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WEEK 5: LECTURE 5 TIME VALUE OF MONEY PRINCIPLES OF FINANCIAL ANALYSIS Lecturer: Chara Charalambous 1 Lecturer: Chara Charalambous 2 LEARNING OUTCOMES 1. Understand what is meant by "the time value of money." 2. Understand the relationship between present and future value. 3. Calculate both the future and present value of: (a) an amount invested today; (b) a stream of equal cash flows (an annuity) 4. Distinguish between an “ordinary annuity” and an “annuity due.” 5. Use interest factor tables and understand how they provide a shortcut to calculations Lecturer: Chara Charalambous 3 Definition of 'Time Value of Money - TVM' • The idea that money available at the present time is worth more than the same amount in the future due to its possible earning capacity. This core principle of finance means that money can earn interest when invested and so any amount of money is worth more the sooner it is received. • A dollar received today worth more than a dollar expected to receive in future because the sooner a dollar received the quicker it can be invested to earn a positive return. The time value of money is the central concept in finance theory. Lecturer: Chara Charalambous 4 • Money received sooner rather than later allows one person to use the funds for investment or consumption purposes. This concept is refer to as the TIME VALUE OF MONEY. Lecturer: Chara Charalambous 5 Time Lines One of the most important tools in time value of money analysis is the time line, which is used to help us picture what is happening in a particular situation. TIME: Time 0 is today, Time 1 is one period from today (e.g. one year), Time 2 is two periods from today Lecturer: Chara Charalambous 6 Time Lines TIME: 5% Cash outflow: -100 Cash inflow: + ? Here the interest rate if I invest my money for each of the four periods is 5%. A cash outflow is a payment of cash for investments and is made at Time 0: because is money given out of my pocket it has a minus sign. At Time 4 I have a cash inflow : a receipt of cash from an investment. The inflow is unknown to me and I have to find it thus I symbolize it with ? And is a positive amount (+) because I will receive money in my pocket. Note that no cash flows occur at Time 1,2 and 3. Lecturer: Chara Charalambous 7 Example • Congratulations!!! You have won a cash prize! You have two payment options: A - Receive $10,000 now OR B - Receive $10,000 in three years. Which option would you choose? Lecturer: Chara Charalambous 8 If you're like most people, you would choose to receive the $10,000 now. After all, three years is a long time to wait. Why would any rational person postpone payment into the future when he or she could have the same amount of money now? For most of us, taking the money in the present is just basic natural. So at the most basic level, the time value of money demonstrates that, all things being equal, it is better to have money now rather than later. Back to our example: by receiving $10,000 today, you are able to increase the future value of your money by investing and gaining interest over a period of time. For Option B, you don't have time on your side, and the payment received in three years would be your future value. To illustrate, we have provided a timeline: Lecturer: Chara Charalambous 9 If you are choosing Option A, your future value will be $10,000 plus any interest acquired over the three years. The future value for Option B, on the other hand, would only be $10,000. So how can you calculate exactly how much more Option A is worth, compared to Option B? Let's take a look. Lecturer: Chara Charalambous 10 Future Value Basics • If you choose Option A and invest the total amount at a simple annual rate of 4.5%, the future value of your investment at the end of the first year is $10,450: • Future value of investment at end of first year: = ($10,000 x 0.045) + $10,000 = $10,450 • Can be written as: $10,000 x [(1 x 0.045) + 1] => => 10,000*(0.045+1)= 10,450 • So the equation for the future value can be written as: FV= PV(i+1) FV = Future Value PV = Present Value – the money invested-the capital i = the interest rate or r = the rate of return on the money invested Lecturer: Chara Charalambous 11 • If the $10,450 left in your investment account at the end of the first year is left untouched and you invested it at 4.5% for another year, how much would you have? To calculate this, you would take the $10,450 and multiply it again by 1.045 (0.045 +1). At the end of two years, you would have $10,920: • Future value of investment at end of second year: = $10,450 x (1+0.045) = $10,920.25 The above calculation, then, is equivalent to the following equation: Future Value = $10,000 x (1+0.045) x (1+0.045) • The equation can be represented as the following: Lecturer: Chara Charalambous 12 • If we were investing our money for 3 years the equation would be: 10,000*(1+0.045) 3 = $11,411.66 • So the equation for the future value can be written as n FV= PV(i+1) FV = Future Value PV = Present Value – the money invested-the capital i = the interest rate n = the number of years or periods I have invested my money The process of going from today’s values, or present values (PV), to future values (FV) is called compounding. Lecturer: Chara Charalambous 13 Present Value Basics • If you received $10,000 today, the present value would of course be $10,000 because present value is what your investment gives you now if you were to spend it today. If $10,000 were to be received in a year, the present value of the amount would not be $10,000 because you do not have it in your hand now, in the present. To find the present value of the $10,000 