Time Value of Money JQY
Agenda • • • • • • • • • • • • •
Time Lines Future Values Present Values Solving for Interest Rate and Time Future Value of an Annuity Present Value of an Annuity Perpetuities Uneven Cash Flow Streams Semiannual and Other Compounding Periods Comparison of Different types of interest rates Fractional Time Periods Amortized Loans Amortization
What is Time Value? • We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return • In other words, “a dollar received today is worth more than a dollar to be received tomorrow” • That is because today’s dollar can be invested so that we have more than one dollar tomorrow
If you have P10,000 today, and you deposit it in the bank, how much will you most likely receive in 10 years? a. b. c.
P9,500 P14,000 P10,000
c.
P10,000
b.P14,000
Timelines An important tool used in the time value of money analysis A graphical representation used to show the timing of cash flows
A timeline is a graphical device used to clarify the timing of the cash flows for an investment Each tick represents one time period
PV
0 Today
FV
1
2
3
4
5
Future Value • The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today • Terms: PV – present value, or beginning amount in your account i – interest rate the bank pays on the account per year INT – amount of interest you earn during the year (aka discount rate, opportunity cost rate) FV – future value or ending amount of your account at the end of n years n – number of periods involved in the analysis
Future Value • Simple Annual Interest Q: Today, Peter invested P1,000 for 5 years with simple annual interest of 10%. How much is its future value? A: FV = PV x [1 + (i*n)] FV = P1,000 x [1 + (10%*5)] FV = P1,000 x 1.5 FV = P1,500
Future Value • Interest compounded Q: Today, Peter invested P100 for 3 years at 10%, compounded annually. How much is its future value? A: FV = PV (1+i)ⁿ 0
1
2
100 After 1 year : FV = PV (1+i)^1 = 100 (1+10%)^1 = 110 After 2 years: FV = PV (1+i)^2 = 100 (1+10%)^2 = 121 After 3 years: FV = PV (1+i)^3 = 100 (1+10%)^3 = 133.10
3 FV = ?
The Magic of Compounding • In 1898, USA bought the Philippines from Spain for $20 million • This happened about 114 years ago, so 5% per year could be earned, the value of the Philippines now (in 2012) would be approximately:
20m (1.05)114 = 5,207,269,916
If they could have earned 10% per year, the Philippines would have been worth:
20m (1.10)114 = 1,046,637,049,000
Agenda • Future Value
• Present Value • Annuities • Rates of Return • Amortization
If you need to have P10,000 in 10 years, how much will you likely have to invest today? a. P7,000 a.P7,000 b. P10,000 b. P10,000 c. P12,000 c. P12,000
Present Value • The value today of a future cash flow or series of cash flows. • Represents the amount that needs to be invested to achieve some desired future value. PV
FVN
1 i
N
Present Value: An Example • Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about $100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal? • N = 13; I = 8%; PV = ?; PMT = 0; FV = 100,000
PV
100,000 . 108
13
$36,769.79
Solving for Interest and Time • I = (FV/PV)^1/n – 1 • Sample problem You can buy a security at a price of $78.35, and it will pay you $100 after 5 years. How much is the interest rate you’d earn if you bought the security? I = (100/78.35)^(1/5) – 1 = 5%
Solving for Interest and Time • N = ln (FV/PV) / ln (1+i) • Sample problem Mr. Amos invested P60,000 in stocks at a 10% interest rate compounded semi-annually. How many years did it take Mr. Amos for his investment to reach P100,000? N = ln (100,000/60,000) / ln (1 + 5%) N = 10.47 / 2 = 5.24 years
Annuities • An annuity is a series of payments of an equal amount at fixed intervals for a specified number of periods. • Annuities are very common: – – – –
Rent Mortgage payments Car payment Pension income
• The timeline shows an example of a 5-year, $100 annuity • Annuity = equal PMT
0
100
100
100
100
100
1
2
3
4
5
Annuities • Ordinary (Deferred) Annuity – An annuity whose payments occur at the end of each period.
