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1 TIME VALUE OF MONEY FACULTY OF BUSINESS AND ACCOUNTANCY Week 5

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2 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

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3 Basic Concepts Time Lines Future Values Present Values Perpetuities Single Sum Annuity Nominal Rate Periodic Rate Effective Annual Rate Required Rate of Return Compounding Periods Amortization Types of Interest Rates

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4 TMV Solution Methods 1.Numerical – using regular calculator w/o financial functions 2.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 3.Financial Calculator 4.Worksheet

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5 Time Lines 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. 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. CF 0 CF 1 CF 3 CF 2 Time 0123 i% End of period 2 & beginning of period 3

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6 Time line illustrations i% % RM100 lump sum due in 2 years End of period 2 PV FV 3 year RM100 ordinary annuity (fixed pmts) 10%

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i% -50 Uneven cash flow stream CF 0 CF 1 CF 2 CF 3

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8 8 KeyDescription Clear all data No of payment per year No of year Annual interest rate Present value Future value (No keyed in) x (P/YR) Begin End Calculates amortization table C ALL P/YR PV FV I/YR N xP/YR BEG/END AMORT

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9 Future and Present Values

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10 Future Value & Present Value Future Value - Ending amount of your account at the end of n periods Present Value – Beginning amount in your account Future Value - Ending amount of your account at the end of n periods Present Value – Beginning amount in your account 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? The process of going from today’s values, or PV to future value is called compounding.

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11 Simple interest; nd period; (Principal) [(2 x 0.05) + 1 ] = RM110 or, [100 (1.05) - 100] + 100(1.05) = RM % % Compounding interest; nd period; (Principal + interest)(1 + i) (RM100 + RM5) ( ) = RM110.25

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12 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? 0 1 i% = ? 20,000 25,000 Given RM20,000 now, in 1 year at ir of 10%, the money deposited will be ? 20, % FV = ?

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13 FV 1 = PV + INT = PV + PV (i) = PV ( 1 + i ) FV 1 = PV + INT = PV + PV (i) = PV ( 1 + i ) Numerical Solution (N/S) ; FV 1 = PV ( 1 + i ) = 20,000 ( ) = RM22,000 FV 1 = PV ( 1 + i ) = 20,000 ( ) = RM22,000 Future Value Interest Factor for i & n (FVIF i,n ) ~ the future value of RM1 left on deposit for n periods at a rate of i percent per period ~ where ( 1 + i) n = FVIF i,n Tabular Solution (T/B) ; FV 1 = PV (FVIF i,n ) = 20,000 ( ) = RM22,000 FV 1 = PV (FVIF i,n ) = 20,000 ( ) = RM22,000

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14 Find the FV of RM20,000 given an interest rate of 10% in one year. DataKey Clear all data 1 No of payment per year 1No of year 10 Annual interest rate 20,000Present value 22, C All P/YR PV FV I/YR N + / -

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15 Present Value : One period case FV 1 = PV ( 1 + i ), so PV = FV 1 (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? 01 PV = ? 10% 25,000

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16 Numerical solution ; PV = FV 1 / (1 + i) = 25,000 = RM22, PV = FV 1 / (1 + i) = 25,000 = RM22, Tabular Solution ; Present Value Interest Factor for i & n (PVIF i,n ) ~ the present value of RM1 due n periods in the future discounted at i percent per period PV = FV (PVIF i,n ) = 25,000 (0.9091) = RM 22, ~ where;

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17 Find the PV of RM25,000 given an interest rate of 10% in one year. DataKey Clear all data 1 Payment per year 1 10 Annual interest rate 25, C ALL P/YR FV PV I/YR N

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18 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 01 2 discounting 5% compounding

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19 The discount rate is often also referred to as the opportunity cost, the required return, and the cost of capital.