you will receive in the future, you need to pretend that the $10,000 is the total future value of an amount that you invested today. In other words, to find the present value of the future $10,000, we need to find out how much we would have to invest today in order to receive that $10,000 in the future. • In order to calculate present value, or the amount that we would have to invest today, we are going to use the FV equation but this time the unknown number it will be the PV. So we will follow the steps below : n • Original equation : FV= PV(i+1) => PV = FV (i+1) n The factor 1 n (i+1) is called the discounting factor Lecturer: Chara Charalambous 14 • Let's walk backwards from the $10,000 offered in Option B. Remember, the $10,000 to be received in three years is really the same as the future value of an investment • So, here is how you can calculate today's present value of the $10,000 expected from a three-year investment earning 4.5%: PV = FV = 10,000 = $8762.97 n 3 (i+1) (0.045+1) Lecturer: Chara Charalambous 15 • So the present value of a future payment of $10,000 is worth $8,762.97 today if interest rates are 4.5% per year. In other words, choosing Option B is like taking $8,762.97 now and then investing it for three years. The equations above illustrate that Option A is better not only because it offers you money right now but because it offers you $1,237.03 ($10,000 - $8,762.97) more in cash! Furthermore, if you invest the $10,000 that you receive from Option A, your choice gives you a future value that is $1,411.66 ($11,411.66 $10,000) greater than the future value of Option B. Lecturer: Chara Charalambous 16 Interest Tables The Future Value Interest Factor for i and n is defined n as (1 + i), and these factors can be found using a table: • • • • • • Period 1% 2% 3% 4% 5% 6% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 Lecturer: Chara Charalambous 17 Interest Tables • The Present Valuen Interest Factor for i and n is defined as 1/(1 + i), and these factors can be found using a table: • Period 1% 2% 3% 4% 5% 6% • 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 • 2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 • 3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 • 4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 • 5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 Lecturer: Chara Charalambous 18 Annuity • An annuity represents series of equal payments (or receipts) occurring for a specified number of equity distant periods. • Ordinary Annuity: Payments or receipts occur at the end of each period. • Annuity Due: Payments or receipts occur at the beginning of each period Lecturer: Chara Charalambous 19 Examples of Annuities • • • • • Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings Lecturer: Chara Charalambous 20 Parts of an Annuity (Ordinary Annuity) End of Period 1 0 Today End of Period 2 End of Period 3 1 2 3 $100 $100 $100 Equal Cash Flows Each 1 Period Apart Lecturer: Chara Charalambous 21 Parts of an Annuity (Annuity Due) Beginning of Period 1 Beginning of Period 2 0 1 2 $100 $100 $100 Today Beginning of Period 3 3 Equal Cash Flows Each 1 Period Apart Lecturer: Chara Charalambous 22 What is the difference between an ordinary annuity and an annuity due? Ordinary Annuity 0 i% 1 2 3 PMT PMT PMT 1 2 3 PMT PMT Annuity Due 0 i% PMT Lecturer: Chara Charalambous 23 Example 1 :If I deposit $1000 at the end of each year for 3 years in a saving account that pays 7% interest per year, how much will I have at the end of the 3 years? Lecturer: Chara Charalambous 24 Example of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $1,000 $1,070 $1,145 $3,215 = FVA3 FVA3 = $1,000(1.07)2 + $1,000(1.07)1 +$1,000(1.07)0= $1,145 +$1070 +$1000= $3,215 Lecturer: Chara Charalambous 25 n-1 n-2 1 0 FVA n = PMT(1+i) + PMT(1+i) +…+ R(1+i) +PMT(1+i) => n FVA n = PV (1+i) – 1 i Lecturer: Chara Charalambous 26 Example 2 :If I deposit $1000 at the beginning of each year for 3 years in a saving account that pays 7% interest per year, how much will I have at the end of the 3 years? Lecturer: Chara Charalambous 27 Example of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7% $1,000 $1,000 $1,000 $1,070 $1,145 $1,225 3 2 $3,440 = FVAD3 FVAD3 = $1,000(1.07) + $1,000(1.07) + $1,000(1.07) = $1,225 + $1,145 + $1,070 = $3,440 1 • FVADn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 + R(1+i)1 => n FVA = PV (1+i) – 1 * (1+i) i Lecturer: Chara Charalambous 29 • Example 3: you are offered a 3-year annuity with payments of $ 1000 at the end of each year. So you have to deposit the payments in a saving account that pays 7% interest per year. How much you have to deposit today? Lecturer: Chara Charalambous 30 Example of the Present Value of an Ordinary Annuity Cash flows occur at the end of the period 0 1 2 3 $1,000 $1,000 4 7% $934.58 $873.44 $816.30 $1,000 $2,624.32 = PVA3 $1,000 / (1.07)1 + $1,000 / (1.07)2 + $1,000 / (1.07)3 = $934.58 + $873.44 + $816.30 = $2,624.32 PVA3 = PVAn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n => PVAn =FV 1-1/(1+i) i n Lecturer: Chara Charalambous 32 • Example 3: you are offered a 3-year annuity with payments of $ 1000 at the beginning of each year. So you have to deposit the payments in a saving account that pays 7% interest per year. How much you have to deposit today? Lecturer: Chara Charalambous 33 Example of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 3 4 7% $1,000.00 $ 934.58 $ 873.44 $1,000 $1,000 $2,808.02 = PVADn PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02 PVAn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 => PVAn =FV 1-1/(1+i) i n * (1+i) Lecturer: Chara Charalambous 35 Exercises Lecturer: Chara Charalambous 36