• Annuity Due – An annuity whose payments occur at the beginning of each period. 100 5-period Annuity Due 5-period Regular Annuity 0
100 100
100 100
100 100
100 100
100
1
2
3
4
5
Future Value of an Ordinary Annuity • Mary deposited P100 at the end of each year for 3 years in a savings account that pays 5% interest per year. How much will she have at the end of three years?
• Fva = 100 {[(1+5%)^3 – 1] / 5%} • Fva = 315.25
Future Value of an Annuity Due • Mary deposited P100 at the beginning of each year for 3 years in a savings account that pays 5% interest per year. How much will she have at the end of three years?
• Fvad = 100 {[(1+5%)^3 – 1] / 5%} (1+5%) = 331.01
Present Value of an Ordinary Annuity • Mary deposited P100 at the end of each year for 3 years in a savings account that pays 5% interest per year. How much is the present value of her payments?
• Pva = 100 [1 – (1/(1+5%)^3) / 5%] • Pva = 272.32
Present Value of an Annuity Due • Mary deposited P100 at the beginning of each year for 3 years in a savings account that pays 5% interest per year. How much is the present value of her payments?
• Pvad = {100 [1 – [1/(1+5%)^(3-1)] / 5%} + 100 • Pvad = 285.94
Ordinary Annuity – Solving for Payment when FV is known • Harold wants to accumulate P50,000 at the end of 5 years. How much should he pay every year assuming that the interest is 5%, and the first payment will be made at the end of the year?
• PMT = 50,000 / [(1+5%)^5 – 1] / 5% = 9,048.74
Annuity Due – Solving for Payment when FV is known • Harold wants to accumulate P50,000 at the end of 5 years. How much should he pay every year assuming that the interest is 5%, and the first payment will be made at the beginning of the year?
• PMTad = 50,000 / {[(1+5%)^5 – 1] / 5%} (1+5%) • PMTad = 8,617.85
Ordinary Annuity – Solving for Payment when PV is known • How much should Sharon pay every year for 5 years assuming that the interest is 5%, and the first payment will be made at the end of the year, if the present value is 10,000?
• PMT = 50,000 / [(1 – (1/(1+5%)^5 )/5%] / 5% = 2,309.75
Annuity Due – Solving for Payment when PV is known • How much should Sharon pay every year for 5 years assuming that the interest is 5%, and the first payment will be made at the beginning of the year, if the present value is 10,000?
• PMTad = 10,000 / {[(1 – (1/(1+5%)^(5-1) )/5%] / 5%} + 1 • PMTad = 2,199.76
Ordinary Annuity – Solving for N when FV is known • Sharon plans to save P100 per year (first payment at end of the year). Assuming that the interest is 5%, how many years does it take for Sharon to accumulate 1,000?
• N = ln[1 – (1,000/-100)5%] / ln (1+5%) = 8.31 years
Annuity Due – Solving for N when FV is known • Sharon plans to save P100 per year (first payment at beginning of the year). Assuming that the interest is 5%, how many years does it take for Sharon to accumulate 1,000?
• Nad = {ln[(1,000 x 5%) / (100 x 1.05)] + 1} / ln (1+5%) • Nad = 7.98 years
Ordinary Annuity – Solving for N when PV is known • Sharon plans to save P100 per year (first payment at end of the year), and the present value if P1,000 Assuming that the interest is 5%, solve for N:
• N = - ln[1 – (1000/100) 5%] / ln (1+5%) = 14.207 years
Annuity Due – Solving for N when PV is known • Sharon plans to save P100 per year (first payment at the beginning of the year), and the present value if P1,000 Assuming that the interest is 5%, solve for N:
• Nad = {- ln[1 + 5% (1 – (1000/100)] / ln (1+5%)} + 1 • Nad = 13.25 years
Ordinary Annuity – Solving for I when FV and PV are known • Can only be calculated through a trial and error process unless financial calculator is used. • However, an approximate equation can be solved, provided that all inputs in the equation below is known: i = {{Annual PMT + [(FV – PV)/Annual N]} / [(40% x FV) + (60% x PV)]}}
Ordinary Annuity – Solving for I when FV and PV are known Belle has 1,000 today. She plans to make an investment where she pays 50 annually at the end of the year. She expects to receive 1,500 at the end of 10 years. How much is the interest rate that is required for this investment so that Belle will receive 1500 after 10 years? i = {{50 + [(1,500 – 1,000)/10]} / [(40% x 1,500) + (60% x 1,000)]}}
Ordinary Annuity: Solving for I when FV is known (CALC) • Frederick will be contributing P10,000 to his girlfriend’s account as a sign of his “love” every month starting next month. He calculates that in 10 years, there will be P1,500,000 in his girlfriend’s account if she does not touch any of the deposit. How much interest is the account earning if Frederick’s calculations are correct?
