CHAPTER 2 Time Value of Money Future value Present value Annuities Rates of return Amortization

What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%? Finding the FV of a cash flow or series of cash flows is called compounding. FV = ? (121x1.1) =133.10 1 2 3 10% 100 (100x1.1)1 =110 (110x1.1)1 =121

Solving for FV: The step-by-step and formula methods After 1 year: FV1 = PV (1 + I) = $100 (1.10) = $110.00 After 2 years: FV2 = PV (1 + I)2 = $100 (1.10)2 =$121.00 After 3 years: FV3 = PV (1 + I)3 = $100 (1.10)3 =$133.10 After N years (general case): FVN = PV (1 + I)N

FVn= PV (1+I)n FV3= PV (1+I)3 =100 (1+10%)3 =100 (1.10)3 =$133.10

or, using FVF table (A1) FV3= PV (Fk% VFn) = 100 (F10% VF3) = 100 (1.331) = $ 133.10

Time lines Show the timing of cash flows. 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.

Solving for FV: The calculator method Solves the general FV equation. Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.) 3 10 -100 INPUTS N I/YR PV PMT FV OUTPUT 133.10

What is the present value (PV) of $100 due in 3 years, if I/YR = 10%? Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding). The PV shows the value of cash flows in terms of today’s purchasing power. 1 2 3 10% PV = ? 82.64/(1.1)1 =75.13 100/(1.1)1 =90.91 90.91/(1.1)1 =82.64 100

Solving for PV: The formula method Solve the general FV equation for PV: PV = FVN / (1 + I)N PV = FV3 / (1 + I)3 = $100 / (1.10)3 = $75.13

FV = PV (1+k)n 100 = PV (1+10%)3 100 = PV (1.1)3 75.13 = PV

or, using table (A-2) PV= FV (Pk% VFn) = 100 (P10% VF3) = 100 (0.7513) = 75.13

Solving for PV: The calculator method Solves the general FV equation for PV. Exactly like solving for FV, except we have different input information and are solving for a different variable. 25 10 100 INPUTS N I/YR PV PMT FV OUTPUT -75.13

Solving for I: What interest rate would cause $100 to grow to $125 Solving for I: What interest rate would cause $100 to grow to $125.97 in 3 years? Solves the general FV equation for I. Hard to solve without a financial calculator or spreadsheet. 3 -100 125.97 INPUTS N I/YR PV PMT FV OUTPUT 8

FV = 125.97 1 2 3 PV=100 n = 3 i = ? PV= 100 FV= 125.97

Solving for N: If sales grow at 20% per year, how long before sales double? Solves the general FV equation for N. Hard to solve without a financial calculator or spreadsheet. 20 -200 400 INPUTS N I/YR PV PMT FV OUTPUT 3.8

FV = 400 ? PV=200 i = 20% PV= 200 FV= 400 n = ? ?

a) Based on the following arrangements, what interest rates are charged by the lending institution to the borrower. You borrow RM 500 and repay RM 551 in two years. FV = PV ( F V F ) n=2 k =? 551 = 500 ( F2 V Fk% ) 551 = ( F2 V Fk% ) 1.10 = ( F2 V Fk% ) k = 5 % from table A-1 PV = 500 FV = 551 n = 2 Pmt = 0 I/yr = ?

b) Borrow RM 100,000 and repay RM 146,932 in 5 years 146,932 = 100,000 ( F V F ) n = 5 k = ? 146,9325 = ( F5 V F k% ) 100,000 1.46 = ( F5V Fk% ) k = 8 % (see table A-1) c) Borrow RM 100,000 and repay RM 300,000 in 10 years 300,000 = 100,000 ( F V F ) n = 10 k = ? 300,000 = ( F10 V Fk% ) 3 = ( F10 V Fk% ) k = 11.6% PV = -100,000 FV = 146,932 n = 5 Pmt = 0 I/yr = ? PV = -100,000 FV = 300,000 n = 10 Pmt = 0 I/yr = ?

