Chapter 4 Introduction to Valuation- The Time Value of Money.pptx

Introduction to Valuation: The Time Value of Money by Leo Vashkor Dewri  Chapter Outline 1. Future Values: Definitions and Formula 2. Present Values 3. PV – Important Relationship 4. Calculate Rates and Number of Periods 5. Calculator Keys 4.2 ...

Introduction to Valuation: The Time Value of Money by Leo Vashkor Dewri  Chapter Outline 1. Future Values: Definitions and Formula 2. Present Values 3. PV – Important Relationship 4. Calculate Rates and Number of Periods 5. Calculator Keys 4.2  What is Time Value of Money (TVM) The time value of money (TVM) is the concept that money you have now is worth more than the identical sum in the future due to its potential earning capacity. This core principle of finance holds that provided money can earn interest, any amount of money is worth more the sooner it is received. ▪ The percentage change in value of a certain amount of money for a certain time period gap is known as time value of money. 2  Cash Flow Timelines Cash flow timeline: An important tool used in time value of money analysis; it is a graphical representation used to show the timing of cash flows (cash outflow: a payment, or disbursement, of cash for expenses, investments, and so forth; cash inflow: a receipt of cash from an investment, an employer, or other sources) o To illustrate the timeline concept, consider the following diagram: ✔ Here BDT. 100 is invested at Time 0, and we want to determine how much this investment will be worth in three years if it earns 6 percent interest each year. 4  Fundamental Concept of TVM future value (FV): The amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate. present value (PV): The value today— that is, current value—of a future cash flow or series of cash flows. 4.5  Future Value versus Present Value Suppose a firm has an opportunity to spend BDT15,000 today on some investment that will produce BDT17,000 spread out over the next five years as follows: Year Cash flow 1 BDT3,000 2 BDT5,000 3 BDT4,000 4 BDT3,000 5 BDT2,000 Future Value versus Present Value (cont.) To make the right investment decision, managers need to compare the cash flows at a single point in time.  Fundamental Concept of TVM Present Value – earlier money on a timeline PV = FVt / ( 1 + r )t Future Value – later money on a timeline FV = PV(1 + r)t Interest rate – “exchange rate” between earlier money and later money – Discount rate – Cost of capital – Opportunity cost of capital – Required return Time value of money: A Taka in hand today is worth more than a Taka promised at some time in the future. 4.8  Simple and Compounded interest rate Simple interest rate: The interest rate that charged only on the principal amount for a specific period is called simple interest rate. Compounded interest rate: Interest earned on interest that is reinvested. In other words, the interest rate that is charged both on principal and interest amount period to period is called compound interest rate. ▪ PV = Present value, or beginning amount ▪ r = Interest rate ▪ INT = Taka of interest you earn during the year = (Beginning amount) * r ▪ FVn = Future value, or value of the account at the end of n periods ▪ 9n = Number of periods  Example of Simple interest rate Suppose you put Tk.5000 in bank for one year at 3% interest rate per year. What is the value of your money in one year? – Interest = 5000*.03 = 150 – Value in one year = principal + interest = 5000+5000*.03 = 5000 + 150 = 5150 or 5000*(1 +.03) = 5150 Suppose you leave the money in for another year. How much will you have two years from now? – FV = 5150 + (5000*.03) = 5150 + 4.10  Example of Compounding interest rate Suppose you put Tk.5000 in bank for one year at 3% interest rate per year. What is the value of your money in one year? – Interest = 5000*.03 = 150 – Value in one year = principal + interest = 5000+5000*.03 = 5000 + 150 = 5150 or 5000*(1 +.03) = 5150 Suppose