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CHAPTER 5 & 6 TIME VALUE OF MONEY

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Basic Principle : A dollar received today is worth more than a dollar received in the future. This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner. This concept is so important in understanding financial management (investment, stock & bond valuation etc..) Time Value Of Money 2 SITI AISHAH BINTI KASSIM (FM2)

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Translate $1 today into its equivalent in the future (compounding) – Future Value SITI AISHAH BINTI KASSIM (FM2) 3 If we can measure this opportunity cost, we can: TodayFuture ? Translate $1 in the future into its equivalent today (discounting)- Present Value TodayFuture ?

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Compound interest occurs when interest paid on the investment during the first period is added to the principal; then, during the second period, interest is earned on this new sum. Compounding is the process of determining the Future Value (FV) of cash flow. The compounded amount (Future Value) is equal to the beginning amount plus interest earned. SITI AISHAH BINTI KASSIM (FM2) 4 Future Value (FV)

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For example : If we place RM 1000 in a savings account paying 5% interest compounded annually. How much will it be worth at the end of each year ? Year 1 = RM1000 (1.05) = RM1050.00 Year 2 = RM1050.00 (1.05) = RM1102.50 Year 3 = RM1102.50 (1.05) = RM1157.63 Year 4 = RM1157.63 (1.05) = RM 1215.51 etc…… SITI AISHAH BINTI KASSIM (FM2) 5 Future Value (FV) RM 1000 04321n.. 5%

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Formula of Future Value (FV) : FVn = PV (1+ i)n or FVn = PV (FVIF i,n ) where; FVn = the FV of the investment at the end of n year n = the number of years i = the annual interest rate PV = original amount invested at beginning of the first year **(1+ i) is also known as compounding factor. SITI AISHAH BINTI KASSIM (FM2) 6 Future Value (FV)

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For example : If we place RM1,000 in a savings account paying 5% interest compounded annually. How much will our account accrue in 4 years? PV=RM1,000, i =5% & n=4 years. SITI AISHAH BINTI KASSIM (FM2) 7 Future Value (FV) a) FVn=PV (1+i)n b) FVn = PV (FVIFi,n) FV4=1,000 (1+0.05)4 FV4 = PV (FVIF5%,4) =1,000 (1.2155) =1,000 (1.2155) = RM1,215.50 =RM1,215.50

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If Adam invests RM10,000 in a bank where it will earn 6% interest compounded annually. How much will it be worth at the end of a) 1 year and b) 5 years? Compounded for 1 year a) FV1= $10000 (1+0.06) 1 b) FV1 = PV (FVIF 6%,1 ) = $10000 (1.06) 1 = $10000 (1.0600) = $10,600.00 = $10,600.00 Compounded for 5 year a) FV5 = $10000 (1+0.06) 5 b) FV5 = PV (FVIF 6%,5 ) = $10000 (1.06) 5 = $10000 (1.3382) = $13,380.00 = $13,382.00 SITI AISHAH BINTI KASSIM (FM2) 8 Example:

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Non-annual periods : not annual compounding but occurs semiannually, quarterly, monthly… If compounding semiannually : FV = PV (1 + i/2) n x 2 or FVn = PV (FVIFi/2, nx2 ) If compounding quarterly : FV = PV (1 + i/4) n x 4 or FVn = PV (FVIFi/4, nx4 ) If compounding monthly : FV = PV (1 + i/12) n x 12 or FVn = PV (FVIFi/12, nx12 ) SITI AISHAH BINTI KASSIM (FM2) 9 Compound Interest With Non-annual Periods

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Present value is the current value of futures sum Finding Present Values (PVs) is called discounting We can calculate PV by using this equation : PV = FVn or PV = FVn (PVIF i,n ) (1+i ) n **[ 1/(1+ i)n ] is also known as discounting factor. SITI AISHAH BINTI KASSIM (FM2) 10 PRESENT VALUE (PV)

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For example : What is the PV of $800 to be received 10 years from today if our discount rate is 10%. PV = 800/ (1.10)10 = $308.43 or PV = $800 (PVIF 10%,10yrs) = $800 (0.3855) = $308.40 SITI AISHAH BINTI KASSIM (FM2) 11 PRESENT VALUE (PV)

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In every single sum future value and present value problem, there are 4 variables: FV, PV, i, and n When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable. Keeping this in mind makes “time value” problems much easier! SITI AISHAH BINTI KASSIM (FM2) 12 Hint for single sum problems:

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For example: What is the PV of an investment that yields $300 to be received in 2 years and $450 to be received in 8 years if the discount rate is 5%? PV = $300(PVIF5%,2) + $450(PVIF5%,8) = $300(0.907) + $450(0.677) = 272.10 + 304.65 = $576.75 SITI AISHAH BINTI KASSIM (FM2) 13 PV with Multiple, Uneven Cash Flows

