Chapter 5The Time Value of Money Some Important Concepts
Topics Covered • Future Values and Compound Interest • Present Values • Multiple Cash Flows • Perpetuities and Annuities • Effective Annual Interest Rates. • Inflation
Future Values Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment.
Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 012345 Interest Earned Value 100 6 106 6 112 6 118 6 124 6 130 Value at the end of Year 5 = $130
Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 012345 Interest Earned Value 100 6 106 6.36 112.36 6.74 119.10 7.15 126.25 7.57 133.82 Value at the end of Year 5 = $133.82
Future Values Future Value of $100 = FV
Future Values Example – Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2006, compounded at 8%. Financial calculator: n=380, i=8, PV=-24, PMT=0 FV= $120.57 trillion
Set up the Texas instrument • 2nd, “FORMAT”, set “DEC=9”, ENTER • 2nd, “FORMAT”, move “↓” several times, make sure you see “AOS”, not “Chn”. • 2nd, “P/Y”, set to “P/Y=1” • 2nd, “BGN”, set to “END” • P/Y=periods per year, • END=cashflow happens end of periods
Present Values • If the interest rate is at 6 percent per year, how much do we need to invest now in order to produce $106 at the end of the year? • How much would we need to invest now to produce $112.36 after two years?
Present Values • Discount Factor (DF) = PV of $1 • Discount Factors can be used to compute the present value of any cash flow. PV: Discounted Cash-Flow (DCF) r: Discount Rate
Present Values Interest Rates
Present Values Example In 2007, Puerto Rico needed to borrow about $2.6 billion for up to 47 years. It did so by selling IOUs, each of which simply promised to pay the holder $1,000 at the end of that time. The market interest rate at the time was 5.15%. How much would you have been prepared to pay for one of these IOUs?
Present Values Example Kangaroo Autos is offering free credit on a $20,000 car. You pay $8,000 down and then the balance at the end of 2 years. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price. If the interest rate is 10%, which company is offering the better deal?
Present Values $12,000 $8,000 PV at Time=0 0 1 2 $8,000 $9,917.36 Total $17,917.36
Present Values Example – Finding the Interest Rate Each Puerto Rico IOU promised to pay holder $1,000 at the end of 47 years. It is sold for $94.4. What is the interest rate? Financial calculator: n=47, FV=1,000 PV=-94.4, PMT=0 i=5.15
Present Values Example – Double your money Suppose your investment advisor promises to double your money in 8 years. What interest rate is implicitly being promised? Financial calculator: n=8, FV=2 PV=-1, PMT=0 i=9.05
Present Value of Multiple Cash Flows Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?
4 0 1 2 3 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV What is the PV of this uneven cash flow stream?
Detailed steps (Texas Instrument calculator) • To clear historical data: • CF, 2nd ,CE/C • To get PV: • CF ,↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2, Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT • “NPV=530.0867427”
Future Value of Multiple Cash Flows Example You plan to save some amount of money each year. You put $1,200 in the bank now, another $1,400 at the end of the first year, and a third deposit of $1,000 at the end of the second year. If the interest rate is 8% per year, how much will be available to spend at the end of the 3 year?
Future Value of Multiple Cash Flows $1,400 $1,200 $1,000 Year FV in year 3 0 1 2 3 $1,080.00 $1,632.96 $1,511.65 $4,224.61
Perpetuities and Annuities • Annuity • Equally spaced level stream of cash flows for a limited period of time. • Perpetuity • A stream of level cash payments that never ends.
Perpetuities and Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?
Perpetuities and Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?
Perpetuities and Annuities Cash Flow Year 0 1 2 3 4 5 6 …Present Value 1. Perpetuity A $1 $1 $1 $1 $1 $1 … 2. Perpetuity B $1 $1 $1 … 3. Three-year annuity $1 $1 $1
Perpetuities and Annuities C = cash payment r = interest rate t = Number of years cash payment is received
Perpetuities and Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.
Perpetuities and Annuities Example – Lottery Suppose you bought lottery tickets and won $295.7 million. This sum was scheduled to be paid in 25 equal annual installments of $11.828 million each. Assuming that the first payment occurred at the end of 1 year, what was the present value of the prize? The interest rate was 5.9 percent. Financial calculator: n=25, i=5.9, PMT=11.828, FV=0, PV=-152.6
Perpetuities and Annuities • Annuity Due • Level stream of cash flows starting immediately.
Perpetuities and Annuities Example – Lottery (Continued) What is the value of the prize if you receive the first installment of $11.828 million up front and the remaining payments over the following 24 years?
Perpetuities and Annuities • Future Value of annual payments
Perpetuities and Annuities Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? Financial calculator: n=20, i=10, PMT=-4,000, PV=0, FV=229,100
Effective Annual Interest Rate Effective Annual Interest Rate - Interest rate that is annualized using compound interest. Annual Percentage Rate (APR) - Interest rate that is annualized using simple interest.
Effective Annual Interest Rate Example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
Classifications of interest rates • 1. Nominal rate (iNOM) – also called the APR,quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly. • 2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly. • iPER = iNOM / m, where m is the number of compounding periods per year. e.g., m = 12 for monthly compounding.
Classifications of interest rates • 3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding. • If the interest rate is compounded m times in a year, the effective annual interest rate is
Example, EAR for 10% semiannual investment • EAR= ( 1 + 0.10 / 2 )2 – 1 = 10.25% • An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.
keys: description: Sets 2 payments per year [↑] [C/Y=] 2 [ENTER] [2nd] [ICONV] Opens interest rate conversion menu [↓][NOM=] 10 [ENTER] Sets 10 APR. [↓] [EFF=] [CPT] 10.25 EAR on a Financial Calculator Texas Instruments BAII Plus
Why is it important to consider effective rates of return? • An investment with monthly payments is different from one with quarterly payments. • Must use EAR for comparisons. • If iNOM=10%, then EAR for different compounding frequency: Annual 10.00% Quarterly 10.38% Monthly 10.47% Daily 10.52%
1 2 3 0 1 2 3 4 5 6 5% 100 100 100 What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually? • Payments occur annually, but compounding occurs every 6 months. • Cannot use normal annuity valuation techniques.
1 2 3 0 1 2 3 4 5 6 5% 100 100 100 110.25 121.55 331.80 Method 1:Compound each cash flow FV3 = $100(1.05)4 + $100(1.05)2 + $100 FV3 = $331.80
Method 2:Financial calculator • Find the EAR and treat as an annuity. • EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%. 3 10.25 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 331.80
When is periodic rate used? • iPER is often useful if cash flows occur several times in a year.
Inflation Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases.
Inflation approximation formula
Inflation Example If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?