you leave the money in for another year. How much will you have two years from now? – FV = [5000 *(1.03)] * (1.03) or 4.11  Example (cont..) 1 year: Value = 5000(1+.03) 2 years: Value = [5000(1+.03)](1+.03) 3 years: Value = {[5000(1+.03)](1+.03)} (1+.03) 4 years: Value = {{[5000(1+.03)](1+.03)} (1+.03)} (1+.03) 4.12  Example (cont..) 1 year: Value = 5000(1+.03) = 5000(1+.03)1 = 5150 2 years: Value = [5000(1+.03)](1+.03) = 5000(1+.03)2 = 5304.5 3 years: Value = {[5000(1+.03)](1+.03)} (1+.03) = 5000(1+.03)3 = 5463.61 4 years: Value = {{[5000(1+.03)](1+.03)} (1+.03)} (1+.03) = 5000(1+.03)4 = 5627.544 4.13  Effects of Compounding Compounding: the process of accumulating interest over time to earn more interest. Compound interest: interest earned on both the initial principal and the interest earned from prior to periods. Compound interest (total interest) =Simple interest+Interest on interest Simple interest: interest on principal 4.14 Illustration on Compounding 2018 2019 2020 2021 Tk.5000 Tk.5150 Tk.5304.5 Interest earned (2004-2005)=Tk.150 Tk.150 Interest earned (2004-2006)=150+(150+4.5)=Tk.304.5 Tk.150 Tk.4.5 Tk.150 4.15 Illustration on Compounding (cont..) 2018 2019 2020 2021 Tk.5000 Tk.5150 Tk.5304.5 Tk.5463.6 Interest earned (18-20)=150+(150+4.5)=Tk.304.5 Interest earned(18-21)=150+(150+4.5)+(150+9.1)=Tk.463.6 Compounding effect=Total interest earned - simple interest =463.6 – 150 – 150 – 150 = 463.6 - 3(150) = 13.6 4.16  Future Values – Example Suppose your ancestor deposit Tk.10 at 5.5% interest 200 years ago. How much would the investment be worth today? FV = 10(1.055)200 = 447,189.84 What is the effect of compounding? Total interest = FV- PV = 447, 189.84 -10 = 447, 179.84 Simple interest = 200[(10)(.055)] = 110 Compounding effect = Total interest – simple interest = 447,179.84 -110= 447,069.84 Compounding has added Tk.447,069.84 to the value of the investment. 4.17  Present Value (PV) Present Value (PV): Present value generates the amount of money today that is worth more than the same amount in the future. In other words, present value demonstrates that money obtained in the future is less valuable today. To calculate present value, it is necessary to assume that the funds could receive a certain rate of return over time. ✔Discounting: The present value of an investment is computed by discounting its predicted cash flows to the present day. 18  Present Value (PV) (Cont’d) Opportunity cost rate: The rate of return on the best available alternative investment of equal risk. – Suppose someone offered to sell you an investment that promises to pay BDT119.10 in three years for BDT 100 today. Should you buy this investment if your opportunity cost is 6 percent? Answer: You should definitely buy it because, as we discovered in the previous section, it would cost you exactly BDT 100 to produce the BDT119.10 in three years if you earn a 6 percent 19 return. On the other hand, if the price is greater  Present Value (PV) (Cont’d) Time Line Solution: Equation (Numerical) Solution: PV = FVn / ( 1 + r )n PV = FV3 / ( 1 + r )3 = BDT119.10 / ( 1+0.06 )3 = BDT100.00 20  Present Value (PV) (Cont’d) ▪ Example # 2: page # 136 – Assume you have the opportunity to purchase an investment that promises to pay you BDT 3,041.63 in five years and your opportunity cost is 4 percent. How much should you be willing to pay for this investment today? How would your answer change if your opportunity cost is 6 percent? (Answers: BDT2,500; BDT 2,272.88) – 21 BDT 3041.63 / ( 1+0.04 ) 5 = 2,500.00  SOLVING FOR INTEREST RATES (r) AND TIME (n) Equations include four variables: PV, FV, r, and n. If we know the values of any three of these variables, we can solve for the value of the fourth. To this point, we have known the number of years, n, and the interest rate, r, plus either the PV or the FV. In many situations, however, we need to solve for either r or n. ▪Suppose you just called the East Key State Bank and found that the balance in your savings account is TK.1200. If you deposited TK.1000 5 years ago, what rate of return have you earned on the savings account? (Answer: 4.0%) 4.22 A = Mature amount (FV) P = Principle amount (PV) r=? n = number of events t = years 4.23 Solving for n 4.24  Solving for n Rule of 72 ▪ The rule of 72 is a means of estimating the number of years it takes for a certain variable to double. This rule is commonly used with an annual compound interest rate to quickly determine how long it would take to double your money. 