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TIME VALUE OF MONEY. Week 5. FACULTY OF BUSINESS AND ACCOUNTANCY. Based on positive time preference ~ a ringgit today is worth more than a ringgit expected in the future. TVM tools are used to;. Calculate deposits required to accumulate a future sum

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## TIME VALUE OF MONEY

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- TIME VALUE OF MONEY Week 5 FACULTY OF BUSINESS AND ACCOUNTANCY
- Based on positive time preference ~ a ringgit today is worth more than a ringgit expected in the future TVM tools are used to; Calculate deposits required to accumulate a future sum Amortize loans by calculating loan payments schedules Determine interest or growth rates of money streams Evaluate perpetuities Find the required rate of return
- Basic Concepts Time Lines Future Values Single Sum Present Values Annuity Perpetuities Types of Interest Rates Nominal Rate Periodic Rate Compounding Periods Amortization Required Rate of Return Effective Annual Rate
- TMV Solution Methods Numerical –using regular calculator w/o financial functions Interest Tables - given with the text book a. Present Value Interest Factor b. Present Value Interest Factor for Annuity c. Future Value Interest Factor d. Future Value Interest Factor for Annuity Financial Calculator Worksheet
- Time Lines End of period 2 & beginning of period 3 Time 0 1 2 3 i% CF0 CF1 CF2 CF3 Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.
- RM100 lump sum due in 2 years 0 1 2 5% 100 0 1 2 3 i% 100 100 100 Time line illustrations 10% End of period 2 PV FV 3 year RM100 ordinary annuity (fixed pmts)
- Uneven cash flow stream 0 1 2 3 i% -50 100 75 50 CF2 CF0 CF1 CF3
- C ALL P/YR N I/YR PV FV xP/YR BEG/END AMORT 8
- Future and Present Values
- Future Value & Present Value Future Value - Ending amount of your account at the end of n periods Present Value – Beginning amount in your account The process of going from today’s values, or PV to future value is called compounding. In determining the final value of a cash flow or series of cash flows, compound interest will be applied. What is compound interest? Is it the same thing as simple interest?
- 0 0 1 1 2 2 5% 5% Simple interest; 100 105 110 2nd period; (Principal) [(2 x 0.05) + 1 ] = RM110 or, [100 (1.05) - 100] + 100(1.05) = RM110 Compounding interest; 100 105 110.25 2nd period; (Principal + interest)(1 + i) (RM100 + RM5) (1 + 0.05) = RM110.25
- Future Value : One period case Case 1: Given that a car dealer offers a car for RM20,000 in cash or RM25,000 on credit for one year. Given an annual i.r. of 10%, which payment is better for you & which for the car dealer? 1 0 i% = ? 25,000 20,000 Given RM20,000 now, in 1 year at ir of 10%, the money deposited will be ? 1 0 10% FV =? 20,000
- Numerical Solution (N/S) ; FV1 = PV + INT = PV + PV (i) = PV ( 1 + i ) FV1 = PV ( 1 + i ) = 20,000 ( 1 + 0.1) = RM22,000 Tabular Solution (T/B) ; Future Value Interest Factor for i & n (FVIFi,n) ~ the future value of RM1 left on deposit for n periods at a rate of i percent per period ~ where ( 1 + i)n = FVIFi,n FV1 = PV (FVIFi,n) = 20,000 ( 1.1000) = RM22,000
- Find the FV of RM20,000 given an interest rate of 10% in one year. C All P/YR N I/YR + / - PV FV 14
- Present Value : One period case FV1 = PV ( 1 + i ) , so PV = FV1 (1 + i) C2 : Given the annual ir of 10%, at what amount of cash would the car dealer be indifferent to receiving RM25,000 at time 1? 0 1 10% 25,000 PV = ?
- Numerical solution ; PV = FV1 /(1 + i) = 25,000 = RM22,727.27 1 + 0.01 Tabular Solution ; Present Value Interest Factor for i & n (PVIFi,n) ~ the present value of RM1 due n periods in the future discounted at i percent per period ~ where; PV = FV (PVIFi,n) = 25,000 (0.9091) = RM 22,727.50
- Find the PV of RM25,000 given an interest rate of 10% in one year. C ALL P/YR N I/YR FV PV
- Future Value & Present Value : Multi – period case Important terms; Compound interest – interest earned on the principal & on the accumulated interest Discount interest rate – the rate that will make the future value equivalent to the present value Fair (Equilibrium) Value – the price at which investors are indifferent btw buying or selling a security compounding 0 1 2 5% discounting
- The discount rate is often also referred to as the opportunity cost, the required return, and the cost of capital.
