculations ▪ Present Value & Future Value ▪ Cash Flows and Timelines ▪ TVM: Annuities and Perpetuities Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  Review/Refresher Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 3.1 The Time Value of Money What Does Time Value of Money Mean? ◼ The concept that money available today is This is because that worth more than the same amount of ▪ a money received today can be money in the future invested to earn interest ◼ This preference rests on the Time value of ▪ Due to money's potential to grow in money. value over time. ◼ Thus, a money received today is worth ▪ Because of this potential, money more than a money received tomorrow, that's available in the present is why? considered more valuable than the same amount in the future Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 3.1 The Time Value of Money Which would you prefer -- $10,000 today or $10,000 in 5 years? Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!! This illustrates that there is an inherent monetary value attached to time. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 3.1 The Time Value of Money ◼ Time Value of Money is dependent not only on the time interval being considered but also the rate of discount used in calculating current or future values. ◼ Based on this, we can use the time value of money concept to calculate how much you need to invest or borrow now to meet a certain future goal Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 3.1 The Time Value of Money Take a simple example: You need to sell your car and you receive offers from three different buyers. – The first offer gives 4,000RM to be paid now – The second offer gives 4,100RM to be paid one year from now – The third offer gives 4,600RM to be paid after five years. Assume that the second and third offers have no credit risk. – The risk free interest rate is 5%. Which offer would you accept? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. First offer gives 4,000RM Second offer gives 4,100RM but after 1 Third offer gives 4,600RM but after 5 year year FV = PV(1 + i) n FV = 4000(1 + 0.05) 1 PV=FV/(1+i)n PV=FV/(1+i)n FV = 4000(1.05) PV=4100/(1+5%)1 PV=4600/(1+5%)5 FV = 4,200 PV= 4100/(1+0.05)1 PV= 4600/(1+0.05)5 PV= 4100/(1.05) PV= 4600/(1.05)5 FV = PV(1 + i) n PV= 3904.7 PV= 4600/(1.27) FV = 4,000(1 + 0.05) 5 FV = 4000(1.27) PV= 3622.0 FV = 5,080 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 3.2 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! Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 4. Interest Rates ◼ A rate which is charged or paid for the use of money. An interest rate is often expressed as an annual percentage of the principal. ◼ Nominal interest rate, is one where the effects of inflation have not been accounted for. (unadjusted for inflation) ◼ Real interest rates are interest rates where inflation has been accounted for. (adjusted for inflation) ◼ Interest on an amount lent or borrowed depends on the principal sum, the interest rate, the compounding frequency, and the length of time over which it is lent, deposited, or borrowed. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 4. Interest Rates Choosing an Interest Rate in Time Value Measurements Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent). Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 4. Interest Rates Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent). When you deposit money into a bank, the bank pays you interest. When you borrow money from a bank, you pay interest to the bank. Simple interest is money paid only Rate of interest is the percent on the principal. charged or earned. I = P  I  t Principal is the amount of money Time that the money is borrowed or invested. borrowed or invested (in years). Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 4. Interest Rates Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent). Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 4. Interest Rates Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Compound interest is interest paid not only on the principal, but also on the interest that has already been earned. The formula for compound interest is below. A = p(1 + r ) t “A” is the final dollar value, “P” is the principal, “r” is the rate of interest, and “t” is the number of compounding periods per year. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 4. Interest Rates Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Timeline Example i= Years 0 1 10 2 3 4 Cash flow −$100 $30 % $20 −$10 $50 The 4-year timeline illustrates the following: – The interest rate is 10%. – A cash outflow of $100 occurs at the beginning of the first year (at time 0), followed by cash inflows of $30 and $20 in years 1 and 2, a cash outflow of $10 in year 3, and cash inflow of $50 in year 4. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Quick Quiz Question 1: Mr. A. currently has RM20,000 in a bank account that pays 6% monthly. Calculate the accumulated sum in the bank account after 5 years using simple interest and compound interest methods. Question 2: Company XYZ, is expanding their food business, they issue bonds to raise money. As an investor you obtain a bond of RM1,000 that will pay you 5% interest annually and it will mature in 5 years. Calculate the amount of interest you will earn on that bond? (Simple & Compound) Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Review Interest Rates What Does Time Value of Money Mean? Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent). Uses of TIME Value of Money I = P  I  t – Bond valuation Compound Interest – Stock valuation Interest paid (earned) on any previous interest earned, – Accept/reject decisions for project management as well as on the principal borrowed (lent). – Financial analysis of firms A = p(1 + r ) t – And many others! Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations ◼ There are two basic types of TVM calculations ◼ The basic types that will be covered are: ◼ Future value of a lump sum ◼ Future Value of Cash Flow Stream ◼ Future Value of annuity ◼ Present value of a lump sum ◼ Present value of cash flow stream ◼ Present value of annuity Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations Basic Rules Example ◼ The following are simple rules that you should How much money will you have in 5 years if always use no matter what type of TVM you invest RM100 today at a 10% rate of problem you are trying to solve: return? Draw a timeline 1. Stop and think: Make sure you understand what the problem is asking. i = 10% RM100 ? 2. Draw a representative timeline and label the cash flows and time periods appropriately. 3. Write out the complete formula using 0 1 2 3 4 5 symbols first and then substitute the actual Write out the formula using symbols: numbers to solve. 4. Check your answers using a calculator. FVt = PV * (1+r)^n Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  One Equation and Four Variables Any time value problem involving lump sums—i.e., a single outflow and a single inflow—requires the use of a single equation consisting of four variables i.e., PV, FV, r, n If three out of four variables are given, we can solve the unknown one. FV = PV × (1 + r)n → solving for future value PV = FV × [1 ÷ (1 + r)n] → solving for present value r = [FV ÷ PV]1÷n − 1 → solving for unknown rate n = [ln(FV ÷ PV) ÷ ln(1 + r)] → solving for # of periods Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Methods of Solving Future Value Problems Method 1: The formula method – Time-consuming, tedious Method 2: The financial calculator approach – Quick and easy Method 3: The spreadsheet method – Most versatile Method 4: The use of Time Value tables: – Easy and convenient but most limiting in scope Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations ❖ The Future value of Money ◼ Future value is the value of an asset at a specific date in the future. ◼ It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return. ◼ Actually, the future value does not include corrections for inflation or other factors that affect the true value of money in the future. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations ❖ The Future value of Money FVn = PV(1 + i)n FV = the future value of the investment at the end of n year i = the annual interest rate PV = the present value, in today’s dollars, of a sum of money ◼ This equation is used to determine the value of an investment at some point in the future. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 1. The Future value of Lump Sum Future value determines the amount that a sum of money invested today will grow to in a given period of time You can think of future value as the opposite of present value The process of finding a future value is called “compounding” FV = PV  (1 + i ) n 7000 6000 0% 5000 5% 10% Interest Rates FV of $100 4000 15% 3000 Future Values of RM100 with Compounding 2000 1000 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Number of Years Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Example of FV of a Lump Sum i = 12% Years 0 1 2… 20 Cash flow −$10,000 Blank Blank Blank Future Value = ? This is a simple future value problem. We can find the future value using Equation. Solve Using a Mathematical Formula Future Value Number of Years ( n ) Present  Annual  in Year n =  1+  Value (PV )  Interest Rate ( i )  (FVn ) FV = $10,000(1.12)20 = $10,000(9.6463) = $96,462.93 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations Example of FV of a Lump Sum How much money will you have in 5 years if you invest RM100 today at a 10% rate of return? 1. Draw a timeline 3. Substitute the numbers into the formula: RM100 i = 10% FV = RM100 * (1+.1)5 ? 4. Solve for the future value: 0 1 2 3 4 5 FV = RM161.05 2. Write out the formula using symbols: FVn = PV or CF0 * (1+r)n Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Example of FV of a Lump Sum In 1867, Secretary of State William H. Seward purchased Alaska from Russia for the sum of RM7,200,000, or about two cents per acre. At the time, the deal was called Seward’s Folly, but from our vantage point today, did Seward get a bargain after all? What would it cost today if the land were in exactly the same condition as it was 148 years ago and the prevailing interest rate over this time were 4%? Solution: At first glance, it seems as if we have a present value problem, not a future value problem, but it all depends on where we are standing in reference to time. Phrasing this question another way, we could ask, “What will the value of RM7,200,000 be in 148 years at an annual interest rattimelineed this way, we can more easily view the problem as a future value problem. A time line is particularly helpful in this instance. We can show the 148-year span from T−148 to T0 or from T0 to T148. FV = PV  (1 + r ) n = $7, 200, 000 1.04148 = $7, 200, 000  313.8442 = $2, 389, 278,156 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Example of FV of a Lump Sum Exercise 1 Let’s say that you have seen your dream house, which is currently listed at RM300,000, but unfortunately, you are not in a position to buy it right away and will have to wait at least another 5 years before you will be able to afford it. If house values are appreciating at the average annual rate of inflation of 5%, how much will a similar house cost after 5 years? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Example of FV of a Lump Sum Exercise 2 Let’s say you want to know how much money you will have accumulated in your bank account after 4 years, if you deposit all RM5,000 of your high-school graduation gifts into an account that pays a fixed interest rate of 5% per year. You leave the money untouched for all four of your college years. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Calculating Future Values Using Non-Annual Compounding Periods If you deposit RM50,000 in an account that pays an annual interest rate of 10% compounded monthly, what will your account balance be in 10 years? i = 10% Months 0 1 2… 120 Cash flow −RM50,000 Blank Blank Blank FV of RM50,000 Compounded for 120 months @ 10%/12 This involves solving for future value of RM50,000. Since the interest is compounded monthly, we will use Equation Using a Mathematical Formula FV = PV (1+i/12)m*12 = RM50,000 (1+.10/12)10*12 = RM50,000 (2.7070) = RM135,352.07 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Compound Interest with Shorter Compounding Periods Banks frequently offer savings account that compound interest every day, month, or quarter. More frequent compounding will generate higher interest income and lead to higher future values. m × (Number of Years ( n ))  Annual  Future Value   Present in Year n =  1+ Interest Rate ( i )  Value (PV )  Compounding  (FVn )  Periods per Year ( m )    The Value of RM100 Compounded at Various Non-Annual Periods and Various Rates For 1 Year at i Percent i = 2% 5% 10% 15% Table shows how shorter compounding periods lead to higher future values. For Compounded annually RM102.00 RM105.0 RM110.00 RM115.00 example, if you invested RM100 at 15% for 0 1 year and the investment was compounded daily rather than annually, you Compounded semiannually 102.01 105.06 110.25 115.56 would end up with RM1.18 more (RM116.18–RM115.00). However, if the Compounded quarterly 102.02 105.09 110.38 115.87 period was extended to 10 years, then the difference would grow to RM43.47 Compounded monthly 102.02 105.12 110.47 116.08 (RM448.03–RM404.56). Compounded weekly (52) 102.02 105.12 110.51 116.16 Compounded daily (365) 102.02 105.13 110.52 116.18 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Compound Interest with Shorter Compounding Periods The Value of RM100 Compounded at Various Non-Annual Periods and Various Rates For 10 Years at i Percent i = 2% 5% 10% 15% Compounded annually RM121.90 RM162.8 RM259.37 RM404.56 9 Compounded semiannually 122.02 163.86 265.33 424.79 Compounded quarterly 122.08 164.36 268.51 436.04 Compounded monthly 122.12 164.70 270.70 444.02 Compounded weekly (52) 122.14 164.83 271.57 447.20 Compounded daily (365) 122.14 164.87 271.79 448.03 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 2. The Future value of 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. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 2. The Future value of Cash Flow Stream Example ◼ For example, consider an investment which promises to pay RM100 one year from now, RM300 two years from now, RM500 three years from now and RM1000 four years from now. How much will be the future value of the cash flow streams at the end of year 4, given that the interest rate is 10%,? ◼ The Future Value at the end of year 4 of the investment can be found as follows in the next slide: Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 2. The Future value of Cash Flow Stream 1. Draw a timeline: RM100 RM300 RM500 RM1000 0 1 2 3 4 ? i = 10% ? ? ? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 2. The Future value of Cash Flow Stream 2. Write out the formula using symbols n FV = ∑ [CFt * (1+r)n-t] t=0 OR FV = [CF1*(1+r)n-1]+[CF2*(1+r)n-2]+ [CF3*(1+r)n3] + [CF4*(1+r)n-4] 3. Substitute the appropriate numbers: FV = [RM100*(1+.1)4-1]+[RM300*(1+.1)4-2]+[RM500*(1+.1)4-3] +[RM1000*(1+.1)4-4] Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 2. The Future value of Cash Flow Stream 4. Solve for the Future Value: FV = RM133.10 + RM363.00 + RM 550.00 + RM1000 FV = RM2046.10 5. Check using the calculator: ◼ Make sure to use the appropriate interest rate, time period and present value for each of the four cash flows. To illustrate, for the first cash flow, you should enter PV=100, n=3, i=10, PMT=0, FV=?. Note that you will have to do four separate calculations. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 2. The Future value of Cash Flow Stream Future Value of an Uneven Cash Flow Stream Jim deposits RM3,000 today into an account that pays 10% per year and follows it up with three more deposits at the end of each of the next 3 years. Each subsequent deposit is RM2,000 higher than the previous one. How much money will Jim have accumulated in his account by the end of 3 years? FV = PV × (1 + r)n FV of cash flow at T0 = RM3,000 × (1.10)3 = RM3,000 × 1.331 = RM3,993.00 FV of cash flow at T1 = RM5,000 × (1.10)2 = RM5,000 × 1.210 = RM6,050.00 FV of cash flow at T2 = RM7,000 × (1.10)1 = RM7,000 × 1.100 = RM7,700.00 FV of cash flow at T3 = RM9,000 × (1.10)0 = RM9,000 × 1.000 = RM9,000.00 Total = RM26,743.00 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 2. The Future value of Cash Flow Stream Exercise 1 ◼ Consider an investment which promises to pay RM100 one year from now, RM300 two years from now, RM500 three years from now and RM1000 four years from now. How much will be the future value of the cash flow streams at the end of year 4, given that the interest rate is 10%,? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 2. The Future value of Cash Flow Stream Exercise 2 ◼ Find the Future Value at the end of year 4 of the following cash flow stream given that the interest rate is 10%. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity What is Annuity? An annuity is a cash flow stream in which the cash flows are all equal and occur at regular intervals. To considered as annuity the following conditions must be present – The periodic payment must be equal in amount – The time period between payments should be constant – The interest rate per year remains constant – The interest is compounded at the end of each time period Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity Types of Annuity ◼ There are different types of annuities, among these are 1. An ordinary annuity is one where the payments are made at the end of each period 2. An annuity due is one where the payments are made at the beginning of each period Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Types of Annuity ◼ Ordinary Annuity: a sequence of equal cash flows, occurring at the end of each period. This is known as an ordinary annuity. Example: ◼ If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. ◼ If you borrow money to buy a house or a car, you will re-pay the loan with 0 1 2 3 4 a stream of equal payments. PV FV ▪ Annuity-due: A sequence of periodic cash flows occurring at the beginning of each period. Example: ◼ Monthly Rent payments: due at the beginning of each month. ◼ Car lease payments. 0 1 2 3 4 ◼ Cable & Satellite TV and most internet service bills. PV FV Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Ordinary) Annuities are equal, periodic outflows/inflows, e.g. rent, lease, mortgage, car loan, and retirement annuity payments. An annuity stream can begin at the start of each period (annuity due) as is true of rent and insurance payments or at the end of each period, (ordinary annuity) as in the case of mortgage and loan payments. The formula for calculating the future value of an annuity stream is as follows: (1 + r ) n − 1 FV = PMT  r FVn = FV of annuity at the end of nth period. PMT = annuity payment deposited or received at the end of each period i = interest rate per period n = number of periods for which annuity will last Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Ordinary) ◼ Example: What amount will accumulate if we deposit RM5,000 at the end of each year for the next 5 years? Assume an interest of 6% compounded annually PV = 5,000 i = 6% n=5 You may like to know how much you need to save each period (i.e., PMT) in order to accumulate a certain amount at the end of n years.  (1 + i )n − 1 FVn = PMT    i  Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Ordinary) ◼ Example: What amount will accumulate if we deposit RM5,000 at the end of each year for the next 5 years? Assume an interest of 6% compounded annually PV = 5,000 i = 6% n=5 Year 1 2 3 4 5 Begin 0 5,000.00 10,300.00 15,918.00 21,873.08 Interest 0 300 618 955.08 1,312.38 Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00 End 5,000.00 10,300.00 15,918.00 21,873.08 28,185.46 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Ordinary)  (1 + r )n − 1  FVordianryAn nuity = CF    r   (1 + 0.06)5 − 1  FVoa = 5000   0.06  FVoa = 28185.4648  (1.06)5 − 1  FVoa = 5000   0.06  1.3382255776 − 1  FVoa = 5000   0.06 FVoa = 5000 0.3382255776   0.06  FVoa = 50005.63709296 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Future Value of an Annuity Stream Ordinary Annuity Stream Jill has been faithfully depositing RM2,000 at the end of each year since the past 10 years into an account that pays 8% per year. How much money will she have accumulated in the account? Example: Answer Future Value of Payment One = RM2,000 × 1.089 = RM3,998.01 Future Value of Payment Two = RM2,000 × 1.088 = RM3,701.86 Future Value of Payment Three = RM2,000 × 1.087 = RM3,427.65 Future Value of Payment Four = RM2,000 × 1.086 = RM3,173.75 Future Value of Payment Five = RM2,000 × 1.085 = RM2,938.66 Future Value of Payment Six = RM2,000 × 1.084 = RM2,720.98 Future Value of Payment Seven = RM2,000 × 1.083 = RM2,519.42 Future Value of Payment Eight = RM2,000 × 1.082 = RM2,332.80 Future Value of Payment Nine = RM2,000 × 1.081 = RM2,160.00 Future Value of Payment Ten = RM2,000 × 1.080 = RM2,000.00 Total Value of Account at the end of 10 years RM28,973.13 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Future Value of an Annuity Stream Ordinary Annuity Stream Example: Answer Formula method Using FVIFA table (A-3) (1 + r ) n − 1 Find the FVIFA in the 8% column and the 10 period FV = PMT  row; FVIFA = 14.486 r FV = 2000 × 14.4865 = RM28.973.13 where, PMT = RM2,000; r = 8%; and n = 10 FVIFA → [((1.08)10 − 1) ÷ 0.08] = 14.486562, FV = RM2000 × 14.486562 → RM28,973.13 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Future Value of an Annuity Stream Ordinary Annuity Stream You are saving money to assist your daughter when she goes to college. How much money will you accumulate by the end of year 10 if you deposit RM3,000 each year for the next 10 years in a savings account that earns 5% per year? Determine the answer by using the equation for computing the FV of an ordinary annuity.  (1 + i )n − 1 FVn = PMT    i  Using Equation FV = RM3000 {[ (1+.05)10 − 1] ÷ (.05)} = RM3,000 { [.63] ÷ (.05) } = RM3,000 {12.58} = RM37,740 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Future Value of an Annuity Stream Ordinary Annuity Stream Exercise 1 You would like to have RM50,000 saved 10 years from now to finance your goal of getting an MBA. If you are going to make annual end-of-year payments to an investment account that pays 8%, how big do these annual payments need to be? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Due) ◼ A form of annuity where periodic receipts or payments are made at the beginning of the period and one period of the annuity term remains after the last payment. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Due) FVad = FVoa (1 + i ) ◼ The Future Value of an Annuity Due is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. ◼ Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (1 + i). Where: FVad = FVoa (1 + i ) FVad = Future Value of an Annuity Due FVoa = Future Value of an Ordinary Annuity i = Interest Rate Per Period Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Due) ◼ Example 1: What amount will accumulate if we deposit RM5,000 at the beginning of each year for the next 5 years? Assume an interest of 6% compounded annually. ◼ PV = 5,000, i =.06, n = 5 Year 1 2 3 4 5 Begin 0 5,300.00 10,918.00 16,873.08 23,185.46 Interest 0 300 618 955.08 1,312.38 Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00 ◼ End FVoa =10,300.00 5,000.00 28,185.46 (1.06) = 29,876.59 15,918.00 21,873.08 28,185.46 Begin 5,000.00 10,300.00 15,918.00 21,873.08 28,185.46 Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00 Interest 300 618 955.08 1,312.38 1,691.13 End 5,300.00 10,918.00 16,873.08 23,185.46 29,876.59 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Due)  (1 + 0.06)5 − 1  = 5000  FVad  0.06  (1.06 )  (1.06)5 − 1  (1.06 ) FVad = 5000   0.06  1.3382255776 − 1 FVad = 5000  (1.06 )  FVad = 29,876.59  0.06 FVad = 5000 0.3382255776   (1.06 )  0.06 FVoa = 50005.63709296 (1.06 ) FVad = (28185.4648)(1.06) Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Due+ Ordinary) ◼ Example 1: What amount will accumulate if we deposit RM1,000 at the End of each year for the next 5 years? Assume an interest of 5% compounded annually. ◼ Example 2: What amount will accumulate if we deposit $1,000 at the beginning of each year for the next 5 years? Assume an interest of 5% compounded annually. ◼ PV = 1,000 i =.05 n=5 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Future value of Annuity (Due) ◼ Example 2: What amount will accumulate if we deposit RM1,000 at the beginning of each year for the next 5 years? Assume an interest of 5% compounded annually. ◼ PV = 1,000 i =.05 n=5 = $1000*5.53*1.05 = $5801.91 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Future Value of an Annuity Stream Annuities Due Exercise 1 You plan to save RM1,000 at the beginning of each year for the next 5 years in an account that earns 5% annual interest, compounded annually. What will be the future value of your savings at the end of 5 years?  (1 + i )n − 1 FVn (annuity due) = PMT   (1 + i )  i  Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Future Value of an Annuity Stream Annuities Due Exercise 2 A property owner receives RM2,500 at the beginning of each month as rental income. They invest the amount in a fund that earns 6% annual interest, compounded monthly for 3 years. What will be the future value of the investment at the end of 3 years?  (1 + i )n − 1 FVn (annuity due) = PMT   (1 + i )  i  Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations Future Value of an Annuity Stream Annuities Due Exercise 3 Let’s say that you are saving up for retirement and decide to deposit RM3,000 each year for the next 20 years into an account which pays a rate of interest of 8% per year. By how much will your accumulated you make each of the 20 deposits at the beginning of the year, starting right away, rather than at the end of each of the next 20 years?  (1 + i )n − 1 FVn (annuity due) = PMT   (1 + i )  i  Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Annuities Due) Let’s say that you are saving up for retirement and decide to deposit RM3,000 each year for the next 20 years into an account which pays a rate of interest of 8% per year. By how much will your accumulated you make each of the 20 deposits at the beginning of the year, starting right away, rather than at the end of each of the next 20 years? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  Review Future value of a lump sum​ Future Value of Cash Flow Stream​ Future Value of annuity​ Present value of a lump sum​ Present value of cash flow stream​ Present value of annuity Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations ❖ The Present value of Money Current value of future payments. What is the value in today's dollars of the sum of money to be received in future? It is a process of discounting the future value of money to obtain its value at zero time period (at present). FV n PV = (1 + i ) n FVn = PV  (1 + i ) n Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations ❖ The Present value of Money To determine present values, we need to know: – The amount of money to be received in the future – The interest rate to be earned on the deposit – The number of years the money will be invested ❖ Discounting and Compounding ❖The mechanism for factoring in the present value of money element is the discount rate. ❑ Compounding converts present cash flows into future cash flows. ❑ The process of finding the equivalent value today of a future cash flow is known as discounting. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations ❖ The Present value of Money Using the PV Formula FV n ◼ Present value of a lump sum PV = ◼ Present value of cash flow stream (1 + i ) n ◼ Present value of annuity Using the Present Value Table – Present value interest factor (PVIF): a factor multiplied by a future value to determine the present value of that amount (PV = FV(PVIF) Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) Assume that you need RM1000 in 2 years. Let's examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 2 7% PVo RM1000 PV= FV/ (1+i)^n PV= 1000/ (1+7%)^2 PV= 1000/ (1+0.07)^2 PV= 1000/ (1.07)^2 PV= 1000/ (1.144) PV = 874 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) Example You would like to accumulate 50,000 in five years by making a single investment today. You believe you can achieve a return from your investment of 8% annually. What is the amount that you need to invest today to achieve your goal?  