eee Quantitative Methods I
THE TIME VALUE OF MONEY Level I 2
Learning Objectives Understand the importance of the time value of money Understand the difference between simple interest and compound interest Know how to solve for present value, future value, time or rate Understand annuities and perpetuities Know how to construct an amortization table RIFT/CFA/Level 1/Quantitative methods/ss2 1
Important Terms Amortize Annuity Annuity due Basis point Cash flows Compound interest Compound interest factor (CVIF) Discount rate Discounting Effective rate Lessee Medium of exchange Mortgage Ordinary annuities Perpetuities Present value interest factor (PVIF) Reinvested Required rate of return Simple interest Time value of money RIFT/CFA/Level 1/Quantitative methods/ss2 2
Types of Calculations Ex Ante: Calculations done before-the-fact It is a forecast of what might happen All forecasts require assumptions It is important to understand the assumptions underlying any formula used to ensure that those assumptions are consistent with the problem being solved. As a forecast, while you may be able to calculate the answer to a high degree of accuracy it is probably best to round off the answer so that users of your calculations are not misled. Ex Post: Calculation done after-the-fact It is an analysis of what has happened It is usually possible, and perhaps wise to express the result as accurately as possible. RIFT/CFA/Level 1/Quantitative methods/ss2 3
The Time Value of Money Concept Cannot directly compare $1 today with $1 to be received at some future date Money received today can be invested to earn a rate of return Thus $1 today is worth more than $1 to be received at some future date The interest rate or discount rate is the variable that equates a present value today with a future value at some later date RIFT/CFA/Level 1/Quantitative methods/ss2 4
Opportunity Cost Opportunity cost = Alternative use The opportunity cost of money is the interest rate that would be earned by investing it. It is the underlying reason for the time value of money Any person with money today knows they can invest those funds to be some greater amount in the future. Conversely, if you are promised a cash flow in the future, it s present value today is less than what is promised! RIFT/CFA/Level 1/Quantitative methods/ss2 5
Choosing from Investment Alternatives You have three choices: 1. $20,000 received today 2. $31,000 received in 5 years 3. $3,000 per year indefinitely To make a decision, you need to know what interest rate to use. This interest rate is known as your required rate of return or discount rate. RIFT/CFA/Level 1/Quantitative methods/ss2 6
Simple Interest Simple interest is interest paid or received on only the initial investment (or principal). At the end of the investment period, the principal plus interest is received. 0 1 2 3 n I 1 I 2 I 3 I n +P RIFT/CFA/Level 1/Quantitative methods/ss2 7
Simple Interest Example PROBLEM: Invest $1,000 today for a fiveyear term and receive 8 percent annual simple interest. How much will you accumulate by the end of five years? Year Beginning Amount Ending Amount 1 $1,000 $1,080 2 1,080 1,160 3 1,160 1,240 4 1,240 1,320 5 1,320 $1,400 Value (time n) P (n P k) Value 5 $1,000 (5$1,000.08) $1,000 (5$80) $1,000 $400 $1,400 RIFT/CFA/Level 1/Quantitative methods/ss2 8
Simple Interest General Formula [1] Value (time n) P (n P k) Where: P = principal invested n = number of years k = interest rate RIFT/CFA/Level 1/Quantitative methods/ss2 9
Compound Interest Compound interest is interest that is earned on the principal amount invested and on any accrued interest. RIFT/CFA/Level 1/Quantitative methods/ss2 10
Compound Interest Example PROBLEM: Invest $1,000 today for a five-year term and receive 8 percent annual compound interest. How much will the accumulated value be at time 5. The solution in one simple step : FV PV ( 1 k) n 0 FV 5 n $1,000(1.08) 5 $1,469.33 RIFT/CFA/Level 1/Quantitative methods/ss2 11