- 0 1 2 3 10% 100 FV = ? C3 : Find the FV o RM100 left for 3 years in an account paying 10 percent, annual compounding; N/S; FV2 = FV1 (1 + i) = PV ( 1 + i ) (1 + i) = PV (1 + i)2 FV3 = PV (1 + i)3 FV1 = PV + INT = PV + PV (i) = PV ( 1 + i ) FV3 = 100 (1.10)3 = 133.10
- FVn = PV (1 + i)n = PV (FVIFi,n) = 100 (1.10)3 = 100 (1.3310) = RM133.10 C ALL P/YR N I/YR +/- PV FV 21
- 0 1 2 3 10% PV = ? 100 C4 : Find the PV of RM100 to be received in 3 years if the appropriate ir is 10 percent, compounded annually: N/S; T/S; PVn = FV (1 + i)n PV3 = FV (1 + i)3 PVn = FVn/ (1 + i)n = FVn 1 n = FVn(PVIFi,n) 1 + i = 100 (1/1.10)3 = 100 (0.7513) = RM75.13
- C ALL P/YR N I/YR FV PV 23
- n (periods) and i (interest rate)
- Solving for n in TVM problems C5: How long will it take a firm’s sales to double, if sales are growing at a 15% rate? n = ? n - 1 0 15% RM2 RM1 T/S; FVn = PV (FVIFi,n) (FVIFi,n) = FV / PV = 2 / 1 = 2 FVn = PV (1 + i)n 2 = 1 (1.15)n 2 = (1.15)n N/S; Look in FVIF Table for (FVIF15%,n) = 2 n 5 periods
- Financial Calculator Solution ; n = ? n - 1 0 15% RM2 RM1 C ALL P/YR +/- PV I/YR FV N
- 0 1 2 3 i = ? 100 125.97 Solving for interest rate C6: What annual ir will cause RM100 to grow to RM125.97 in 3 years? 100 (1 + i) 100 (1 + i)2 100 (1 + i)3 100 (1 + i)3 = 125.97 100 (FVIFi,3) = 125.97 FVIFi,3 = 1.2597 Look at Row 3 of FVIF Table. 1.2597 is in the 8% column T/S;
- Annuities & Perpetuities
- Annuities An annuity is a series of equal payments made at fixed intervals for a specific number of periods Ordinary annuity - payments occur at the end of each period - eg. Students loan Annuity due – payments are made at the beginning of each period - eg. Mthly rentals, insurance premiums
- Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT What is the difference between an ordinary annuity and an annuity due? Both are 3-yr annuities ( 3 pmts)
- Future Value of an Ordinary Annuity C7: What is the future value of an ordinary annuity of RM100 per period for 3 yrs if the ir is 10 percent, compounded annually? Time line approach; 3 0 1 2 10% 100 100 100 110 121 331 + 100(1 + i) Twice compounding 100(1 + i)2
- FVA3 = PMT (1 + i) + PMT (1 + i)1 + PMT (1 + i)2 = 100 (1) + 100 (1.10) + 100 (1.21) = RM331 N/S; FVAn = PMT (FVIFAi,n) FVA3 = 100 (FVIFA10%,3) = 100 (3.3100) = RM331 T/S; FC; Make sure no BGN sign 1 10 3 -100 = 331.00 P/YR PMT I/YR N FV
- Future Value of an Annuity Due C8: What is the future value of RM100 payments made at beginning of each year for 3 yrs in a saving account that pays 10 percent, compounded annually? Triple compounding Time line approach; 3 0 2 1 10% 0 110 121 133.10 100 100 100 100(1 + i) + 100(1 + i)2 100 (1 + i)3 364.10
- FVAD3 = PMT (1 + i)3 + PMT (1 + i)2 + PMT (1 + i)1 = 100 (1.331) + 100 (1.21) + 100 (1.10) = RM364.10 N/S; FVADn = FVA3 (1 + i) or = PMT (FVIFA10%,3) (1 + i) = 331 (1.10) = 100 (3.3100) (1.10) = RM364.10 = 364.10 T/S; Make sure BGN sign FC; BEG/END 1 -100 10 3 = 364.10 P/YR PMT I/YR N FV
- Present Value of an Ordinary Annuity C9: What is the PV of an annuity of RM100 per period for 3 years if the ir is 10 percent annually? Time line approach; 0 10% 1 2 3 100 100 100 0 90.91 82.64 75.13 100 / (1 + i) + 100 /(1 + i)2 Triple discounting 100 / (1 + i)3 248.68
- Present Value of an Annuity Due C10: How much lump sum today to make it equivalent with a 3 year annuity paying RM100 at beginning of each year? Time line approach; Make sure BGN sign 0 10% 1 2 3 100 100 100 90.91 82.64 + 100 / (1 + i) Double discounting 100 /(1 + i)2 273.55
- An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.
