The Time Value of Money Which would you prefer -- $10,000 today or $10,000 in 5 years? Obviously, $10,000 today. Time value of money -- Money to be paid out or received in the future is not equivalent to money paid out or received today You already recognize that there is TIME VALUE TO MONEY!! What is Time Value? • We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return • In other words, “a dollar received today is worth more than a dollar to be received tomorrow” • That is because today’s dollar can be invested so that we have more than one dollar tomorrow Intuition Behind Present Value • There are three reasons why a dollar tomorrow is worth less than a dollar today – Individuals prefer present consumption to future consumption. To induce people to give up present consumption you have to offer them more in the future. – When there is monetary inflation, the value of currency decreases over time. The greater the inflation, the greater the difference in value between a dollar today and a dollar tomorrow. – If there is any uncertainty (risk) associated with the cash flow in the future, the less that cash flow will be valued. • Other things remaining equal, the value of cash flows in future time periods will decrease as – the preference for current consumption increases. – expected inflation increases. – the uncertainty in the cash flow increases. The Terminology of Time Value • Present Value - An amount of money today, or the current value of a future cash flow • Future Value - An amount of money at some future time period • Period - A length of time (often a year, but can be a month, week, day, hour, etc.) • Interest Rate - The compensation paid to a lender (or saver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate) Abbreviations • PV - Present value • FV - Future value • Pmt - Per period payment amount • N - Either the total number of cash flows or the number of a specific period • i - The interest rate per period Importance • Shareholders’ wealth maximization requires to address timing of cash flow associated with any investment alternatives. So, it is the basic foundation for the goal. • To compare the initial cash out flow (investment) and the future cash flow generated from it in long run. Such cash flow occurs at several point of time in future so, to make it comparable it must be discounted to present value. • This concept also facilitates investors of securities to evaluate the price of securities before investing on them. • Also used to evaluate the cost benefit of credit. • Helps to compare cost of different sources of financing and help in financing decisions. • To calculate effective rate on different cash flow compounded for different period. Cash flow time line Graphical presentation of cash flow at different point of time. Each tick represents one time period 0 period means today i=10% PV 0 1 2 3 4 5 FV 6 -100 20 25 30 40 25 30 Today Calculating the Future Value • Suppose that you have an extra $100 today that you wish to invest for one year. If you can earn 10% per year on your investment, how much will you have in one year? -100 ? 0 1 2 3 4 5 FV1 1001 0.10 110 Calculating the Future Value (cont.) • Suppose that at the end of year 1 you decide to extend the investment for a second year. How much will you have accumulated at the end of year 2? -110 ? 0 1 2 3 4 5 FV2 1001 0.101 010 121 . or FV2 1001 0.10 121 2 Generalizing the Future Value • Recognizing the pattern that is developing, we can generalize the future value calculations as follows: • FV=PV(FVIF,i%,n) FVN PV1 i N If you extended the investment for a third year, you would have: FV3 1001 010 13310 3 . . Compound Interest • Note from the example that the future value is increasing at an increasing rate • In other words, the amount of interest earned each year is increasing – Year 1: $10 – Year 2: $11 – Year 3: $12.10 • The reason for the increase is that each year you are earning interest on the interest that was earned in previous years in addition to the interest on the original principle amount Calculating the Present Value • So far, we have seen how to calculate the future value of an investment • But we can turn this around to find the amount that needs to be invested to achieve some desired future value: PV=FV(PVIF,i%,n) FVN PV 1 i N Present Value: An Example • Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about $100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal? 100,000 PV $36,769.79 108 . 13 • Find the future value of an initial Rs. 500 compounded for 1 year at 10%. • Find the future value of an initial Rs. 500 compounded for 5 year at 10%. • Find the present value of Rs. 500 due in 1 year at 10%. • Find the present value of Rs. 500 due in 5 years at 10%. • Repeat all above if compounding is for each six months. • If you deposit Rs. 20000 at XYZ bank’s saving account which provide 10%, how long will it take to be the deposit double? • If you borrow Rs. 700 today and promise to pay back Rs. 749 at the end of year what is the interest rate you are paying? Annuities • An annuity is a series of nominally equal payments occurred at equal interval of time for a given number of periods. • There are two types of annuities: a) ordinary annuity : Series of equal payment at the end of each period. b) Annuity due: Series of equal payment at the beginning of each period. • Annuities are very common: – Rent – Mortgage payments – Car payment – Pension income • The timeline shows an example of a 5-year, $100 annuity 100 100 100 100 100 0 1 2 3 4 5 The Principle of Value Additivity • How do we find the value (PV or FV) of an annuity? • First, you must understand the principle of value additivity: – The value of any stream of cash flows is equal to the sum of the values of the components • In other words, if we can move the cash flows to the same time period we can simply add them all together to get the total value Present Value of an Ordinary Annuity • 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 1 i 2 1 i N Present Value of an Ordinary Annuity (cont.) • Using the example, and assuming a discount rate of 10% per year, we find that the present value is: 100 100 100 100 100 PVA 379.08 110 . 