Annuity Due: Solving for I when FV is known (CALC) • Frederick contributes P10,000 to his girlfriend’s account as a sign of his “love” every month starting at the beginning of this month. He calculates that in 10 years, there will be P1,500,000 in his girlfriend’s account if she does not touch any of the deposit. How much interest is the account earning if Frederick’s calculations are correct?
Summary • Rules regarding annuity, ceteris paribus: Ordinary Annuity
Annuity Due
Future Value
Lower
Higher
Present Value
Lower
Higher
Payment
Higher
Lower
N
Higher
Lower
Interest
Higher
Lower
Perpetuities • A stream of equal payments expected to continue forever • PV (Perpetuity) = Payment / Interest rate • Suppose each consol (British government perpetual bonds) promised to pay $100 in perpetuity, if the discount rate or opportunity cost rate is 5% and 10%: • PV (Perpetuity) = 100/5% = $2,000 • PV (Perpetuity) = 100/10% = $1,000
Uneven Cash Flow Stream • A series of cash flows in which the amount varies from one period to the next • PMT = equal cash flows coming at regular intervals • CF = uneven cash flows
Uneven Cash Flows: An Example (PV) • Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment?
0
PV
100
200
300
1
2
3
100 . 107
1
200 . 107
2
4
300 . 107
3
5
513.04
Uneven Cash Flows: An Example (FV) • Terminal Value = The future value of an uneven cash flow stream • Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year?
0
300
500
700
1
2
3
4
5
FV 300105 . 500105 . 700 1,555.75 2
1
Non-annual Compounding • We could assume that interest is earned semi-annually, quarterly, monthly, daily, or any other length of time • The only change that must be made is to make sure that the rate of interest is adjusted to the period length
Non-annual Compounding (cont.) • Suppose that you have $1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds? Bank First National Second National Third National
Interest Rate 10% 10% 10%
Compounding Annual Monthly Daily
Non-annual Compounding (cont.) • We can find the FV for each bank as follows:
First National Bank:
Second National Bank:
Third National Bank:
FV 1,000110 . 1100 , 1
010 . FV 1,000 1 12
12
010 . FV 1,000 1 365
365
Obviously, you should choose the Third National Bank
1104 , .71 110516 , .
Continuous Compounding • There is no reason why we need to stop increasing the compounding frequency at daily • We could compound every hour, minute, or second • We can also compound every instant (i.e., continuously):
F Pe
rt
Here, F is the future value, P is the present value, r is the annual rate of interest, t is the total number of years, and e is a constant equal to about 2.718
Continuous Compounding (cont.) • Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. What is the future value of your $1,000 investment?
F 1,000e
0 .10 1
110517 , .
This is even better than daily compounding The basic rule of compounding is: The more frequently interest is compounded, the higher the future value
Continuous Compounding (cont.) • Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. If you plan to leave the money in the account for 5 years, what is the future value of your $1,000 investment?
