1 6-1 Chapter 6 Time Value of Money Concepts 6-2 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in one year. 6-3 Simple Interest Interest amount = P i n Assume you invest $1,000 at 6% simple interest for 3 years. You would earn $180 interest. ($1, = $180) (or $60 each year for 3 years)
2 Compound Interest When we compound interest, we assume you earn interest on both principal and interest. Principal Interest 6-5 Compound Interest Assume we will save $1,000 for three years and earn 6% interest compounded annually. What is the balance in our account at the end of three years? 6-6 Compound Interest Original balance $ 1, First year interest Balance, end of year 1 $ 1, Balance, beginning of year 2 $ 1, Second year interest Balance, end of year 2 $ 1, Balance, beginning of year 3 $ 1, Third year interest Balance, end of year 3 $ 1,191.02
3 Future Value of a Single Amount Multiply a year s beginning principal by the interest rate and add that year s interest to the account balance. $1, = $1, and $1, = $1, and $1, = $1, Future Value of a Single Amount Writing in a more efficient way, we can say.... $1, = $1, or $1,000 [1.06] 3 = $1, Future Value of a Single Amount $1,000 [1.06] 3 = $1, We can generalize this as... FV = PV (1 + i) n Future Present Interest Value Value Rate Number of Periods
4 Future Value of a Single Amount Find the Future Value of $1 table in your textbook Future Value of a Single Amount Find the factor for 6% and 3 periods. Solve our problem like this... FV = $1, FV = $1, FV $ Future Value of a Single Amount You invest $10,000 today and earn 8% interest for 8 years. What will the balance in your account be at the end of 8 years if.... A. Interest is simple. B. Interest is compounded annually.
5 Future Value of a Single Amount A - Simple Interest $10, = $6,400 $10,000 + $6,400 = $16,400 B - Compound Annually $10, = $18, Present Value of a Single Amount Instead of asking what is the future value of a current amount, we might want to know what amount we must invest today to accumulate a known future amount. This is a present value question Present Value of a Single Amount Remember our equation? FV = PV [1 + i] n We can solve for PV and get.... FV PV = [1 + i] n
6 Present Value of a Single Amount We can rearrange the equation... PV = FV or 1 [1 + i] n PV = FV [1 + i] -n 6-17 Present Value of a Single Amount Find the Present Value of $1 table in your textbook. Hey, it looks familiar! 6-18 Present Value of a Single Amount Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time. What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?
7 Present Value of a Single Amount i =.08, n = 5 Present Value Factor = $20, = $13, If you deposit $13, now, at 8% annual interest, you will have $20,000 at the end of 5 years Time Value of Money Example If you deposit $5,000 in a bank at 8% interest compounded annually, how much will you have in 5 years?... in 10 years? 5 Years 10 Years a. $7,387 $8,144 b. $7,347 $10,795 c. $7,347 $9,471 d. $6,984 $9, Time Value of Money Example If you deposit $5,000 in a bank at 8% interest compounded annually, how much will you have in 5 years?... in 10 years? 5 Years 10 Years a. $7,387 $8,144 b. $7,347 $10,795 c. $7,347 $9,471 d. $6,984 $9,186 Future Value of $1 Table $5, = $ 7, $5, = $10,794.60
8 Time Value of Money Example What amount must you deposit today at 6% interest compounded annually, to have $10,000 for your first year of college 5 years from now? a. $7,462 b. $7,921 c. $7,473 d. $7, Time Value of Money Example What amount must you deposit today at 6% interest compounded annually, to have $10,000 for your first year of college 5 years from now? a. $7,462 b. $7,921 c. $7,473 d. $7,581 Present Value of $1 Table $10,000 x = $7, Present Value On June 1, your company purchases equipment by paying $5,000 down and issuing a $27,000 noninterest-bearing note payable that is due in three years. Similar transactions carry a stated interest rate of 6%. What is the purchase price of the equipment?
9 Present Value Face of note $ 27,000 Present value of $1 (i = 6%, n = 3) Present value of note $ 22,670 Cash paid 5,000 Cost of equipment $ 27,670 Journal entry to record the note and equipment 6-26 Present Value GENERAL JOURNAL Page 25 Date Description Post. Ref. Debit Credit Jun. 1 Equipment 27,670 Discount on Notes Payable 4,330 Notes Payable 27,000 Cash 5, Solving for Other Values FV = PV (1 + i) n Future Present Interest Value Value Rate Number of Periods There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined.
10 Solving for Other Variables Example Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? a. 3.5% b. 4.0% c. 4.5% d. 5.0% 6-29 Solving for Other Variables Example Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to? a. 3.5% b. 4.0% c. 4.5% d. 5.0% Present Value of $1 Table $1,000 = $1,092? $1,000 $1,092 = Search the PV of $1 table in row 2 (n=2) for this value No Explicit Interest Some notes do not include a stated interest rate. We call these notes noninterest-bearing notes. Even though the agreement states it is a noninterest-bearing note, the note does, in fact, include interest. We impute an appropriate interest rate for a loan of this type to use as the interest rate.