1 110 . 2 110 . 3 110 . 4 110 . 5 62.09 68.30 75.13 82.64 90.91 379.08 100 100 100 100 100 0 1 2 3 4 5 Present Value of an Ordinary Annuity (cont.) • Actually, there is no need to take the present value of each cash flow separately • We can use a closed-form of the PVA equation instead: • PVA=Pmt(PVIFA,i%,n) 1 1 N 1 i N 1 i Pmt t PVA t Pmt t 1 i Present Value of an Ordinary Annuity (cont.) • We can use this equation to find the present value of our example annuity as follows: 1 1 PVA Pmt 110 379.08 . 5 . 010 This equation works for all regular annuities, regardless of the number of payments Future Value of an Ordinary Annuity • We can also use the principle of value additivity to find the future value of an annuity, by simply summing the future values of each of the components: N Pmt t 1 i Pmt 1 1 i Pmt 2 1 i Nt N 1 N 2 FVA Pmt N t 1 The Future Value of an Ordinary Annuity (cont.) • Using the example, and assuming a discount rate of 10% per year, we find that the future value is: FVA 100110 100110 100110 100110 100 610.51 4 3 2 1 . . . . } 146.41 133.10 121.00 = 610.51 110.00 at year 5 100 100 100 100 100 0 1 2 3 4 5 The Future Value of an Annuity (cont.) • Just as we did for the PVA equation, we could instead use a closed-form of the FVA equation: • FVA=Pmt(FVIFA,i%,n) N 1 i N 1 Pmt 1 i Nt FVA t Pmt t 1 i This equation works for all regular annuities, regardless of the number of payments The Future Value of an Annuity (cont.) • We can use this equation to find the future value of the example annuity: 1105 1 . FVA 100 610.51 010 . Annuities Due • Thus far, the annuities that we have looked at begin their payments at the end of period 1; these are referred to as regular annuities • A 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 Present Value of an Annuity Due • 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: • PVAD=Pmt(PVIFA,i%,n-1)+Pmt or • PVAD=Pmt(PVIFA,i%,n)(1+i) 1 1 N 1 Pmt 1 i Pmt PVAD i Present Value of an Annuity Due (cont.) • Therefore, the present value of our example annuity due is: 1 1 51 100 110 100 416.98 . PVAD 010. Note that this is higher than the PV of the, otherwise equivalent, regular annuity Future Value of an Annuity Due • To calculate the FV of an annuity due, we can treat it as regular annuity, and then take it one more period forward: • FVAD=Pmt(FVIFA,i%,n)(1+i) 1 i N 1 FVAD Pmt 1 i i Pmt Pmt Pmt Pmt Pmt 0 1 2 3 4 5 Future Value of an Annuity Due (cont.) • The future value of our example annuity is: 1105 1 . FVAD 100 110 67156 . . 010 . Note that this is higher than the future value of the, otherwise equivalent, regular annuity Deferred Annuities • 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 deferred annuity 100 100 100 100 100 0 1 2 3 4 5 6 7 PV of a Deferred Annuity • We can find the present value of a deferred annuity in the same way as any other annuity, with an extra step required • Before we can do this however, there is an important rule to understand: When using the PVA equation, the resulting PV is always one period before the first payment occurs PV of a Deferred Annuity (cont.) • To find the PV of a deferred annuity, we first find use the PVA equation, and then discount that result back to period 0 • Here we are using a 10% discount rate PV2 = 379.08 PV0 = 313.29 0 0 100 100 100 100 100 0 1 2 3 4 5 6 7 PV of a Deferred Annuity (cont.) 1 1 PV2 100 110 379.08 . 5 Step 1: 010 . 379.08 Step 2: PV0 313.29 110 . 2 FV of a Deferred Annuity • The future value of a deferred annuity is calculated in exactly the same way as any other annuity • There are no extra steps at all Uneven Cash Flows • Very often an investment offers a stream of cash flows which are not either a lump sum or an annuity • We can find the present or future value of such a stream by using the principle of value additivity Uneven Cash Flows: An Example (1) • Assume that an investment offers the following cash flows. If your 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 513.04 107 . 1 107 . 2 107 . 3 Uneven Cash Flows: An Example (2) • Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year? 300 500 700 0 1 2 3 4 5 FV 300105 500105 700 1,555.75 2 1 . . Class Discussion • Here are the three contracts offered by different companies for an Engineer who has to select one. The cash flow (Rs. In million) from each contract is as follows: Year CF Contract 1 CF Contract 2 CF Contract 3 1 3 2 7 2 3 3 1 3 3 4 1 4 3 5 1 • If discounting interest rate is 10%, which contract should the engineer accept? Class Discussion • Assume that it is now Jan-1, 2010 and you need Rs. 1000 on Jan-1, 2014. Your bank account interest rate is 8% annual. If you want to make equal payment on each Jan-1, 2011 through 2014 to accumulate the Rs. 1000, how large must each deposit be? If your father offer either to make the payments calculated above or to give you a lump sum of Rs. 750 on Jan-1, 2011, which would you choose? • If you won a lottery which will pay you Rs. 1.75 million per year over next 20 years and the first installment is received immediately, find the following: – If interest rate is 8% what is the present value of the lottery? – If interest rate is 8% what is the future value of the lottery? – How would your answer change if the payments were received at the end of each year? Non-annual