- Perpetuities is a stream of equal payments expected to continue forever - a type of annuity PV (Perpetuity) = payment = PMT interest rate i the current price
- C11: A perpetual bond promised to pay RM100 per year in perpetuity. What would the bond’s worth today if the opportunity cost, or discount rate was 5 percent PV (Perpetuity) = RM100 = RM2000 0.05 As the interest rate increases, the perpetuity’s value will drop. When ir = 10%; PVp = 100 = RM1000 0.1
- Uneven Cash Flow Stream Payment (PMT) - equal cash flows at regular intervals Cash flow (CF) - uneven cash flows Examples of uneven cash flows; - common stock’s dividend - returns from fixed asset investments ~ production income ~ rentals
- Present Value of an Uneven Cash Flow Stream C12: Find PV of the following cash flows stream, discounted at 10% 0 1 2 3 4 10% 0 100 300 300 -50 CF0 CF1 CF2 CF3 CF4 PV = CF0 1 0 + CF1 1 1 + CF2 1 2 1 + i 1 + i 1 + i + CF3 1 3 + CF4 1 4 1 + i 1 + i
- For cash flow calculation ; C ALL P/YR CFj Nj IRR/YR NPV
- C ALL C ALL CFj CFj CFj CFj CFj CFj CFj Nj +/- CFj +/- CFj I/YR I/YR NPV NPV 43 43
- C13: Find the PV of the following c/f discounted at 10% 0 1 2 3 4 5 10% 0 100 50 200 200 200 90.91 41.32 411.03 100(1/1 + i)1 50(1/1 + i)2 200 (PVIFA10%,3) = 200 (2.4869) 497.38 or 497.38(1/1 + i)2 543.26 150.26 136.60 124.18 200 (PVIFA10%,3) or 200 (1/1 + i)3 200 (PVIFA10%,4) or 200 (1/1 + i)4 200 (PVIFA10%,5) or 200 (1/1 + i)5
- Future Value of an Uneven Cash Flow Stream C14: Find FV of the following cash flows stream,compoundedat 10% 0 1 2 3 4 5 10% 0 100 50 200 200 200 CF0 CF1 CF2 CF3 CF4 CF5 FV = CF5 (1 + i)0 + CF4 (1 + i)1 + CF3 (1 + i)2 + CF2 (1 + i)3 + CF1 (1 + i)4 + CF0 (1 + i)5
- 0 1 2 3 4 5 10% 0 100 50 200 200 200 220 242 66.55 146.41 200 (1 + i) 462 200 (1 + i)2 50 (1 + i)3 = 50 (1.331) 100 (1 + i)4 = 100 (1.4641) NFV = 874.96 200 (FVIFA10%,2) = 200 (2.1) 462 420 ( 1 + i)
- 0 1 2 3 4 5 10% NFV=? 0 100 50 200 200 200 CF0CF1 CF2 CF3 CF4 CF5 C ALL I/YR P/YR NPV CFj N CFj PMT CFj FV CFj Using formula; NFV = NPV ( 1 + i)n = 543.26 (1 + 0.1)5 = 874.93 Nj
- Different Compounding Periods annually / semiannually / quarterly / monthly / daily compounding the quoted interest rate is normally the annual one. If bank promised 10% annual interest rate semiannually what does it means ? ~ interest will be added each 6 months but will the interest be 10% as quoted or more/less?
- Say that you want to deposit your money in the bank which offer you the highest return. As you shopped around, you come up with these rates: Bank A : 15 percent, compounded daily Bank B : 15.5 percent, compounded quarterly Bank C : 16 percent, compounded annually Which one has the best rates for deposits?