Compound Interest Example PROBLEM: Invest $1,000 today for a five-year term and receive 8 percent annual compound interest. The Interest-earned-on-Interest Effect: Interest (year 1) = $1,000.08 = $80 Interest (year 2 ) =($1,000 + $80).08 = $86.40 Interest (year 3) = ($1,000+$80+$86.40).08 = $93.31 Year Beginning Amount Ending Amount Interest earned in the year 1 $1,000.00 $1,080.00 $80.00 2 1,080.00 1,166.40 $86.40 3 1,166.40 1,259.71 $93.31 4 1,259.71 1,360.49 $100.78 5 1,360.49 1,469.33 $108.84 RIFT/CFA/Level 1/Quantitative methods/ss2 12
Compound Interest General Formula [2] FVn PV 0( 1 k) n Where: FV= future value P = principal invested n = number of years k = interest rate RIFT/CFA/Level 1/Quantitative methods/ss2 13
Compound Interest General Formula [2] FVn PV 0( 1 k) n ( 1 k) n is known as the compoundinterest factor CVIF RIFT/CFA/Level 1/Quantitative methods/ss2 14
DOLLARS Compound Interest Compounding of interest magnifies the returns on an investment. Returns are magnified: The longer they are compounded The higher the rate they are compounded 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Simple Compound RIFT/CFA/Level 1/Quantitative methods/ss2 15
Compound Interest Compounding of interest magnifies the returns on an investment. Returns are magnified: The longer they are compounded The higher the rate they are compounded Table 1: Ending Wealth of $1,000 Invested From 1938 to 2005 in Various Asset Classes Annual Arithmetic Average (%) Annual Geometric Mean (%) Yeark-End Value, 2005 ($) Government of Canada treasury bills 5.20 5.11 $29,711 Government of Canada bonds 6.62 6.24 61,404 Canadian stocks 11.79 10.60 9,46,009 U.S. stocks 13.15 11.76 19,23,692 Source: Data are from the Canadian Institute of Actuaries RIFT/CFA/Level 1/Quantitative methods/ss2 16
Compound Interest Input the following variables: 0 ; -1,000 ; 10 ; and 5 PMT PV I/Y N Press CPT (Compute) and then FV PMT refers to regular payments FV is the future value I/Y is the period interest rate N is the number of periods PV is entered with a negative sign to reflect investors must pay money now to get money in the future. Answer = $1,610.51 RIFT/CFA/Level 1/Quantitative methods/ss2 17
Compound Interest Underlying Assumptions Notice the compound interest assumptions that are embodied in the basic formula: FV 2 = $1,000 (1+k 1 ) (1+k 2 ) FV n = PV 0 (1+k) n Assumptions: The rate of interest does not change over the periods of compound interest Interest is earned and reinvested at the end of each period The principal remains invested over the life of the investment The investment is started at time 0 (now) and we are determining the compound value of the whole investment at the end of some time period (t= 1, 2, 3, 4, ) RIFT/CFA/Level 1/Quantitative methods/ss2 18
Compound Interest Time = 0 Time = 1 Time = 2 Time = 0 Time = 1 Time = 2 Time of Investment RIFT/CFA/Level 1/Quantitative methods/ss2 19
Compound Interest Formula FV n =PV 0 (1+k) n Where: FV n = the future value (sum of both interest and principal) of the investment at some time in the future PV 0 = the original principal invested k= the rate of return earned on the investment n = the time or number of periods the investment is allowed to grow RIFT/CFA/Level 1/Quantitative methods/ss2 20
CVIF k,n Tables of Compound Value Interest Factors can be created: CVIF k 5%, n10 years (1.05) 1.6289 10 Period 1% 2% 3% 4% 5% 6% 7% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 RIFT/CFA/Level 1/Quantitative methods/ss2 21
CVIF k,n The table shows that the longer you invest the greater the amount of money you will accumulate. It also shows that you are better off investing at higher rates of return. Period 1% 2% 3% 4% 5% 6% 7% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 RIFT/CFA/Level 1/Quantitative methods/ss2 22