Compounding • So far we have assumed that the time period is equal to a year • However, there is no reason that a time period can’t be any other length of time • We could assume that interest is earned semi-annually, quarterly, monthly, daily, or any other length of time • The only change that must be made is to make sure that the rate of interest is adjusted to the period length Non-annual Compounding (cont.) • Suppose that you have $1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds? Bank Interest Rate Compounding First National 10% Annual Second National 10% Monthly Third National 10% Daily Non-annual Compounding (cont.) • To solve this problem, you need to determine which bank will pay you the most interest • In other words, at which bank will you have the highest future value? • To find out, let’s change our basic FV equation slightly: Nm i FV PV 1 m In this version of the equation ‘m’ is the number of compounding periods per year Non-annual Compounding (cont.) • We can find the FV for each bank as follows: FV 1,000110 1100 1 First National Bank: . , 12 010 . Second National Bank: FV 1,000 1 1104.71 , 12 365 010 . Third National Bank: FV 1,000 1 110516 , . 365 Obviously, you should choose the Third National Bank Effective Annual Rate • If a rate is quoted at 16%, compounded semiannually, then the actual rate is 8% per six months. Is 8% per six months the same as 16% per year? • Normally there are two kinds of interest rate: simple interest rate and effective annual rate. • Simple interest rate also called annual percentage rate (APR) is the quoted interest rate which is used to compute the interest paid per period. • Effective annual rate (EAR) is the annual interest rate actually being earned, as opposed to the quoted rate, considering the compounding of interest. • EAR is calculated to compare between two simple interest rate when the compounding period is different. m i EAR 1 1 m Class Discussion • Suppose Bank of Kathmandu pays 7 percent interest, compounded annually on saving deposit. Everest Bank Ltd pays 6.5 percent interest compounded quarterly. Based on the effective annual rate in which bank would you prefer to deposit your money? • You have an option to invest at a development bond issued by Nepal Rastra Bank paying interest 8% but the interest is paid semiannually. Similarly, a bank is offering you a fixed deposit scheme with 7.5% interest rate compounded monthly. If you have Rs. 100000 investable fund where will you invest and what will be the total amount with you after one year? • A bank charges 1% per month on car loans. What is the APR? What is the EAR? Loan Amortization • One of the most important applications of compound interest involves loans that are paid off in installments over time. • Example: automobile loans, student loans, home mortgage loan etc. • If a loan is to be repaid in equal periodic amounts (monthly, quarterly or annually) then it is said to be amortized loan. Loan Amortization (Cont.) • Suppose a loan of Rs. 10000 is to be repaid in 4 equal installments including principal and 10% interest per annum. • The time line for the loan will be: Inst1 Inst 2 Inst 3 Inst 4 10000 0 1 2 3 4 Loan Amortization (Cont.) • All four installment will be equal and they contain principal repayment as well as interest amount. • Installment amount is calculated by using the following formula: Amount of Loan 10000 Installment 3154 .67 (PVIFA, i%, n) 3.1699 • The installment of Rs. 3154 includes both principal and interest. • But the interest and principal amount differs each year. • Loan amortization schedule provides the amount of interest and principal repayment for each period. Loan Amortization (Cont.) Year (A) Loan (B) Installment Interest (D) Principal Loan (C) =Bx10% Repayment Balance (E)=C-D 1 10000 3154.67 1000 2154.67 7845.30 2 7845.30 3154.67 784.53 2370.14 5475.16 3 5475.16 3154.67 547.52 2607.15 2868.01 4 2868.01 3154.67 286.66 2868.01 0 Class Discussion • ABC inc. just borrowed Rs. 25000. Loan is to be repaid in equal installment at the end of each of the next 5 years and the interest rate is 10%. Set up an amortization schedule for the loan. How large must each annual payment be if the loan is for Rs. 50000? How large be the installment be if the loan is Rs. 50000 and equal installment would be paid at the end of next 10 years? Although the loan is for the same amount but the payment is spread over twice as many periods. Why are these payments not half as large as the payment of the loan in the earlier case? Class Discussion • You want to buy a Mazda. It costs $25,000. With a 10% down payment, the bank will loan you the rest at 12% per year (1% per month) for 60 months. What will your monthly payment be? • Consider Bill’s retirement plan Assume he just turned 40, but, recognizing that he has a lot of time to make up for, he decides to invest in some high-risk ventures that may yield 20% annually. (Or he may lose his money completely!) Anyway, assuming that Bill still wishes to accumulate $1 million by age 65, and will begin making equal annual deposits in one year and make the last one at age 65, now how much must each deposit be? Class Discussion • To complete your last year in Business school and then go through Law school, you need Rs. 10000 per year for next 4 years starting from next year i.e. you need to withdraw first 10000 one year from now. Your rich uncle offers to put you through the school and he will deposit in a bank paying 7% interest a sum of money that will be sufficient to provide the four payments of Rs. 10000 each. His deposit will be made today. How large must the deposit be? How much will be in the account immediately after you make the first withdrawal? After the last withdrawal?