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20 C3 : Find the FV o RM100 left for 3 years in an account paying 10 percent, annual compounding; FV = ? % 100 FV 1 = PV + INT = PV + PV (i) = PV ( 1 + i ) FV 1 = PV + INT = PV + PV (i) = PV ( 1 + i ) FV 2 = FV 1 (1 + i) = PV ( 1 + i ) (1 + i) = PV (1 + i) 2 FV 3 = PV (1 + i) 3 FV 2 = FV 1 (1 + i) = PV ( 1 + i ) (1 + i) = PV (1 + i) 2 FV 3 = PV (1 + i) 3 N/S; FV 3 = 100 (1.10) 3 =

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21 DataKey C ALL P/YR PV FV I/YR N FV n = PV (1 + i) n = PV (FVIF i,n ) = 100 (1.10) 3 = 100 (1.3310) = RM = 100 (1.10) 3 = 100 (1.3310) = RM /-

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22 C4 : Find the PV of RM100 to be received in 3 years if the appropriate ir is 10 percent, compounded annually: % PV = ? PV n = FV (1 + i) n PV 3 = FV (1 + i) 3 PV n = FV (1 + i) n PV 3 = FV (1 + i) 3 N/S; PV n = FV n / (1 + i) n = FV n 1 n = FV n (PVIF i,n ) 1 + i PV n = FV n / (1 + i) n = FV n 1 n = FV n (PVIF i,n ) 1 + i = 100 (1/1.10) 3 = 100 (0.7513) = RM75.13 = 100 (1/1.10) 3 = 100 (0.7513) = RM75.13 T/S;

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23 DataKey C ALL P/YR FV PV I/YR N

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24 n (periods) and i (interest rate)

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25 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? 0 15% n = ? n - 1 RM1 RM2 FV n = PV (1 + i) n 2 = 1 (1.15) n 2 = (1.15) n FV n = PV (1 + i) n 2 = 1 (1.15) n 2 = (1.15) n FV n = PV (FVIF i,n ) (FVIF i,n ) = FV / PV = 2 / 1 = 2 FV n = PV (FVIF i,n ) (FVIF i,n ) = FV / PV = 2 / 1 = 2 T/S; Look in FVIF Table for (FVIF 15%,n ) = 2 n 5 periods N/S;

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% n = ? n - 1 RM1 RM2 Financial Calculator Solution ; DataKey C ALL P/YR FV PV I/YR N +/-

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27 Solving for interest rate C6: What annual ir will cause RM100 to grow to RM in 3 years? i = ? 100 T/S; 100 (1 + i)100 (1 + i) (1 + i) (1 + i) 3 = (FVIF i,3 ) = FVIF i,3 = (1 + i) 3 = (FVIF i,3 ) = FVIF i,3 = Look at Row 3 of FVIF Table is in the 8% column

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28 Annuities & Perpetuities

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29 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

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30 What is the difference between an ordinary annuity and an annuity due? Both are 3-yr annuities ( 3 pmts) Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due

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31 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; % (1 + i) 2 100(1 + i) + Twice compounding

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32 N/S; FVA 3 = PMT (1 + i) + PMT (1 + i) 1 + PMT (1 + i) 2 = 100 (1) (1.10) (1.21) = RM331 FVA 3 = PMT (1 + i) + PMT (1 + i) 1 + PMT (1 + i) 2 = 100 (1) (1.10) (1.21) = RM331 T/S; FVA n = PMT (FVIFA i,n ) FVA 3 = 100 (FVIFA 10%,3 ) = 100 (3.3100) = RM331 FC; = Make sure no BGN sign PMTI/YRNFVP/YR 1

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33 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? Time line approach; % (1 + i) 2 100(1 + i) + Triple compounding (1 + i)

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34 N/S; FVAD 3 = PMT (1 + i) 3 + PMT (1 + i) 2 + PMT (1 + i) 1 = 100 (1.331) (1.21) (1.10) = RM FVAD 3 = PMT (1 + i) 3 + PMT (1 + i) 2 + PMT (1 + i) 1 = 100 (1.331) (1.21) (1.10) = RM T/S; FVAD n = FVA 3 (1 + i) or = PMT (FVIFA 10%,3 ) (1 + i) = 331 (1.10) = 100 (3.3100) (1.10) = RM = FC; = Make sure BGN sign P/YR PMTI/YRN 1 FV BEG/END

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35 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; % /(1 + i) / (1 + i) Triple discounting / (1 + i)

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36 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; % /(1 + i) / (1 + i) Double discounting Make sure BGN sign

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37 An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

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38 Perpetuities - is a stream of equal payments expected to continue forever - a type of annuity PV (Perpetuity) = payment = PMT interest rate i PV (Perpetuity) = payment = PMT interest rate i the current price