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An annuity is a series of equal payments for a specified numbers of years. There are 2 types of annuities*: - ordinary annuity - annuity due *in finance, ordinary annuities are used much more frequently than are annuities due SITI AISHAH BINTI KASSIM (FM2) 14 ANNUITY 100 03124

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Ordinary Annuity is an annuity which the payments occur at the end of each period. a) Present Value of Annuity (PVA) Present Value of Annuity (PVA) can be calculated by using these equations: PVAn = PMT / (1+i) n or PVAn = PMT (PVIFA i,n ) For example: Find the PV of $500 received at the end of each year of the next 3 years discounted back to the present at 10%? SITI AISHAH BINTI KASSIM (FM2) 15 Ordinary Annuity

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a) PVA 3 = (500/1.10) 1 + (500/1.10) 2 + (500/1.10) 3 = 454.55 + 413.22 + 375.66 = $ 1243.43 OR b) PVA 3 = 500 (PVIFA 10%,3 ) = 500 (2.487) = $ 1243.50 Solutions: SITI AISHAH BINTI KASSIM (FM2) 16 Ordinary Annuity

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b) Future Value of Annuity (FVA) Compound Annuity / Future Value of Annuity (FVA) can be calculated by using these equations : FVAn = PMT (1+ i) n or FVAn = PMT (FVIFA i,n ) For example : We are going to deposit $15,000 at the end of each year for the next 5 years in a bank where it will earn 9% interest. How much will we get at the end of 5 years? SITI AISHAH BINTI KASSIM (FM2) 17 Ordinary Annuity

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a) At the end of 5 years, we will get… FVA 5 = 15000(1.09) 4 + 15000(1.09) 3 +15000(1.09) 2 + 15000(1.09) 1 +15000 = 21173.72 +19425.44 + 17821.50 + 16350 + 15000 = $89,770.66 OR b) By using FVIFA table FVA 5 = 15000 (FVIFA 9%,5 ) = 15000 (5.9847) = $89,770.50 SITI AISHAH BINTI KASSIM (FM2) 18 Ordinary Annuity Solutions:

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Annuity Due is an annuity in which the payments occur at the beginning of each period. a) Future Value of Annuity Due (FVAD) FVADn = PMT (FVIFA i,n ) (1+i) b) Present Value of Annuity Due (PVAD) PVADn = PMT (PVIFA i,n ) (1+i) Annuity Due SITI AISHAH BINTI KASSIM (FM2) 19

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For example (FVAD): We are going to deposit $1,000 at the beginning of each year for the next 5 years in a bank where it will earn 5% interest. How much will we get at the end of 5 years? FVADn = PMT (FVIFA i,n ) (1+i) = 1000 (5.526) (1 + 0.05) = $5802.30 SITI AISHAH BINTI KASSIM (FM2) 20 Annuity Due

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For example (PVAD) : Find the PV of $500 received at the beginning of each year of the next 5 years discounted back to the present at 6%? PVADn = 500 (PVIFA 6%,5 ) (1+0.06) = 500 (4.212) (1 + 0.06) = $2,232.36 SITI AISHAH BINTI KASSIM (FM2) 21 Annuity Due

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Amortized loan is a loan that paid off in equal installments. To determine the installment (payment) we can use this formula : PMT = Loans PVIFA i,n Each installment consists partly of interest and partly of repayment of principal. This breakdown is given in the amortization schedule. Amortized Loan SITI AISHAH BINTI KASSIM (FM2) 22

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For example: Daniel wants to accumulate RM75,000 by the end of five (5) years. Assume that the fund will earn an interest at 9.5% compounded annually. PMT = $75000/ PVIFA 9.5%,5 = $10000 / 6.0446 = $ 12,407.73 SITI AISHAH BINTI KASSIM (FM2) 23 Amortized Loan

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Year Beginning Balance (1) Annual Deposit (2) Interest Generated (3) Accumulated Amount (1)+(2)+(3)=(4) 1012,407.730 2 1,178.7425,994.20 3 12,407.732,469.4540,871.38 4 12,407.733,882.7857,161.89 5 12,407.735,430.38 75,000 Loan Amortization Schedule SITI AISHAH BINTI KASSIM (FM2) 24

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Perpetuity is an annuity that continues forever. The equation representing the present value of annuity: PV = PP i where, PV= PV of the perpetuity PP= Constant dollar amount provided by perpetuity i = interest rate Perpetuity SITI AISHAH BINTI KASSIM (FM2) 25

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26 SITI AISHAH BINTI KASSIM (FM2) THE END

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