CVIF k,n How long does it take to double or triple your investment? At 5%...at 10%? Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105 6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716 7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487 8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.1436 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3579 10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937 11 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531 12 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384 13 1.1381 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523 14 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975 15 1.1610 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.1772 16 1.1726 1.3728 1.6047 1.8730 2.1829 2.5404 2.9522 3.4259 3.9703 4.5950 RIFT/CFA/Level 1/Quantitative methods/ss2 23
The Rule of 72 If you don t have access to time value of money tables or a financial calculator but want to know how long it takes for your money to double use the rule of 72! Number of 72 years to double Annual compoundinterest rate If you expect to earn a 4.5% rate on your money it will 72 4.5 16 years double in : RIFT/CFA/Level 1/Quantitative methods/ss2 24
FV of $1.00 The Rule of 72 Let us predict what happens with an investment if it is invested at 5% show the accumulated value after t=1, t=2, t=3, etc. Period 1% 2% 3% 4% 5% 1 1.0100 1.0200 1.0300 1.0400 1.0500 2 1.0201 1.0404 1.0609 1.0816 1.1025 3 1.0303 1.0612 1.0927 1.1249 1.1576 4 1.0406 1.0824 1.1255 1.1699 1.2155 5 1.0510 1.1041 1.1593 1.2167 1.2763 6 1.0615 1.1262 1.1941 1.2653 1.3401 7 1.0721 1.1487 1.2299 1.3159 1.4071 8 1.0829 1.1717 1.2668 1.3686 1.4775 9 1.0937 1.1951 1.3048 1.4233 1.5513 10 1.1046 1.2190 1.3439 1.4802 1.6289 FV 1.8000 1.6000 1.4000 1.2000 1.0000 0.8000 0.6000 0.4000 0.2000 0.0000 1 2 3 4 5 6 7 8 9 10 Year RIFT/CFA/Level 1/Quantitative methods/ss2 25
FV of $1.00 The Rule of 72 Let us predict what happens with an investment if it is invested at 5% and 10% show the accumulated value after t=1, t=2, t=3, etc. Future Value Period 5% 10% 1 1.0500 1.1000 2 1.1025 1.2100 3 1.1576 1.3310 4 1.2155 1.4641 5 1.2763 1.6105 6 1.3401 1.7716 7 1.4071 1.9487 8 1.4775 2.1436 9 1.5513 2.3579 10 1.6289 2.5937 8.0000 7.0000 6.0000 5.0000 4.0000 3.0000 2.0000 1.0000 0.0000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time Notice: compound interest creates an exponential curve and there will be a substantial difference over the long term when you can earn higher rates of return. RIFT/CFA/Level 1/Quantitative methods/ss2 26
Types of Compounding Problems There are really only four different things you can be asked to find using this basic equation: FV n =PV 0 (1+k) n Find the initial amount of money to invest (PV 0 ) Find the Future value (FV n ) Find the rate (k) Find the time (n) RIFT/CFA/Level 1/Quantitative methods/ss2 27
Types of Compounding Problems Your have asked your father for a loan of $10,000 to get you started in a business. You promise to repay him $20,000 in five years time. What compound rate of return are you offering to pay? This is an ex ante calculation. FV t =PV 0 (1+k) n $20,000= $10,000 (1+r) 5 2=(1+r) 5 2 1/5 =1+r 1.14869=1+r r = 14.869% RIFT/CFA/Level 1/Quantitative methods/ss2 28
Types of Compounding Problems You have $150,000 in your RRSP (Registered Retirement Savings Plan). Assuming a rate of 8%, how long will it take to have the plan grow to a value of $300,000? This is an ex ante calculation FV t =PV 0 (1+k) n $300,000= $150,000 (1+.08) n 2=(1.08) n ln 2 =ln 1.08 n 0.69314 =.07696 n t = 0.69314 /.076961041 = 9.00 years RIFT/CFA/Level 1/Quantitative methods/ss2 29
Types of Compounding Problems You have $150,000 in your RRSP (Registered Retirement Savings Plan). Assuming a rate of 8%, how long will it take to have the plan grow to a value of $300,000? This is an ex ante calculation. FV t =PV 0 (1+k) n $300,000= $150,000 (1+.08) n 2=(1.08) n log 2 =log 1.08 n 0.301029995 = 0.033423755 n t = 9.00 years RIFT/CFA/Level 1/Quantitative methods/ss2 30