- C15: A bank declares that it pays a 6% annual ir semiannually & you want to deposit RM 100. What is FV at the end of 3rd year? Annual compounding 0 1 2 3 6% -100 FV = ? FV3 = PV (1 + i)3 = 100 (1 + 0.06)3 = RM119.10 Semiannual compounding 0 1 2 3 4 5 6 i% -100 FV = ? i% = 6% /2 = 3% n = 3 x 2 = 6 FV6 = PV (1 + i)6 = 100 (1 + 0.03)6 = RM119.41
- Different compounding periods are used for different types of investment In order to compare securities with different compounding periods, need to put them on a common basis. Types of interest rates; Nominal or quoted interest rates Annual percentage rates (APR) Effective annual rates (EAR)
- 1. Nominal or Quoted interest rates , inom Is the contracted, or stated, or declared ir. The rate which is given by the bank or issuer. Annual Percentage Rate (APR) The interest rate charged per period multiplied by the number of periods per year. C16: If a bank is charging 1.2% per month on car loans, what is the APR? APR = 1.2% x 12 = 14.4% It is the nominal rates for loan that some government requires the bank to display to customers.
- Periodic Rate Periodic rate is the nominal rate at each period; where m is the no of compounding periods per year Eg: 6% compounded quarterly. periodic rate, iper = inom / m = 6% / 4 = 1.5%
- So periodic rate is the rate charged by a lender or paid by a borrower in each period. C17: A bank charges 18% annual interest rates monthly on credit card loans, what is the periodic rate? iper = inom / m = 18% / 12 = 1.5% Bank will charge 1.5% of interest monthly or per month So if we delayed paying our credit card debt for a year, will the debt be the same as we take a loan of the same amount at 18% annual interest?
- Notice that iper is the rate that is shown on time lines and used in certain calculations, not the annual rate. C18: How much would you have at the end of the 2nd year when you make RM100 deposit in an account that pays 12% interest rate semiannually. 0 1/2 1 1/2 2 (x2) = N 6% -100 FV = ? For calculation of FV given only PV, must use this rate not the annual rate given in this case. AND maintain P/YR = 1.
- 2. Effective Annual Rates (EAR) or (EFF) The rate which would produce the same ending (future) value if annual compounding has been used. ~ (the interest rate expressed as if it were compounded once per year) Annual compounding EAR ; The annual rate that produces the same FV as if we had compounded at a given periodic rate m times per year 0 1 2 3 6% -100 119.10 Semiannual compounding 0 1 2 3 4 5 6 3% -100 119.41 0 3 i = ? -100 119.41
- C19: EFF% or EAR for 10% semiannual investment. An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually. Why?
- C20: What is the FV of RM100 compounded semiannually for 3 years if inom = 10%? Would it be different if it were compounded quarterly? Semiannual compounding 0 1 2 3 5% -100 FV = ? 134.01 Quarterly compounding 0 1 2 3 2.5% -100 FV = ? 134.49
- FVn = PV 1 + inommn m FV3 = 100 1 + 0.1 2(3) 2 = 100(1.05)6 = 134.01 Semiannual compounding EAR = ( 1 + inom / m )m - 1 = ( 1 + 0.10 / 2 )2 – 1 = 10.25% Quarterly compounding FVn = PV 1 + inommn m FV3 = 100 1 + 0.1 4(3) 4 = 100(1.025)12 = 134.49 EAR = ( 1 + inom / m )m - 1 = ( 1 + 0.10 / 4 )4 – 1 = 10.38%
- F/C ; Given annual i.r. of 10%, compounded semiannually; To find APR, given the EAR of 10.25%; NOM% 10 10.25 EFF% 2 P/YR 2 P/YR EFF% 10.25 EAR NOM% 10.00 The APR formula; APR = 1 + EFF 1/m - 1 x m x 100 100
- Compounding More Frequently than Annually Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently. As a result, the effective interest rate is greater than the nominal (annual) interest rate. Furthermore, the effective rate of interest will increase the more frequently interest is compounded.
- Nominal & Effective Rates The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. The effective interest rate is the rate actually paid or earned. The effective rate > nominal rate whenever compounding occurs more than once per year EAR > inom If compounding occurs only once a year, then; EAR = inom = iper
- Nominal & Effective Rates C21: What is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly? EAR = (1 + .18/12) 12 -1 EAR = 19.56%
- Why is it important to consider effective rates of return? An investment with monthly payments is different from one with quarterly payments. Must put each return on an EFF% basis to compare rates of return. Must use EFF% for comparisons. See the following values of EFF% rates at various compounding levels. EARANNUAL 10.00% EARQUARTERLY 10.38% EARMONTHLY 10.47% EARDAILY (365) 10.52%
- Fractional Time Periods Before, payments only occur at beginning or end of periods. What if, payments occur at some date within a period? 0 1st 2nd month 20th day month 1% -100 FV = ?