F 1,000e
0.10 5
1,648.72
Different Rates • Nominal (Quoted, Stated, Annual Percentage) Interest Rate – The rate charged by banks and other financial institutions – For example, 6% compounded quarterly, 5% compounded monthly
• Effective (Equivalent Annual) Rate – The annual rate of interest actually being earned
– EAR = (1+iNOM/m)^m – 1 – If payment is only once a year, EAR = nominal rate
• Periodic Rate – Rate charged by a lender or paid by a borrower each period
– iPER = iNOM/m – If Nominal rate is quoted at 18%, payable monthly, periodic rate is 18%/12 or 1.5% – If payment is only once a year, Nominal rate = periodic rate. Landbank charges 10% interest rate, compounded quarterly. How much is the nominal, EAR, and periodic rate? Nominal = 10%, EAR = 10.38%, Periodic = 2.5%
Fractional Time Periods • If you deposit $100 in a bank that uses daily compounding and pays a nominal rate of 10% with a 365 days, how much is the FV after 9 months? N = 365 x 9/12 ; I = 10% / 365 ; PV = 100 FV = PV x [(1 + i)^n] = 100 x [(1 + 0.000273973)^274] = 107.79
• You borrow $100 that charges 10% simple interest but you borrow only for 274 days. How much interest do you owe? Interest owed = 100 x 10% x 274/365 = $7.51
Amortized Loans • A loan that is repaid in equal payments over its life. • If a firm borrows $1,000 and the loan is to be repaid in 3 equal payments at the end of each of the next three years, and the lender charges 6% on the loan balance, how much is the periodic payment, and construct the loan amortization schedule. • N = 3; I = 6%; PV = 1000; PMT = ?; FV = 0
• PMT = 1,000 / [(1 – (1/(1+6%)^3)/6%] / 6% = 374.11
Amortization Schedule Payment Interest
Principal Repayment
Beg
Balance 1,000.00
Y1
374.11
60.00
314.11
685.89
Y2
374.11
41.15
332.96
352.93
Y3
374.11
21.18
352.93
(0.00)
Balloon Loan • A long-term loan, often a mortgage, that has one large payment due upon maturity. • Advantage: very low interest payments, requiring very little capital outlay during the life of the loan. • Disadvantage: An undisciplined borrower will be in trouble because he has to make a large single payment upon maturity.
Partial Amortization: Balloon Loans • A house is worth $200,000, and a bank agrees to lend the potential home buyer $175,000 secured by a mortgage on the house. However, the buyer only has $5,000 and he is unable to make the full $25,000 downpayment. The seller may take a note of 20k, 8% interest rate and payments at the end of the year based on a 20 year amortization schedule but with loan maturing at the end of the 10th year. • N = 20; I = 8%; PV = 20,000; PMT = ? • Annual PMT of the note = 2,037.04
Additional Problem: • To save money for a new house, you want to begin contributing money to a brokerage account. Your plan is to make 40 contributions to the brokerage account. Each contribution will be for $1,500. The first contribution will occur today and then every quarter, you will contribute another $1,500 to the brokerage account. Assume that the brokerage account pays a 6 percent return with annual compounding. How much money do you expect to have in the brokerage account in ten years (Quarter 40)? How much money do you expect to have in the brokerage account in Quarter 39?
Additional Problem: • Today you opened up a local bank account. Your plan is make five $1,000 contributions to this account. The first $1,000 contribution will occur today and then every six months you will contribute another $1,000 to the account. (So your final $1,000 contribution will be made two years from today). The bank account pays a 6 percent nominal annual interest, and interest is compounded monthly. After two years, you plan to leave the money in the account earning interest, but you will not make any further contributions to the account. How much will you have in the account 8 years from today?
Problem 8-30 • Erika and Kitty, who are twins, just received $30,000 each for their 25th birthday. They both have aspirations to become millionaires. Each plans to make a $5,000 annual contribution to her “early retirement fund” on her birthday, beginning a year from today. Erika opened an account with the Safety First Bond Fund, a mutual fund that invests in high-quality bonds whose investors have earned 6% per year in the past. Kitty invested in the New Issue Bio-Tech Fund, which invests in small, newly issued bio-tech stocks and whose investors have earned an average of 20% per year in the fund’s relatively short history. • Requirement 1: If the two women’s funds earn the same returns in the future as in the past, how old will each be when she becomes a millionaire? • Requirement 2: How large would Erika’s annual contributions have to be for her to become a millionaire at the same age as Kitty, assuming their expected returns are realized? • Requirement 3: Is it rational or irrational for Erika to invest in the bond fund rather than in stocks?
Thank you for listening!