Types of Compounding Problems You have $650,000 in your pension plan today. Because you have retired, you and your employer will not make any further contributions to the plan. However, you don t plan to take any pension payments for five more years so the principal will continue to grow. Assuming a rate of 8%, forecast the value of your pension plan in 5 years. This is an ex ante calculation. FV t =PV 0 (1+k) n FV 5 = $650,000 (1+.08) 5 FV 5 = $650,000 1.469328077 FV 5 = $955,063.25 RIFT/CFA/Level 1/Quantitative methods/ss2 31
Types of Compounding Problems You hope to save for a down payment on a home. You hope to have $40,000 in four years time; determine the amount you need to invest now at 6% This is a process known as discounting This is an ex ante calculation FV n =PV 0 (1+k) n $40,000= PV 0 (1.1) 4 PV 0 = $40,000/1.4641=$27,320.53 RIFT/CFA/Level 1/Quantitative methods/ss2 32
Compound Interest [3] PV FV n 0 n n n (1 k) FV 1 ( 1 k) RIFT/CFA/Level 1/Quantitative methods/ss2 33
Annuity An annuity is a finite series of equal and periodic cash flows. RIFT/CFA/Level 1/Quantitative methods/ss2 34
Annuities and Perpetuities [4] PV 0 1 PMT 1 (1 k) k n RIFT/CFA/Level 1/Quantitative methods/ss2 35
Ordinary Annuity Time = 0 Time = 1 Time = 2 Time = 3 Time = n Time of Investment n=0 PMT 1 PMT 2 PMT 3 PMT n An annuity is a finite series of equal and periodic cash flows where PMT 1 =PMT 2 =PMT 3 = =PMT n RIFT/CFA/Level 1/Quantitative methods/ss2 36
Future Value of An Ordinary Annuity An example of a compound annuity would be where you save an equal sum of money in each period over a period of time to accumulate a future sum. RIFT/CFA/Level 1/Quantitative methods/ss2 37
Annuities and Perpetuities Compound Value Annuity Formula (CVAF) [5] FV n PMT ( 1 k) k n 1 PMT(CVAF) RIFT/CFA/Level 1/Quantitative methods/ss2 38
Future Value of An Annuity FV n PMT ( 1 k) k n 1 Example: How much will you have at the end of three years if you save $1,000 each year for three years at a rate of 10%? FV 3 = $1,000 {[(1.1) 3-1].1} =$1,000 3.31 = $3,310 RIFT/CFA/Level 1/Quantitative methods/ss2 39
Future Value of An Annuity Example: How much will you have at the end of three years if you save $1,000 each year for three years at a rate of 10%? FV 3 = $1,000 {[(1.1) 3-1] /.1} =$1,000 3.31 = $3,310 What does the formula assume? $1,000 1 (1.1) (1.1) = $1,210 + $1,000 2 (1.1) = $1,100 + $1,000 3 = $1,000 Sum = = $3,310 RIFT/CFA/Level 1/Quantitative methods/ss2 40
Future Value of An Annuity FVA 3 = $1,000 {[(1.1) 3-1].1} =$1,000 3.31 = $3,310 What does the formula assume? $1,000 1 (1.1) (1.1) = $1,210 + $1,000 2 (1.1) = $1,100 + $1,000 3 = $1,000 Sum = = $3,310 If these assumptions don t hold you can t use the formula. The CVAF assumes that time zero (t=0) (today) you decide to invest, but you don t make the first investment until one year from today. The Future Value you forecast is the value of the entire fund (a series of investments together with the accumulated interest) at the end of some year n = 1 or n = 2 in this case n = 3. NOTE: the rate of interest is assumed to remain unchanged throughout the forecast period. RIFT/CFA/Level 1/Quantitative methods/ss2 41
Future Value of An Annuity The time value of money formula can be applied to any situation what you need to do is to understand the assumptions underlying the formula then adjust your approach to match the problem you are trying to solve. In the foregoing problem ít isn t too logical to start a savings program and then not make the first investment until one year later!!! RIFT/CFA/Level 1/Quantitative methods/ss2 42
Example of Adjustment You plan to invest $1,000 today, $1,000 one year from today and $1,000 two years from today. What sum of money will you accumulate at time 3 if your money is assumed to earn 10%. This is known as an annuity due rather than a regular annuity. RIFT/CFA/Level 1/Quantitative methods/ss2 43