CFt  PV = 50,000(PVIF)5,8% PV =   t   (1 + r )  =50,000 x 0.681 = 34050 50,000 PV = 5 = 34029.16 (1.08) Or We can use table Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) XYZ Company expects to receive a cash flow of $2 million in five years. If the competitive market interest rate is fixed at 4% per year, how much can they borrow today in order to be able to repay the loan in its entirety with that cash flow? Solution: First set up your timeline. The cash flows for the loan are represented by the following timeline: Thus, XYZ Company will be able to repay the loan with its expected $2 million cash flow in five years. To determine the value today, we compute the present value using Eq. 3.2 and our interest rate of 4%. Execute: Evaluate: The loan is much less than the $2 million the company will pay back because $2,000,000 of the time value of money PV= =$1,643,854.21 (1.04 ) 5 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) Your retirement goal is RM2,000,000. The bank is offering you a certificate of deposit that is good for 40 years at 6.0%. What initial deposit do you need to make today to reach your RM2,000,000 goal at the end of 40 years? Solution: The following time line illustrates the problem. We designate today as T0 and our future date 40 years later as T40. Using the equation 1 PV = $2, 000, 000  = $2, 000, 000  0.0972 = $194, 444.38 1.0640 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) Let’s say you just won a jackpot of RM50,000 at the casino and would like to save a portion of it so as to have RM40,000 to put down on a house after 5 years. Your bank pays a 6% rate of interest. How much money will you have to set aside from the jackpot winnings? Example 5: Answer FV = amount needed = RM40,000 Calculator method: N = 5 years; Interest rate = 6%; FV 40,000; N = 5; I ÷ Y = 6%; PMT = 0; CPT PV = −RM29,890.33 PV = FV × 1 ÷ (1 + r)n Spreadsheet method: Rate = 0.06; Nper = 5; Pmt = 0; Fv = $40,000; Type = 0; PV = RM40,000 × 1 ÷ (1.06)5 Pv = −RM29,890.33 Time value table method: PV = RM40,000 × 0.747258 PV = FV(PVIF, 6%, 5) = 40,000 × (0.7473) = RM29,892 where (PVIF, 6%,5) = Present value interest factor listed under the 6% column and in the 5-year row of the Present Value of RM1 table PV = RM29,890.33 → Amount needed to set aside today = 0.7473 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) Exercise 1 You are considering investing in a savings bond that will pay RM15,000 in 10 years. If the competitive market interest rate is fixed at 6% per year, what is the bond worth today? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) Exercise 2 Let’s say you just won a jackpot of RM50,000 at the casino and would like to save a portion of it so as to have RM40,000 to put down on a house after 5 years. Your bank pays a 6% rate of interest. How much money will you have to set aside from the jackpot winnings? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 1. The Present value of Lump Sum (Single Cashflow) Exercise 3 You are considering investing in a savings bond that will pay $15,000 in 10 years. If the competitive market interest rate is fixed at 6% per year, what is the bond worth today? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 2. The Present value Stream of Cash flow) Valuing a lump sum (single) amount is easy to evaluate because there is one cash flow. What do we need to do if there are multiple cash flow? – Equal Cash Flows: Annuity or Perpetuity – Unequal/Uneven Cash Flows Uneven cash flows exist when there are different cash flow streams each year Treat each cash flow as a Single Sum problem and add the PV amounts together. FV1 FV2 FV N PV = + +    + (1 + i )1 (1 + i )2 (1 + i )N N FVt PV =  t =1 (1 + i ) t Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 2. The Present value Stream of Cash flow) What is the present value of the preceding cash flow stream using a 12% discount rate? ? 1000 2000 3000 0 1 2 3 n =3 100 200 300  n FVt PV =   + + (1 + 0.12) (1 + 0.12)3  PV =  t =0 (1 + r ) t = 0  (1 + 0.12) n 1 2 PV =  893 + 1594 + 2135 PV = 4622 Yr 1 RM1,000 / (1.12)1 = RM893 Yr 2 RM2,000 / (1.12)2 = 1,594 Yr 3 RM3,000 / (1.12)3 = 2,135 RM4,622 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 2. The Present value Stream of Cash flow) Assume that an investment offers the following cash flows. The required return is 7%, what is the maximum price that you would pay for this investment? 100 200 300 0 1 2 3 4 5 100 200 300 PV = + + = 513.04 (1.07) 1 (1.07) 2 (1.07) 3 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  Time Value of Money (TVM) Calculations 2. The Present value Stream of Cash flow Exercise 1 A company expects to receive the following annual cash flows from a project: Year 1: RM5,000 Year 2: RM7,000 Year 3: RM10,000 The discount rate is 8% per year. What is the Present Value (PV) of the cash flow stream? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 2. The Present value Stream of Cash flow Exercise 2 Amirah has just purchased some equipment for her beauty salon. She plans to pay the following amounts at the end of the next 5 years: RM8,250, RM8,500, RM8,750, RM9,000, and RM10,500. If she uses a discount rate of 10%, what is the cost of the equipment that she purchased today? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Ordinary Due) The present value of an annuity is the value now of a series of equal amounts to be received (or paid out) for some specified number of periods in the future. It is computed by discounting each of the equal periodic amounts. An annuity is a series of nominally equal payments equally spaced in time The timeline shows an example of a 4-year, RM250 annuity with interest rate of 8%. 