11 Expected Cash Flow Approach Statement of Financial Accounting Concepts No. 7 Using Cash Flow Information and Present Value in Accounting Measurements The objective of valuing an asset or liability using present value is to approximate the fair value of that asset or liability. Expected Cash Flow Risk-Free Rate of Interest Present Value 6-32 Basic Annuities An annuity is a series of equal periodic payments Ordinary An annuity with payments at the end of the period is known as an ordinary annuity. End End
12 Due An annuity with payments at the beginning of the period is known as an annuity due. Beginning Beginning Beginning 6-35 Future Value of an Ordinary The equation to find the future value of an annuity is... n FVA = (1 + i) i Because this is the equation for the FV of $1, the equation for FVA should be easy to remember Future Value of an Ordinary To find the future value of an ordinary annuity, multiply the amount of a single payment or receipt by the future value factor.
13 Future Value of an Ordinary We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded annually, on all invested funds. What will be the fund balance at the end of 10 years? 6-38 Future Value of an Ordinary Amount of annuity $ 2, Future value of ordinary annuity of $1 (i = 8%, n = 10) Balance $ 36, Find the Future Value of Ordinary of $1 factor in your textbook Future Value of an Due Compute the future value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually.
14 Future Value of an Due Compute the future value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. Amount of annuity $ 10,000 FV of annuity due of $1 (i=6%, n=4) Present value of annuity $ 46, Present Value of an Ordinary Rats! More Present Value Present Value of an Ordinary The equation to find the present value of a series of $1 payments is.... PV = 1-1 [1+i] i n This is the equation for the PV of $1
15 Present Value of an Ordinary You wish to withdraw $10,000 at the end of each of the next 4 years from a bank account that pays 10% interest compounded annually. How much do you need to invest today to meet this goal? 6-44 Present Value of an Ordinary Today $10,000 $10,000 $10,000 $10,000 PV1 PV2 PV3 PV Present Value of an Ordinary PV Present Factor Value PV1 $ 10, $ 9, PV2 10, , PV3 10, , PV4 10, , Total $ 31, If you invest $31, today you will be able to withdraw $10,000 at the end of each of the next four years.
16 Present Value of an Ordinary PV Present Factor Value PV1 $ 10, $ 9, PV2 10, , PV3 10, , PV4 10, , Total $ 31, Can you find this value in the Present Value of Ordinary of $1 table? 6-47 Present Value of an Ordinary How much must a person 65 years old invest today at 8% interest compounded annually to provide for an annuity of $20,000 at the end of each of the next 15 years? a. $153,981 b. $171,190 c. $167,324 d. $174, Present Value of an Ordinary How much must a person 65 years old invest today at 8% interest compounded annually to provide for an annuity of $20,000 at the end of each of the next 15 years? a. $153,981 b. $171,190 c. $167,324 d. $174,680 PV of Ordinary $1 Payment $ 20, PV Factor Amount $171,189.60
17 Present Value of an Ordinary Assume the person only has $140,000. What annuity will this amount provide at the end of each of the next 15 years if it is invested today at 8% interest compounded annually? a. $15,891 b. $16,356 c. $17,742 d. $18, Present Value of an Ordinary Assume the person only has $140,000. What annuity will this amount provide at the end of each of the next 15 years if it is invested today at 8% interest compounded annually? a. $15,891 b. $16,356 c. $17,742 d. $18,123 PV of Ordinary of $1 Amount $ 140,000 Divided by $16, Present Value of an Due Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually.
18 Present Value of an Due Compute the present value of $10,000 received at the beginning of each of the next four years with interest at 6% compounded annually. Amount of annuity $ 10,000 PV of annuity due of $1 (i=6%, n=4) Present value of annuity $ 36, Present Value of Annuities Western Gas, Inc. lost a lawsuit requiring the company to pay $2,250,000 immediately or $260,000 ($3,900,000 total) at the end of each of the next 15 years. Assume Western Gas can earn 9% on all funds available. Which settlement option would you recommend? 6-54 Present Value of Annuities Annual payment $ 260,000 PV factor for ordinary annuity, n = 15, i = 9% PV of annuity payments $ 2,095,779 Because the present value of the payments is less than the lump sum payment, you would recommend that Western Gas make the annual payments of $260,000.
19 Present Value of a Deferred In a deferred annuity, the first cash flow is expected to occur more than one period after the date of the agreement Present Value of a Deferred On January 1, 2003, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, If you require a 12% return on your investments, how much are you willing to pay for this investment? Cost Today? $12,500 $12,500 1/1/03 12/31/03 12/31/04 12/31/05 12/31/06 12/31/ Present Value of a Deferred Payment PV of $1 i = 12% PV n 1 $ 12, $ 8, , ,944 4 $ 16,841 Cost Today? $12,500 $12,500 1/1/03 12/31/03 12/31/04 12/31/05 12/31/06 12/31/
FIN 301 Homework Solution Ch4 Chapter 4: Time Value of Money 1. a. 10,000/(1.10) 10 = 3,855.43 b. 10,000/(1.10) 20 = 1,486.44 c. 10,000/(1.05) 10 = 6,139.13 d. 10,000/(1.05) 20 = 3,768.89 2. a. $100 (1.10)
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