Annuity Due Time = 0 Time = 1 Time = 2 Time = 3 Time = n PMT 1 PMT 2 PMT 3 PMT n No PMT An annuity due is a finite series of equal and periodic cash flows where PMT 1 =PMT 2 =PMT 3 = =PMT n but the first payment occurs at time=0. RIFT/CFA/Level 1/Quantitative methods/ss2 44
Example of Adjustment You plan to invest $1,000 today, $1,000 one year from today and $1,000 two years from today. What sum of money will you accumulate in three years if your money is assumed to earn 10%. $1,000 1 (1.1) (1.1) (1.1) = $1,331 + $1,000 2 (1.1) (1.1) = $1,210 + $1,000 3 (1.1) = $1,100 Sum = = $3,641 You should know that there is a simple way of adjusting a normal annuity to become an annuity due just multiply the normal annuity result by (1+k) and you will convert to an annuity due! FV 3 (Annuity due)= $1,000 {[(1.1)3-1].1} (1+ k) =$1,000 3.31 1.1 = $3,310 1.1 = $3,641 RIFT/CFA/Level 1/Quantitative methods/ss2 45
Annuities and Perpetuities [6] FV n ( 1 PMT k) k n 1 ( 1 k) RIFT/CFA/Level 1/Quantitative methods/ss2 46
Annuities and Perpetuities [7] 1 1 n (1 k) PV0 PMT (1 k k) RIFT/CFA/Level 1/Quantitative methods/ss2 47
What is Discounting? Discounting is the inverse of compounding. 1 1 PVIF k, n CVIF k, n (1 k) n RIFT/CFA/Level 1/Quantitative methods/ss2 48
Example of Discounting You will receive $10,000 one year from today. If you had the money today, you could earn 8% on it. What is the present value of $10,000 received one year from now at 8%? PV 0 =FV 1 PVIF k,n = $10,000 (1/ 1.08 1 ) PV 0 = $10,000 0.9259 = $9,259.26 NOTICE: A present value is always less than the absolute value of the cash flow unless there is no time value of money. If there is no rate of interest then PV = FV RIFT/CFA/Level 1/Quantitative methods/ss2 49
PVIF k,n Tables of present value interest factors can be created: PVIF k, n 1 (1 k) n Period 1% 2% 3% 4% 5% 6% 7% 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 RIFT/CFA/Level 1/Quantitative methods/ss2 50
PVIF k,n Notice the farther away the receipt of the cash flow from today the lower the present value Notice the higher the rate of interest the lower the present value. 1 (1.07) PVIF k 7%, n 10 10 0.5083 Period 1% 2% 3% 4% 5% 6% 7% 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 RIFT/CFA/Level 1/Quantitative methods/ss2 51
PVIF k,n If someone offers to pay you a sum 50 or 60 years hence that promise is pretty-much worthless!!! PVIF k, n 1 (1 k) n Period 5% 10% 15% 20% 25% 30% 35% 60 0.0535 0.0033 0.0002 0.0000 0.0000 0.0000 0.0000 70 0.0329 0.0013 0.0001 0.0000 0.0000 0.0000 0.0000 80 0.0202 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 90 0.0124 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 100 0.0076 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 110 0.0047 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 The present value of $10 million promised 100 years from today at a 10% discount rate is = $10,000,000 * 0.0001 = $1,000!!!! RIFT/CFA/Level 1/Quantitative methods/ss2 52
The Nature of Compound Interest When we assume compound interest, we are implicitly assuming that any credited interest is reinvested in the next period, hence, the growth of the fund is a function of interest on the principal, and a growing interest upon interest stream. This principal is demonstrated when we invest $10,000 at 8% per annum over a period of say 4 years the future value of this investment can be decomposed as follows... RIFT/CFA/Level 1/Quantitative methods/ss2 53
FV 4 of $10,000 @ 8% Rate of Interest = 8.00% Time Principal at Beginning of the Year End of Period Value of the Fund (Principal plus Interest) Interest 1 $10,000.00 $800.00 $10,800.00 2 $10,800.00 $864.00 $11,664.00 3 $11,664.00 $933.12 $12,597.12 4 $12,597.12 $1,007.77 $13,604.89 Of course we can find the answer using the formula: FV 4 =$10,000(1+.08) 4 =$10,000(1.36048896) =$13,604.89 RIFT/CFA/Level 1/Quantitative methods/ss2 54
Annuity Assumptions When using the unadjusted formula or table values for annuities (whether future value or present value) we always assume: the focal point is time 0 the first cash flow occurs at time 1 intermediate cash flows are reinvested at the rate of interest for the remaining time period the interest rate is unchanging over the period of the analysis. RIFT/CFA/Level 1/Quantitative methods/ss2 55