Finding Implied Interest And Number Of Factoring Years How long does it take for the followings to happen a) RM 856 to grow into RM1,122 at 7 % FV = PV ( Fk% V Fn ) RM 1,122 = RM 856 ( F 7%V Fn=? ) RM 1,122 = ( F7% V Fn=? ) RM 856 1.31 = ( F7% V Fn=? ) 4 = n (use table A1) b) RM10,000 to grow into RM100,000 at 8 % 100,000 = 10,000 ( F V F ) 100,000 = ( F8% V Fn=? ) 10,000 10 = ( F8% V Fn=? ) n = 30 (use table A1) PV = -856 FV = 1,122 k = 7% Pmt = 0 n = ? PV = -10,000 FV = 100,000 k = 8% Pmt = 0 n = ?

c) RM10,000 to grow into RM200,000 at 8 % 200,000 = 10,000 ( F V F ) 200,000 = ( F8% V Fn=? ) 10,000 20 = ( F8% V Fn=? ) n = 38 (use table A1) PV = -10,000 FV = 200,000 k = 7% Pmt = 0 n = ?

Example of present value problem 1. En. Kamal is planning ahead for his son’s education. The boy is eight now and will start college in 12 years time. How much must he set aside now to have RM 100,000 when his son starts schooling. The interest rate is 8 % k = 8% n= 12 years FV = RM 100,000 PV = ? FV = PV ( 1 + k )n PV = RM 100,000 (1 + 0.08)12 = RM 100,000 2.518 = RM 39,711 FV = 100,000 k = 8% Pmt = 0 n = 12 PV = ?

or, using present value factor table PV = FV ( P V F ) K% n PV = RM 100,000 ( P V F ) 8 % 12 PV = RM 100,000 ( 0.3971 ) from Table A-2 PV = RM 39,710 ~ RM 39,71

Drawing time lines 100 (100x1.1)1 =110 (110x1.1)1 =121 1 2 3 1 2 3 I% FV 3 year $100 ordinary annuity @ 10% Σ 100 + 110 + 121 = 331

Solving for FV: 3-year ordinary annuity of $100 at 10% $100 payments occur at the end of each period, but there is no PV. 3 10 -100 INPUTS N I/YR PV PMT FV OUTPUT 331

Problems of future value annuity (for all future value annuity problems, please refer to table A-3 ) 1. Assume that you have deposited RM100 each year in a bank account that pays 15% interest rate. How much will you have in your account at the end of the fifth year? F VF = P m t ( F V F A ) k% n k = 15% n = 5 F VF A = 100 ( F 15%V F A 5 ) = 100 (6.7424 ) (use table A-3) F V A = 674.24 PV = 0 k = 15% Pmt = -100 n = 5 FV = ?

2. Assume you have deposited RM200 each year in a bank account that pays 20% interest rate. How much will you have in your account after 25 years. F VF = P m t ( F V F A ) k% n k = 20% n =2 5 F VF A = 200 ( F 20% V F A25 ) = 200 (471.98 ) (use table A-3) F V A = 94,396 PV = 0 k = 20% Pmt = -200 n = 25 FV = ?

3. Assume you have deposited RM1000 each year in a bank account that pays 20% interest rate. How much will you have in your account at the end of the 25 year F VF = P m t ( F V F A ) k % n k = 20% n =2 5 F VF A = 1000( F 20%V F A 25 ) = 1000 (471.98 ) F V A = 471,980 PV = 0 k = 20% Pmt = -1,000 n = 25 FV = ?