72 / r = n ▪ Problem 4-4 (Brigham: Page-167): To the closest year, how long will it take a BDT200 investment to double if it earns 7 percent interest? How long will it take if the investment earns 7 percent? 25 = 200/(1.07)^10.28 = 400  Annuity Payments ▪ Annuity: A series of payments of an equal amount at fixed, equal intervals for a specified number of periods. ▪ Ordinary (deferred) annuity: An annuity with payments that occur at the end of each period. ▪ Annuity due: An annuity with payments that occur at the beginning of each period. Note: If nothing is mentioned in a math then ALWAYS consider Ordinary 26 (deferred) annuity. 4.27  Cash Flow Timeline Ordinary Annuity: Annuity Due: 28  Future Value of an Ordinary Annuity TIME LINE SOLUTION: One way to determine the future value of an ordinary annuity, which we designate FVAn, is to compute the future value of each annuity payment using Equation and then sum the results. Using this approach, the computation of the future value of Alice’s deposits (annuity) is as follows: 4.29 4.30 EQUATION (NUMERICAL) SOLUTION: 4.31 Suppose Xavier would like to save BDT 1,000,000 for 10 years from now to pay towards his down payment on a new house. If he is going to make equal annual end-of each year to an investment account that pays 7 percent. In this example, the series of payments is a regular annuity in which the payments are made at the end of each period. The expected future value of this payment stream using the above formula is as follows: Here we know, - PVn = BDT 100,000; - n = 10 years; - i = 7% - we need to determine FV 4.32 - FVAn= PMT {[ (1+.07)10 -1 ] ÷ (0.07)} = PMT {[0.96715] ÷ (0.07)} = PMT {13.8164} - BDT 1,381,644.80 = BDT 100,000 x 13.8164 4.33  Future Value of an Annuity Due, FVA(DUE) Suppose, Alvin intends to make his deposits at the beginning of each year, his series of deposits represents an annuity due. To determine the future value of an annuity due, we must adjust our computation to recognize that each annuity payment earns interest for one additional year (period). 4.34 Suppose Xavier would like to save BDT 1,000,000 for 10 years from now to pay towards his down payment on a new house. If he is going to make equal annual end-of each year to an investment account that pays 7 percent. In this example, the series of payments is a regular annuity in which the payments are made at the end of each period. The expected future value of this payment stream using the above formula is as follows: 4.35 - FVA(Due)n= PMT {[ (1+.07)10 -1 ] ÷ (0.07) * (1+0.07)} = PMT {[0.96715] ÷ (0.07) * (1+0.07)} = PMT {13.8164 *(1+0.07) } = PMT {14.7836} 4.36  PRESENT VALUE OF AN ANNUITY, PVA The term “present value of annuity” refers to the series of equal future payments that are discounted to the present day. However, the payment can be received either at the beginning or at the end of each period and accordingly there are two different formulations. In case the cash flow is to be received at the beginning, then it is known as the present value of an annuity due and the formula can be derived based on the periodic payment, interest rate, number of years and frequency of occurrence in a year. 4.37 Present value of an annuity, PVA Present Value of an Ordinary Annuity: −𝑛 1− ( 1+𝑟 ) ^ 𝑃𝑉𝐴= 𝑃𝑀𝑇 [ 𝑟 ] −3 1 − ( 1+.05 ) ^ 𝑃𝑉𝐴=100 [.05 ] 2.723248029 4.38 4.39  Present value of an annuity, PVA Present Value of an Annuity