- C22: Deposits RM100 in a bank that pays 12% ir compounded monthly. How much the amount will be then in 1 month and 20 days? N/S; FVn = PV (1 + i)n = PV (1 + i)1+ 20/30 = 100(1 + 0.1)1.67 = RM101.68 If use 360-day year; iper = 0.12 /360 = 0.00033333 per day No of days deposited; = 50 days FVn = PV (1 + iper)n = 100 ( 1.0003333)50 = RM 101.68 F/C ;
- Note of Caution; Rule 1: Money saved/deposited/invested should be in negative sign. Money withdrawn/received should be in positive sign. C23: RM1000 is deposited today for a semiannual payment of RM300 for 3 years. Given an interest rate of 10% semiannually, how much would be left in the account in 3 years time? 0 1 2 3 5% -1000 300 300 300 300 300 300 FV = ?
- Rule 2 : If there is only PV & no PMT, either; If use periodic ir, keep P/YR = 1. If use nominal rate, change P/YR accordingly. C24: RM1000 is deposited today. Given an interest rate of 10% semiannually, how much would be in the account in 3 years time? 0 1 2 3 5% -100 FV = ?
- Rule 3: For case with PMT or PMT and PV, N = no of payment made, I/YR = annual interest rate, P/YR = no of payments made per year. C25: Interest is 8% compounded quarterly. Initial deposit is RM100, and regular payments of RM150 will be made every semiannually. 0 1 2 3 8% -100 -150 -150 -150 -150 -150 -150 FV = ?
- An Example of Everything C26: Someone offers to sell you a note calling for the pmt of RM1000, 15 months from today for RM850. You have RM850 in the bank, which pays a 7% nominal rate with daily compounding. Should you buy the note or leave your money in the bank. iper = 7%/365 = 0.0192% 0 456 days -850 1,000 How to solve this? Have to compare both investments on similar grounds; Fvnote vs. FVbank PVnote vs. PVbank EARnote vs. EARbank
- Fvnote vs. Fvbank Bank : FV = 850 (1.000192)456 = 927.67 Note : FV = 1,000 Buy note (more value in future) PVnote vs. Pvbank Bank : PV = RM850 Note : PV = 1,000 (1.000192)-456 = 916.27 Buy note (more value now) EARnote vs. EARbank Bank : iper = 0.0192% Note : 1,000 = 850 (1 + i)456, solving i = 0.0356% Buy note (higher iper means higher EAR)
- C27: Cost of note = RM850 PMT = RM190 quarterly for 5 quarters inom= 7% compounded daily Is this a good investment? 0 91 182 274 366 456 days -850 190 190 190 190 190 iper for bank = 7% / 365 = 0.0192%
- PVAnote = 190 (1.000192)-91 + 190 (1.000192)-182 + … + 190 (1.000192)-456 = 901.68 PVApocket = 850 Buy note (more value now) EARnote ; finding iper; inom = (iper) (m) = (3.83) (4) = 15.3% So for daily rate = inom / 365 = 0.0419% Buy note coz iper,note > iper, bank = 0.0192% Quarterly iper
- Loan Types Pure Discount Loans - the borrower receives money today & repays a single sum at some time in the future - eg. A 1-year, 10% RM100 pure discount loan, would require the borrower to repay RM110 in one year. 2. Interest-Only Loans - a loan that has a repayment plan that calls for the borrower to pay interest each period & repay the entire principal (original loan amount) at some point in the future - eg. With a 3-year, 10%, interest-only loan of RM1000, the borrower would pay RM1000(0.1) = 100 in interest at the end of 1st & 2nd years. At the end of 3rd year, the borrower would return the principal along with RM100 in interest for that year.
- 3. Amortized Loans a loan that is repaid in equal payments over its life. - eg. Car & home loans C28: Say, borrow RM1,000 at 10% interest and have to pay equally at the end of each of the next 3 years. 0 1 2 3 10% -1,000 PMT PMTPMT F/C ; T/S ; PVAn = PMT (PVIFAi,n) 1,000 = PMT(PVIFA10%,3) PMT = 1,000 / (2.4869) = RM402.11
- Constructing an amortization table:Repeat steps 1 – 4 until end of loan (2) – (3) (1) – (4) PVn(1 + i) Interest paid declines with each payment as the balance declines.

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