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39 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 = RM PV (Perpetuity) = RM100 = RM As the interest rate increases, the perpetuity’s value will drop. When ir = 10%; PV p = 100 = RM When ir = 10%; PV p = 100 = RM

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40 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 Examples of uneven cash flows; - common stock’s dividend - returns from fixed asset investments ~ production income ~ rentals

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41 Present Value of an Uneven Cash Flow Stream C12: Find PV of the following cash flows stream, discounted at 10% % PV = CF CF CF i 1 + i 1 + i + CF CF i 1 + i CF 0 CF 1 CF 2 CF 3 CF 4

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42 For cash flow calculation ; Key Clear all No of periods per year Cash flow j No of consecutive times CFj occurs Internal rate of return per year Net present value C ALL P/YR Nj CFj IRR/YR NPV

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43 Key C ALL I/YR CFj NPV CFj +/- 43 Key C ALL I/YR CFj NPV CFj Nj +/- CFj

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% C13: Find the PV of the following c/f discounted at 10% (PVIFA 10%,3 ) = 200 (2.4869) 100(1/1 + i) 1 50(1/1 + i) (1/1 + i) 2 or 200 (PVIFA 10%,3 ) or 200 (1/1 + i) (PVIFA 10%,4 ) or 200 (1/1 + i) (PVIFA 10%,5 ) or 200 (1/1 + i)

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45 Future Value of an Uneven Cash Flow Stream C14: Find FV of the following cash flows stream,compounded at 10% % FV = CF 5 (1 + i) 0 + CF 4 (1 + i) 1 + CF 3 (1 + i) 2 + CF 2 (1 + i) 3 + CF 1 (1 + i) 4 + CF 0 (1 + i) 5 FV = CF 5 (1 + i) 0 + CF 4 (1 + i) 1 + CF 3 (1 + i) 2 + CF 2 (1 + i) 3 + CF 1 (1 + i) 4 + CF 0 (1 + i) 5 CF 0 CF 1 CF 2 CF 3 CF 4 CF 5

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% ( 1 + i) 200 (FVIFA 10%,2 ) = 200 (2.1) 200 (1 + i) (1 + i) 50 (1 + i) 3 = 50 (1.331) 100 (1 + i) 4 = 100 (1.4641) NFV =

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% DataKey C ALL P/YR I/YR CFj CF 0 CF 1 CF 2 CF 3 CF 4 CF 5 NFV=? DataKey CFj Nj CFj NPV N PMT FV CFj Using formula; NFV = NPV ( 1 + i) n = ( ) 5 = Using formula; NFV = NPV ( 1 + i) n = ( ) 5 =

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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?

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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 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?

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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 3 rd year? P/YR 1 PV(-)100 n 3 I% 6 FV P/YR12 PV(-)100 N 66 I% 6/26 FV % Annual compounding Semiannual compounding i% FV 3 = PV (1 + i) 3 = 100 ( ) 3 = RM FV 3 = PV (1 + i) 3 = 100 ( ) 3 = RM FV = ? FV 6 = PV (1 + i) 6 = 100 ( ) 6 = RM FV 6 = PV (1 + i) 6 = 100 ( ) 6 = RM i% = 6% /2 = 3% n = 3 x 2 = 6

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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)

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1. Nominal or Quoted interest rates, i nom 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% APR = 1.2% x 12 = 14.4% It is the nominal rates for loan that some government requires the bank to display to customers.

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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, i per = i nom / m = 6% / 4 = 1.5% Periodic Rate

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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? i per = i nom / 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?

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Notice that i per 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 2 nd 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.

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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 Semiannual compounding % i = ? EAR ; The annual rate that produces the same FV as if we had compounded at a given periodic rate m times per year

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An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually. Why? C19: EFF% or EAR for 10% semiannual investment.

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C20: What is the FV of RM100 compounded semiannually for 3 years if i nom = 10%? Would it be different if it were compounded quarterly? Quarterly compounding Semiannual compounding % -100 FV = ? FV = ? 2.5%

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FV n = PV 1 + i nom mn m FV 3 = (3) 2 = 100(1.05) 6 = EAR = ( 1 + i nom / m ) m - 1 = ( / 2 ) 2 – 1 = 10.25% Semiannual compounding Quarterly compounding FV n = PV 1 + i nom mn m FV 3 = (3) 4 = 100(1.025) 12 = EAR = ( 1 + i nom / m ) m - 1 = ( / 4 ) 4 – 1 = 10.38%

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F/C ; Given annual i.r. of 10%, compounded semiannually; To find APR, given the EAR of 10.25%; The APR formula; APR = 1 + EFF 1/m - 1 x m x APR = 1 + EFF 1/m - 1 x m x NOM% 10 2 P/YR EFF% EAR NOM% P/YR EFF%

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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.