Due, PVA(DUE) −𝑛 1− ( 1+𝑟 ) ^ 𝑃𝑉𝐴=𝑃𝑀𝑇 [ 𝑟 ] ∗(1 +𝑟 ) * (1+.05) 2.859410431 40 4.41  PVA - Example Example (Brigham: Page-149): Suppose you are considering an investment that pays BDT5,000 per year for 10 years and your opportunity cost is 7 percent interest. If you don’t receive the first BDT5,000 payment until one year from today, what is the most you should be willing to pay for the investment? How much would you be willing to pay if you receive the first BDT5,000 payment today? (Answers: BDT35,118; BDT37,576) 42  Perpetuities An annuity is a stream of cash flows. A perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate. KEY TAKEAWAYS A perpetuity, in finance, refers to a security that pays a never-ending cash stream. The present value of a perpetuity is determined using a formula that divides cash flows by some discount rate. An example of a perpetuity is the British consol, which were discontinued in 2015. 43 Consol A perpetual bond issued by the British government to consolidate past debts; in general, any perpetual bond. Example: Suppose each consol promised to pay $100 per year in perpetuity. (Actually, interest was stated in pounds.) What would each bond be worth if the opportunity cost rate, or discount rate, was 5 percent? The answer is: 4.44  Perpetuity Example with growth measure For example, if a company is projected to make $100,000 for 10 years, and the company’s cost of capital is 8%, with a long-term growth rate of 3%, the value of the perpetuity is as follows: This means that $100,000 paid into a perpetuity, assuming a 3% rate of growth with an 8% cost of capital, is worth $2.06 million in 10 years. Now, a person must find the value of that $2.06 million in future. 4.45  Uneven Cash Flow Streams Uneven cash flow stream -- A series of cash flows in which the amount varies (unequal / nonconstant) from one period to the next. – Payment (PMT) -- This term designates constant cash flows— that is, the amount of an annuity payment. – Cash flow (CF) -- This term designates cash flows in general, including uneven 46 cash flows. Present Value of an Uneven Cash Flow Stream PV – Uneven CF: 47  Future Value of an Uneven Cash Flow Stream FV – Uneven CF: 4.48  Present and Future Value of an Uneven Cash Flow Stream (Cont’d) Terminal value -- The future value of a cash flow stream. Example (Brigham: Page-155): – Consider the following uneven cash flow stream: BDT1,000 at the end of Year 1, BDT5,000 at the end of Year 2, BDT800 at the end of Year 3, and BDT2,000 at the end of Year 4. If your opportunity cost is 10 percent, what is the most you should pay for an investment with these payoffs? If you were to invest each 49 cash flow when you receive it, how much would your investment be worth at the end of Semiannual and Other Compounding Periods Annual compounding -- The process of determining the future (or present) value of a cash flow or series of cash flows when interest is paid once per year. Semiannual compounding -- The process of determining the future (or present) value of a cash flow or series of cash flows when interest is paid twice per year. 50  Comparisons of Different Interest Rates (Cont’d) 51  Amortized Loans (Cont’d) Amortized loan -- A loan that requires equal payments over its life; the payments include both interest and repayment of the debt/loan. Amortization schedule -- A schedule showing precisely how a loan will be repaid. It gives the required payment on each payment date and a breakdown of the payment, showing how much is interest and how much is repayment of principal. 52  Amortized Loans Suppose a firm borrows BDT15,000, and the loan is to be repaid in three equal payments at the end of each of the next three years. The lender will charge 8 percent interest on the loan balance that is outstanding at the beginning of each year. Step 1: Find out Payment (PMT) by using PVA (ordinary) formula. Here PMT = BDT 5,066.81 Step 2: Prepare loan Amortization schedule 53