250 250 250 250 0 1 2 3 4 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Ordinary Due) The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present and adding up the present values. We can use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components: N  (1 + i) Pmt t Pmt 1 Pmt 2 Pmt N PVA = = + +  + t =1 t (1 + i) (1 + i) 1 2 (1 + i) N Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Ordinary Due) Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years] Thus, there is no need to take the present value of each cash flow separately The closed-form of the PVA equation is: 1 − 1   (1 + r ) N  PV = FV    r      Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Ordinary Due) We can use this equation to find the present value of our annuity example as follows:  1  Using the example, and assuming a discount rate of 1 −  (1 + 0.1)  5 10% per year, we find that the present value is: PV = 100   0.1  100 100 100 100 100   PVA = + + + + = 379.08  1  (110. ) 1 (110. ) 2 (110. ) 3 (110. ) 4 (110. ) 5 1 − (1.1) 5  PV = 100   0.1  62.09   68.30 75.13  0.3791 PV = 100  0.1  82.64 90.91 379.08 100 100 100 100 100 PV = 1003.791 PV = 379.1 0 1 2 3 4 5 This equation works for all regular (ordinary) annuities, regardless of the number of payments Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Ordinary Due) Exercise 1 John wants to make sure that he has saved up enough money prior to the year in which his daughter begins college. Based on current estimates, he figures that college expenses will amount to RM40,000 per year for 4 years (ignoring any inflation or tuition increases during the 4 years of college). How much money will John need to have accumulated in an account that earns 7% per year, just prior to the year that his daughter starts college?   1  1 −  n       (1 + r )   PV = PMT  r Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Annuities Due) The annuities that we begin their payments at the end of period 1 are referred as regular annuities (ordinary annuities) An annuity due is the same as a regular annuity, except that its cash flows occur at the beginning of the period rather than at the end 5-period Annuity Due 100 100 100 100 100 5-period Regular Annuity 100 100 100 100 100 0 1 2 3 4 5 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Annuities Due) The formula for the present value of an annuity due, sometimes referred to as an immediate annuity, is used to calculate a series of periodic payments, or cash flows, that start immediately We can find the present value of an annuity due in the same way as we did for a regular annuity, with one exception Note from the timeline that, if we ignore the first cash flow, the annuity due looks just like a four-period regular annuity Therefore, we can value an annuity due with: 1 − 1   (1 + i ) n−1  PV = CF   + CF  i    Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Annuities Due) Using the example, and assuming a discount rate of 10% per year, CF RM100 for 5years, we find that the present value is:  1  1 −  (1 + 0.1)5−1  PV AD = 100  + 100 = 416.98  0.1    Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. 5. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Annuities Deferred) A deferred annuity is the same as any other annuity, except that its payments do not begin until some later period The timeline shows a five-period payment deferred annuity 100 100 100 100 100 0 1 2 3 4 5 6 7 We can find the present value of a deferred annuity in the same way as any other annuity, with an extra step required Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations 3. The Present value Annuity (Annuities Due) Exercise 1 What will be the PV of annuity due if you receive RM500 at the beginning of each of year for 5 years and are able to earn at an interest rate of 6%? What will be the PV if it was ordinary annuity? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Time Value of Money (TVM) Calculations ❖ Perpetuity A perpetuity is an annuity whose payment take place forever, i.e. it is an annuity whose term is infinite The PV of a perpetuity is calculated by dividing the cash payment by Example the interest rate: What is the value of a bond that promises to pay RM15 every year for ever? C Cash Payment PV of a perpetuity = = The interest rate is 10-percent. r Interest rate RM15 RM15 RM15 The interest rate on a perpetuity is calculated by dividing the cash payment by the PV: … 0 1 2 3 C RM15 Interest rate on a perpetuity = = Cash Payment PV of a perpetuity = 0.10 PV Present Value PV of a perpetuity = RM150 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Quick Quiz Question 1: What is the present value of RM500 to be received 10 years from today if our discount rate is 6%? Question 2: What is the present value of an investment that yields RM500 to be received in 5 years and RM1000 to be received in 10 years if the discount rate is 4%? Question 3: What is the present value of mixed stream involving 2 annuities discounted at 5%? N  (1 + i) Pmt t Pmt 1 Pmt 2 Pmt N PVA = = + +  + t =1 t (1 + i) 1 (1 + i) 2 (1 + i) N 1 − 1   (1 + r ) N  PV = FV    r      Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  Chapter Quiz 1. If crude oil trades in a competitive market, would an oil refiner that has a use for the oil value it differently than another investor would? 2. How do we determine whether a decision increases the value of the firm? 3. Is the value today of money to be received in one year higher when interest rates are high or when interest rates are low? 4. What do you need to know to compute a cash flow’s present or future value? How is the future value of a single cash flow computed? How is the present value of a series of cash flows computed. What is a perpetuity? An annuity? Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Copyright This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  Example: Present Value of a Single Future Cash Flow Problem You are considering investing in a savings bond that will pay $15,000 in 10 years. If the competitive market interest rate is fixed at 6% per year, what is the bond worth today? Solution First, set up your timeline. The cash flows for this bond are represented by the following timeline: Thus, the bond is worth $15,000 in 10 years. To determine the value today, we compute the present value using Equation 3.2 and our interest rate of 6%. 15,000 PV= 10 =$8375.92 today 1.06 Evaluate The bond is worth much less today than its final payoff because of the time value of money. Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Example: Present Value of a Single Future Cash Flow Problem: XYZ Company expects to receive a cash flow of $2 million in five years. If the competitive market interest rate is fixed at 4% per year, how much can they borrow today in order to be able to repay the loan in its entirety with that cash flow? Solution: First set up your timeline. The cash flows for the loan are represented by the following timeline: Thus, XYZ Company will be able to repay the loan with its expected $2 million cash flow in five years. To determine the value today, we compute the present value using Eq. 3.2 and our interest rate of 4%. Execute: Evaluate: $2,000,000 The loan is much less than the $2 million the PV= =$1,643,854.21 (1.04 ) company will pay back because of the time 5 value of money Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Example 5.14 Frey Company, a shoe manufacturer, has the opportunity to receive the following mixed stream of cash flows over the next five years. If the firm discounts cash flows at 9%, the following timeline shows that the cash flow stream is worth $1,904.76 today. Timeline for present value of a mixed stream of end-of-year cash flows, discounted at 9% to time 0 Time Cash flow 0 $ 0 1 400 2 800 3 500 4 400 5 300 Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Review of Learning Goals LG 1 – Discuss the role of time value in finance, the use of computational tools, and the basic patterns of cash flow. ▪ Financial managers and investors use time-value-of-money techniques when assessing the value of expected cash flow streams ▪ Alternatives can be assessed by either compounding to find future value or discounting to find present value ▪ Financial managers rely primarily on present-value techniques ▪ Financial calculators and electronic spreadsheets streamline the time-value calculations ▪ Cash flow patterns are of three types: a single amount or lump sum, an annuity, or a mixed stream LG 2 – Understand the concepts of future value and present value, their calculation for single amounts, and the relationship between them. ▪ Future value (FV) relies on compound interest to translate current dollars into future dollars ▪ The initial principal or deposit in one period, along with the interest earned on it, becomes the beginning principal of the following period ▪ The present value (PV) of a future amount is the amount of money today that is equivalent to the given future amount, considering the return that can be earned ▪ Present value is the inverse of future value Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.  Review of Learning Goals (3 of 8) LG 3 – Find the future value and the present value of both an ordinary annuity and an annuity due, and find the present value of a perpetuity. ▪ An annuity is a pattern of equal periodic cash flows ▪ For an ordinary annuity, the cash flows occur at the end of the period ▪ For an annuity due, cash flows occur at the beginning of the period ▪ The future or present value of an ordinary annuity can be found by using algebraic equations, a financial calculator, or a spreadsheet program – Find the future value and the present value of both an ordinary annuity and an annuity due, and find the present value of a perpetuity. ▪ The value of an annuity due is always r% greater than the value of an identical annuity ▪ The present value of a perpetuity—an infinite-lived annuity—equals the annual cash payment divided by the discount rate ▪ The present value of a growing perpetuity equals the initial cash payment divided by the difference between the discount rate and the growth rate Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Review of Learning Goals (5 of 8) LG 4 – Calculate both the future value and the present value of a mixed stream of cash flows. ▪ A mixed stream consists of unequal periodic cash flows that reflect no particular pattern ▪ The future value of a mixed stream is the sum of the future values of the individual cash flows ▪ Similarly, the present value of a mixed stream is the sum of the present values of the individual cash flows LG 5 – Understand the effect that compounding interest more frequently than annually has on future value and on the effective annual rate of interest. ▪ Interest can compound at intervals ranging from annually to daily and even continuously ▪ The more often interest compounds, the larger the future amount that will be accumulated, and the higher the effective, or true, annual rate (EAR) ▪ The annual percentage rate (APR)—a nominal annual rate—is quoted on credit cards and loans ▪ The annual percentage yield (APY)—an effective annual rate—is quoted on savings products Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved. Review of Learning Goals (7 of 8) LG 6 – Describe the procedures involved in (1) determining deposits needed to accumulate a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods. ▪ (1) The periodic deposit to accumulate a given future sum can be found by solving the equation for the future value of an annuity for the annual payment ▪ (2) A loan can be amortized into equal periodic payments by solving the equation for the present value of an annuity for the periodic payment – Describe the procedures involved in (1) determining deposits needed to accumulate a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods. ▪ (3) Interest or growth rates can be estimated by finding the unknown interest rate in the equation for the present value of a single amount or an annuity ▪ (4) The number of periods can be estimated by finding the unknown number of periods in the equation for the present value of a single amount or an annuity Copyright © 2022, 2019, 2015 Pearson Education, Inc. 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