Drawing time lines 100 100/(1.1)1 =90.91 1 2 3 1 2 3 I% PV 3 year $100 ordinary annuity @ 10% 100/(1.1)1 =90.91 90.91/(1.1)1 =82.64 82.64/(1.1)1 =75.13 100 100/(1.1)1 =90.91 90.91/(1.1)1 =82.64 Σ 90.91 + 82.64 + 75.13 = 248.68

Solving for PV: 3-year ordinary annuity of $100 at 10% $100 payments still occur at the end of each period, but now there is no FV. 3 10 100 INPUTS N I/YR PV PMT FV OUTPUT -248.69

Example: We would like to know what is the present value of four payments in the amount of RM1,000 each to be received in the next four years. The current market interest rate is 10% P V F = P m t ( P V F A ) Pmt = RM1,000, k%= 10, n = 4 k % n PVF = 1000 (P10%VFA4) (please refer to table A-4 for this value) PVF = 1000 (3.1699) PVF = 3,169.9

What is the difference between an ordinary annuity and an annuity due? PMT 1 2 3 i% PMT 100 100(1.1) =110 100(1.1)1 110 (1.1)1 =121 1 2 3 i% Annuity Due 100(1.1)1 110 (1.1)1 121 (1.1)1

Solving for FV: 3-year annuity due of $100 at 10% Now, $100 payments occur at the beginning of each period. FVAdue= FVAord(1+I) = $331(1.10) = $364.10. Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity: BEGIN 3 10 -100 INPUTS N I/YR PV PMT FV OUTPUT 364.10

FV Annuity due 3 year annuity due of $100 at 10%, n= 3, Pmt=100, k=10%, FV=? FVA = Pmt (F10% VFA3) = 100 (3.31) table A-3 FVA = 331 (1 + 10%) FVAdue = 331 (1.10) = 364.10

Present value annuity due The process of discounting the future payment back to the present time in the case of Present value annuity due is different than the normal present value annuity. The figure below illustrate the difference Present value annuity due P m t 1 P m t 2 P m t 3 PVA 1/(1 + k)1 1/ (1 + k)2 Present value annuity 0 P m t 1 P m t 2 P m t 3 PVA 1/(1 + k)1 1/(1 + k)2 1/(1 + k)3

Because the first payment took place at the beginning of year one and not the end of year one, that payment will be discounted one less period. Hence that factor has to be taken into consideration when we compute the present value annuity due amount. Illustrated below, we have added (1+k) factor to the original formula use to compute an ordinary present value annuity amount P V A due = P m t ( P V F A ) ( 1 + k ) Annuity due factor N K%

PV Annuity due 3 year annuity due of $100 at 10% n= 3, Pmt= 100, k=10%, PV= ? PVA = Pmt ( PV10%FA3) = 100 ( 2.486 ) table A-4 = 248.6 PVAdue = 248.6 ( 1.10 ) = 273.46

Solving for PV: 3-year annuity due of $100 at 10% Again, $100 payments occur at the beginning of each period. PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55. Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity: BEGIN 3 10 100 INPUTS N I/YR PV PMT FV OUTPUT -273.55

What is the present value of a 5-year $100 ordinary annuity at 10%? Be sure your financial calculator is set back to END mode and solve for PV: N = 5, I/YR = 10, PMT = 100, FV = 0. PV = $379.08

Present value annuity Pmt =100, n =5, i =10%, FV =0, PV =? PVA = 100 ( PVk% FAn ) = 100 ( PV10% FA5 ) = 100 ( 3.7908 ) Table A-4 PVA = 379.08

What if it were a 10-year annuity? A 25-year annuity? A perpetuity? N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $614.46. 25-year annuity N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $907.70. Perpetuity PV = PMT / I = $100/0.1 = $1,000.

A perpetuity ∞ 1 100 2 ∞ PV =Σ ? Pmt = 100 i = 10 PV = ? Pmt = 100 = 1,000

Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant? LARGER, as the more frequently compounding occurs, interest is earned on interest more often. Yr= 1 Yr= 2 Yr= 3 10% 100 133.10 Annually: N= 3, PV= -100, I/YR= 10, PMT=0, FV =133.10 1 2 3 5% 4 5 6 134.01 Yr= 1 Yr= 2 Yr= 3 100 Semiannually: N= 2x3, PV= -100, I/YR= 10/2, PMT=0, FV =134.01