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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 > i nom EAR = i nom = i per If compounding occurs only once a year, then;

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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) EAR= 19.56%

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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. EAR ANNUAL 10.00% EAR QUARTERLY 10.38% EAR MONTHLY 10.47% EAR DAILY (365) 10.52%

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Fractional Time Periods Before, payments only occur at beginning or end of periods. What if, payments occur at some date within a period? 0 1 st 2 nd month 20 th day month 1% -100 FV = ?

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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; FV n = PV (1 + i) n = PV (1 + i) 1+ 20/30 = 100( ) 1.67 = RM FV n = PV (1 + i) n = PV (1 + i) 1+ 20/30 = 100( ) 1.67 = RM F/C ; N20÷30+1 I/YR1 PV(-)100 P/YR1 FV If use 360-day year; i per = 0.12 /360 = per day No of days deposited; = 50 days FV n = PV (1 + i per ) n = 100 ( ) 50 = RM

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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? % FV = ?

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Rule 2 : If there is only PV & no PMT, either; a.If use periodic ir, keep P/YR = 1. b.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? % -100 FV = ? N3x2 = 6 I/YR10÷2 = 5 PV(-)100 PMT0 P/YR1 FV N3x2 = 6 I/YR10 PV(-)100 PMT0 P/YR2 FV134.01

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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 %8% FV = ? C25: Interest is 8% compounded quarterly. Initial deposit is RM100, and regular payments of RM150 will be made every semiannually. 2P/YR 3x2 = 6N 8.08I/YR (-)100PV (-)150PMT FV 8NOM% 4P/YR 8.24I/YR 8.24EFF% 2P/YR 8.08NOM%

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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. An Example of Everything days ,000 i per = 7%/365 = % How to solve this? Have to compare both investments on similar grounds; Fv note vs. FV bank PV note vs. PV bank EAR note vs. EAR bank

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Fv note vs. Fv bank Bank : FV = 850 ( ) 456 = Note : FV = 1,000 Buy note (more value in future) PV note vs. Pv bank Bank : PV = RM850 Note : PV = 1,000 ( ) -456 = Buy note (more value now) EAR note vs. EAR bank Bank : i per = % Note : 1,000 = 850 (1 + i) 456, solving i = % Buy note (higher i per means higher EAR)

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C27: Cost of note = RM850 PMT = RM190 quarterly for 5 quarters i nom = 7% compounded daily Is this a good investment? C27: Cost of note = RM850 PMT = RM190 quarterly for 5 quarters i nom = 7% compounded daily Is this a good investment? days i per for bank = 7% / 365 = %

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PVA note = 190 ( ) ( ) … ( ) -456 = PVA pocket = 850 Buy note (more value now) EAR note ; finding i per ; i nom = (i per ) (m) = (3.83) (4) = 15.3% So for daily rate = i nom / 365 = % Buy note coz i per,note > i per, bank = % -850CF j 190CF j 5NjNj IRR Quarterly i per

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Loan Types 1.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 1 st & 2 nd years. At the end of 3 rd year, the borrower would return the principal along with RM100 in interest for that year.

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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 ,000 PMT PMT PMT 10% T/S ; PVA n = PMT (PVIFA i,n ) 1,000 = PMT(PVIFA 10%,3 ) PMT = 1,000 / (2.4869) = RM F/C ; 3 N 10 I/YR PV 0 FV PMT402.11

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Constructing an amortization table: Repeat steps 1 – 4 until end of loan Interest paid declines with each payment as the balance declines. YearBeginning Balance (1) PMT (2) Interest (3) Princip Repmt (4) End Balance 1RM1,000RM402RM100RM302RM Total1, ,000- PV n (1 + i) (2) – (3)(1) – (4)

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77 1P/YR 3 N 10 I/YR PV 0 FV PMT INPUT AMORT1-1 = PRIN = INT = BAL 1INPUT2 AMORT1-2 = PRIN = INT = BAL

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