FV of an Annuity Demonstrated When determining the Future Value of an Annuity we assume we are standing at time zero, the first cash flow will occur at the end of the year and we are trying to determine the accumulated future value of a series of five equal and periodic payments as demonstrated in the following time line... 0 1 2 3 4 5 $2,000 $2,000 $2,000 $2,000 $2,000 RIFT/CFA/Level 1/Quantitative methods/ss2 56
FV of an Annuity Demonstrated We could be trying find out how much we would accumulate in a savings fund if we saved $2,000 per year for five years at 8% but we won t make the first deposit in the fund for one year... 0 1 2 3 4 5 $2,000 $2,000 $2,000 $2,000 $2,000 RIFT/CFA/Level 1/Quantitative methods/ss2 57
FV of an Annuity Demonstrated The time value of money formula assumes that each payment will be invested at the going rate of interest for the remaining time to maturity. 0 1 2 3 4 5 This final $2,000 is contributed to the fund, but is assumed not to earn any interest. $2,000 $2,000 $2,000 $2,000 $2,000 $2,000 invested at 8% for 2 years $2,000 invested at 8% for 3 years $2,000 invested at 8% for 1 year $2,000 invested at 8% for 4 years RIFT/CFA/Level 1/Quantitative methods/ss2 58
FV of an Annuity Demonstrated Annuity Assumptions: A demonstration - focal point is time zero - the first cash flow occurs at time one Future value of a $2,000 annuity at the end of five years at 8%: Time Cashflow CVIF Future Value 0 1 $2,000 1.3605 $2,720.98 2 $2,000 1.2597 $2,519.42 3 $2,000 1.1664 $2,332.80 4 $2,000 1.0800 $2,160.00 5 $2,000 1.0000 $2,000.00 Future Value of Annuity = FV(5) $11,733.20 CVIF for 4 years at 8% (4 years is the remaining time to maturity.) Notice that the final cashflow is just received, it doesn't receive any interest. RIFT/CFA/Level 1/Quantitative methods/ss2 59
FV of an Annuity Demonstrated Annuity Assumptions: A demonstration - focal point is time zero - the first cash flow occurs at time one You can always discount or compound the individual cash flows however it may be quicker to use an annuity formula. Future value of a $2,000 annuity at the end of five years at 8%: Time Cashflow CVIF Future Value 0 1 $2,000 1.3605 $2,720.98 2 $2,000 1.2597 $2,519.42 3 $2,000 1.1664 $2,332.80 4 $2,000 1.0800 $2,160.00 5 $2,000 1.0000 $2,000.00 Future Value of Annuity = FV(5) $11,733.20 CVIF for 4 years at 8% (4 years is the remaining time to maturity.) Notice that the final cashflow is just received, it doesn't receive any interest. Using the formula: FV(5) = PMT(CVAF t=5, r=8%) = $2,000 [(((1 + r)t)-1) / r] = $2,000(5.8666) = $11,733.20 RIFT/CFA/Level 1/Quantitative methods/ss2 60
FV of an Annuity Demonstrated In summary the assumptions are: focal point is time zero we assume the cash flows occur at the end of every year we assume the interest rate does not change during the forecast period the interest received is reinvested at that same rate of interest for the remaining time until maturity. RIFT/CFA/Level 1/Quantitative methods/ss2 61
PV of an Annuity Demonstrated Annuity Assumptions: A demonstration - focal point is time zero - the first cash flow occurs at time one You can always discount or compound the individual cash flows however it may be quicker to use an annuity formula. Present value of a five year $2,000 annual annuity at 8%: PVIF for 1 year at 8% Time Cashflow PVIF Present Value 0 1 $2,000 0.9259 $1,851.85 2 $2,000 0.8573 $1,714.68 3 $2,000 0.7938 $1,587.66 4 $2,000 0.7350 $1,470.06 5 $2,000 0.6806 $1,361.17 Present Value of Annuity = $7,985.42 Using the formula: PV = PMT(PVIFA n=5, k=85) = $2,000 [1-1/(1 + k)n] / k = $2,000(3.9927) = $7,985.40 RIFT/CFA/Level 1/Quantitative methods/ss2 62