Quaterly: N= 4x3, PV= 100, I/yr= 10/4, FV=. = 12 = 2. 5 134 Quaterly: N= 4x3, PV= 100, I/yr= 10/4, FV= ? = 12 = 2.5 134.49 Monthly: N= 12x3, PV= 100, I/yr= 10/12, FV= ? = 36 = 0.833 134.80

Will the PV of a lump sum be larger or smaller if discounted more often, holding the stated I% constant Annually FV= 100, I/yr= 10%, n= 3, PV= ? 75.13 Semiannually FV= 100, I/yr= 10%/2, n= 3x2, PV= ? = 5 = 6 74.62

Quarterly FV= 100, I/yr= 10%/4, n= 3x4, PV=. = 2. 5 = 12 74 Quarterly FV= 100, I/yr= 10%/4, n= 3x4, PV= ? = 2.5 = 12 74.36 Monthly FV= 100, I/yr= 10%/12, n= 3x12, PV= ? = 0.833 = 36 74.18

Yr 1 Yr 1 Yr 1 3 100 1 2 “ “ 100 (1+i )n Yr 1 Yr2 Yr 3 1 2 3 4 5 6 100 “ “ “ “ “ 100 (1+i )n

The Power of Compound Interest A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer. At the end of the year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%. How much money will she have when she is 65 years old?

20 21 22 23 64 65 1,095 1,095 1,095 1,095 1,095 1,095 n= 65-20 = 45 i / yr = 12% PV = 0 Pmt = 1,095 FV = ?

Solving for FV: If she begins saving today, how much will she have when she is 65? If she sticks to her plan, she will have $1,487,261.89 when she is 65. INPUTS OUTPUT N I/YR PMT PV FV 45 12 -1095 1,487,262

Solving for FV: If you don’t start saving until you are 40 years old, how much will you have at 65? If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20. Lesson: It pays to start saving early. 25 12 -1095 INPUTS N I/YR PV PMT FV OUTPUT 146,001

40 41 42 43 44 64 65 1,095 1,095 1,095 n= 65-40 = 25 i / yr = 12% PV = 0 Pmt = ? FV = 146,000.59

Solving for PMT: How much must the 40-year old deposit annually to catch the 20-year old? To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT. 25 12 1,487,262 INPUTS N I/YR PV PMT FV OUTPUT -11,154.42

40 41 42 43 44 64 65 ? ? ? n= 65-40 = 25 i / yr = 12% PV = 0 Pmt = ? FV = 1,487,261.89

The Effective Annual rate(EAR) Example Borrow: RM 100, Payback time: 1yr @ 12% Compounding Final balance Annual 112.00 Semiannual 112.36 Quarterly 112.55 Monthly 112.68

When is each rate used? INOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on timelines. EAR Used to compare returns on investments with different compounding periods. i.e. A credit card holder must pay 18% monthly compounding interest on the amount outstanding. What is the EAR of the credit card?

Effective Annual Rate (EAR) EAR = ( 1 + I nom )m – 1.0 m = (1 + 0.18/12 )12 – 1.0 = 0.1956 ≈ 19.56% ie: A credit card holder must pay 18% monthly compounding interest on the amount outstanding. What is the EAR of the credit card? I nom = 18% annually m = Compounding period

Loan amortization Amortization tables are widely used for home mortgages. Ie. A home owner borrows, RM 200,000 on a mortgage loan. The loan is to be repaid in 180 ( 12months x 15 years) equal payments at the end of each month. The bank charges 12% on the loan per year ( 1% monthly). What is the monthly repayment?

? ? ? ? ? ? 200,000 Pmt 1 Pmt 2 Pmt 3 Pmt 178 Pmt 179 Pmt 180 n= 12 months x 15 = 180 i / yr = 12% / 12 = 1% PV = 200,000 FV = 0 Pmt = ?

Step 1: Find the required annual payment All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan. 180 1 -200,000 INPUTS N I/YR PV PMT FV OUTPUT 2,400