The Reinvestment Rate Assumption It is crucial to understand the reinvestment rate assumption that is built-in to the time value of money. Obviously, when we forecast, we must make assumptions however, if that assumption not realistic it is important that we take it into account. This reinvestment rate assumption in particular, is important in the yield-to-maturity calculations in bonds and in the Internal Rate of Return (IRR) calculation in capital budgeting. RIFT/CFA/Level 1/Quantitative methods/ss2 63
Perpetuities A perpetuity is an infinite annuity An infinite series of payments where each payment is equal and periodic. Examples of perpetuities in financial markets includes: Common stock (with a no growth in dividend assumption) Preferred stock Consol bonds (bonds with no maturity date) RIFT/CFA/Level 1/Quantitative methods/ss2 64
Perpetuities Time = 0 Time = 1 Time = 2 Time = 3 Time = α Time of Investment n=0 PMT 1 PMT 2 PMT 3 PMT α A perpetuity is an infinite series of equal and periodic cash flows where PMT 1 =PMT 2 =PMT 3 = =PMT α RIFT/CFA/Level 1/Quantitative methods/ss2 65
Perpetuities [8] PV 0 PMT k Where: PV 0 = Present value of the perpetuity PMT = the periodic cash flow k = the discount rate RIFT/CFA/Level 1/Quantitative methods/ss2 66
Perpetuity: An Example While acting as executor for a distant relative, you discover a $1,000 Consol Bond issued by Great Britain in 1814, issued to help fund the Napoleonic War. If the bond pays annual interest of 3.0% and other long U.K. Government bonds are currently paying 5%, what would each $1,000 Consol Bond sell for in the market? RIFT/CFA/Level 1/Quantitative methods/ss2 67
Perpetuity: Solution PV 0 PMT k $1, 0000.03 0.05 $30 0.05 $600 RIFT/CFA/Level 1/Quantitative methods/ss2 68
Nominal Versus Effective Interest Rates So far, we have assumed annual compounding When rates are compounded annually, the quoted rate and the effective rate are equal As the number of compounding periods per year increases, the effective rate will become larger than the quoted rate RIFT/CFA/Level 1/Quantitative methods/ss2 69
Nominal Versus Effective Interest Rates [9] k QR m ( 1 ) 1 m RIFT/CFA/Level 1/Quantitative methods/ss2 70
Calculating the Effective Rate k Effective m QR 1 1 m Where: k Effective = Effective annual interest rate QR = the quoted interest rate M = the number of compounding periods per year RIFT/CFA/Level 1/Quantitative methods/ss2 71
Example: Effective Rate Calculation A bank is offering loans at 6%, compounded monthly. What is the effective annual interest rate on their loans? k Effective m QR 1 1 m 12.06 1 1 12 6.17% RIFT/CFA/Level 1/Quantitative methods/ss2 72
Continuous Compounding When compounding occurs continuously, we calculate the effective annual rate using e, the base of the natural logarithms (approximately 2.7183) [10] k e QR 1 RIFT/CFA/Level 1/Quantitative methods/ss2 73
10% Compounded At Various Frequencies Compounding Frequency Effective Annual Interest Rate 2 10.25% 4 10.3813% 12 10.4713% 52 10.5065% 365 10.5156% Continuous 10.5171% RIFT/CFA/Level 1/Quantitative methods/ss2 74
Calculating the Quoted Rate If we know the effective annual interest rate, (k Eff ) and we know the number of compounding periods, (m) we can solve for the Quoted Rate, as follows: 1 QR 1 keff m 1m RIFT/CFA/Level 1/Quantitative methods/ss2 75
When Payment & Compounding Periods Differ When the number of payments per year is different from the number of compounding periods per year, you must calculate the interest rate per payment period, using the following formula k Per Period m f QR 1 1 m Where: f = the payment frequency per year RIFT/CFA/Level 1/Quantitative methods/ss2 76
Nominal versus Effective Rates [11] k m f QR ( 1 ) -1 m RIFT/CFA/Level 1/Quantitative methods/ss2 77
Loan Amortization A blended payment loan is repaid in equal periodic payments However, the amount of principal and interest varies each period Assume that we want to calculate an amortization table showing the amount of principal and interest paid each period for a $5,000 loan at 10% repaid in three equal annual instalments. RIFT/CFA/Level 1/Quantitative methods/ss2 78
Blended Interest and Principal Loan Where: Principal Principal PMT(PVAF 1 PMT k,n 1 (1 k) k Pmt = the fixed periodic payment t= the amortization period of the loan r = the rate of interest on the loan ) n RIFT/CFA/Level 1/Quantitative methods/ss2 79
Blended Interest and Principal Loan Principal 1 PMT 1 (1 k) r 1 1 20 (1.08) $10,000 Pmt.08 $10,000 Pmt $1,018.52 9.818147 n Calculator Approach: 10,000 PV 0 FV 20 N 8 I/Y CPT PMT $1,018.52 Where: Pmt = unknown t= 20 years r = 8% RIFT/CFA/Level 1/Quantitative methods/ss2 80
How are Loan Amortization Tables Used? To separate the loan repayments into their constituent components. Each level payment is made of interest plus a repayment of some portion of the principal outstanding on the loan. This is important to do when the loan has been taken out for the purposes of earning taxable income as a result, the interest is a tax-deductible expense. RIFT/CFA/Level 1/Quantitative methods/ss2 81
Loan Amortization Tables Principal = $100,000 Rate = 8.0% Term = 5 PVAF = 3.99271 Payment = $25,045.65 Retired Ending Year Principal Interest Payment Principal Balance 1 100,000.00 8,000.00 25,045.65 17,045.65 82,954.35 2 82,954.35 6,636.35 25,045.65 18,409.30 64,545.06 3 64,545.06 5,163.60 25,045.65 19,882.04 44,663.02 4 44,663.02 3,573.04 25,045.65 21,472.60 23,190.41 5 23,190.41 1,855.23 25,045.65 23,190.41 0.00 RIFT/CFA/Level 1/Quantitative methods/ss2 82
Loan or Mortgage Arrangements [11] k Eff m f QR ( 1 ) -1 m RIFT/CFA/Level 1/Quantitative methods/ss2 83
Loan Amortization First calculate the annual payments PV Annuity PMT n 1 1k PMT k PVAnnuity n 11k k 5,000 1 1.10 3 0.10 $2, 010.57 Calculator Approach: 5,000 PV 0 FV 3 N 10 I/Y CPT PMT $2,010.57 RIFT/CFA/Level 1/Quantitative methods/ss2 84
Amortization Table Period Principal: Start of Period Payment Interest Principal Principal: End of Period 1 5,000.00 2,010.57 500.00 1,510.57 3,489.43 2 3,489.43 2,010.57 348.94 1,661.63 1,827.80 3 1,827.80 2,010.57 182.78 1,827.78 0 RIFT/CFA/Level 1/Quantitative methods/ss2 85
Calculating the Balance O/S At any point in time, the balance outstanding on the loan (the principal not yet repaid) is the PV of the loan payments not yet made. For example, using the previous example, we can calculate the balance outstanding at the end of the first year, as shown on the next page RIFT/CFA/Level 1/Quantitative methods/ss2 86
Calculating the Balance O/S after the 1 st Year PV t1 n 1 1k PMT k 2 1 1.10 2, 010.57.10 $3, 489.42 RIFT/CFA/Level 1/Quantitative methods/ss2 87
Calculating the Monthly Payment Now, calculate the monthly payment on the mortgage PV t0 PMT n 1 1k PMT k PVt 0 n 11k k 100, 000 1 1.004938622.004938622 $639.81 300 Calculator Approach: 100,000 PV 0 FV 300 N.4938622 I/Y CPT PMT $639.81 RIFT/CFA/Level 1/Quantitative methods/ss2 88
Monthly Mortgage Loan Amortization Table Principal = $100,000 Quoted rate = 6.0% Effective annual Rate = 6.090% (Assuming semi-annual compounding) Effective monthly Rate = 0.49386% Term = 25 years Term in months = 300 PVAF = 156.297225 Payment = $639.81 Retired Ending Month Principal Interest Payment Principal Balance 1 100,000.00 493.86 639.81 145.94 99,854.06 2 99,854.06 493.14 639.81 146.67 99,707.39 3 99,707.39 492.42 639.81 147.39 99,560.00 4 99,560.00 491.69 639.81 148.12 99,411.88 5 99,411.88 490.96 639.81 148.85 99,263.03 RIFT/CFA/Level 1/Quantitative methods/ss2 89
Summary and Conclusions What you have learned: To compare cash flows that occur at different points in time To determine economically equivalent future values from values that occur in previous periods through compounding. To determine economically equivalent present values from cash flows that occur in the future through discounting To find present value and future values of annuities, and To determine effective annual rates of return from quoted interest rates. RIFT/CFA/Level 1/Quantitative methods/ss2 90