COMP2022 Programming for FinTech Applications
Spring 2020 Professor: Dr. Grace Wang
Supplementary material: Time Value of Money
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Basic Definitions
qPresent Value – earlier money on a time line
qFuture Value – later money on a time line qInterest rate – “exchange rate” between
earlier money and later money
vDiscount rate
vCost of capital vOpportunity cost of capital vRequired return
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Future Values
q Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year?
vInterest = 1,000(.05) = 50
vValue in one year = principal + interest =
1,000 + 50 = 1,050
vFuture Value (FV) = 1,000(1 + .05) = 1,050
q Suppose you leave the money in for another year. How much will you have two years from now?
vFV = 1,000(1.05)(1.05) = 1,000(1.05)2 = 1,102.50
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Future Values: General Formula
qFV = PV(1 + r)t
v FV = future value
v PV = present value
v r = period interest rate, expressed as a decimal v t = number of periods
qFuture value interest factor = (1 + r)t
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Effects of Compounding
q Simple interest
q Compound interest
q Consider the previous example
v FV with simple interest = 1,000 + 50 + 50 = 1,100
v FV with compound interest = 1,102.50
v The extra 2.50 comes from the interest of .05(50) = 2.50 earned on the first interest payment
q Compounding: the process by which a sum of money grows over time as interest earnings are added to the principal amount and earn interest in addition to the interest on the base amount
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Future Values – Example 2
qSuppose you invest the $1,000 from the previous example for 5 years. How much would you have?
v 1,276.28
qThe effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1,250, for a difference of $26.28.)
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Future Values – Example 3
qSuppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?
v 447,189.84
qWhat is the effect of compounding?
vSimple interest = 10 + 200(10)(.055) = 120.00
vCompounding added $447,069.84 to the value of the investment
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Future Value as a General Growth Formula
qSuppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you sell 3 million widgets in the current year, how many widgets do you expect to sell in the fifth year?
v5 N;15 I/Y; 3,000,000 PV
vCPT FV = -6,034,072 units (remember the sign convention)
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Examples: Compounding $100…
q$100 invested at 4% interest for 2 years vAnnualcompounding FV=𝑃𝑉(1+𝑖)1
v $100(1 + 0.04)’=108.16
vSemi-annualcomputing FV=𝑃𝑉(1+𝑖/2)’1 v$100(1 + 0.04/2)+=108.24
v Continuous computing FV= lim 𝑃𝑉 1 + 8; 81 = 𝑃𝑉𝑒;1 v $100(𝑒)’∗).)+=108.33 8→:
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Examples: Compounding $100…
2 yrs
5 yrs
10 yrs
4%
$108.16
$121.67
$148.02
10%
$121.00
$161.05
$259.37
20%
$144.00
$248.83
$619.17
Annual compounding
Continuous compounding
2yrs
5yrs
10 yrs
4%
$108.33
$122.14
$149.18
10%
$122.14
$101.65
$271.83
20%
$149.18
$271.83
$738.91
Principle:
As interest rates increase or the time over which interest compounds increases, the future value (FV) of the original investment increases at an increasing rate
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Present Values
q How much do I have to invest today to have some amount in the future?
v FV = PV(1 + r)t
vRearrange to solve for PV = FV / (1 + r)t
q When we talk about discounting, we mean finding
the present value of some future amount
q Discounting: the process of converting future cash flows into current values by reversing the math of compounding
q When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.
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Present Value – One Period Example
qSuppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?
qPV = 10,000 / (1.07)1 = 9,345.79
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Present Values – Example 2
qYou want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?
vN = 17; I/Y = 8; FV = 150,000
vCPT PV = -40,540.34 (remember the sign convention)
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Present Values – Example 3
qYour parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?
vN = 10; I/Y = 7; FV = 19,671.51 vCPT PV = -10,000
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Present Value – Important Relationship I
qFor a given interest rate – the longer the time period, the lower the present value
vWhat is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%
v5 years: N = 5; I/Y = 10; FV = 500 CPT PV = -310.46
v10 years: N = 10; I/Y = 10; FV = 500 CPT PV = -192.77
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Present Value – Important Relationship II
qFor a given time period – the higher the interest rate, the smaller the present value
vWhat is the present value of $500
received in 5 years if the interest rate
is 10%? 15%?
• Rate=10%:N=5;I/Y=10;FV=500 CPT PV = -310.46
• Rate=15%;N=5;I/Y=15;FV=500 CPT PV = -248.59
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The Basic PV Equation – Refresher
qPV = FV / (1 + r)t
qThere are four parts to this equation
vPV, FV, r and t
vIf we know any three, we can solve for the
fourth
qIf you are using a financial calculator, be sure and remember the sign convention or you will receive an error (or a nonsense answer) when solving for r or t
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Discount Rate
qOften we will want to know what the implied interest rate is on an investment
qRearrange the basic PV equation and solve for r
vFV = PV(1 + r)t vr=(FV/PV)1/t –1
qIf you are using formulas, you will want to make use of both the yx and the 1/x keys
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Discount Rate – Example 1
qYou are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest?
vr = (1,200 / 1,000)1/5 – 1 = .03714 = 3.714%
vCalculator – the sign convention matters!!! • N=5
• PV = -1,000 (you pay 1,000 today)
• FV = 1,200 (you receive 1,200 in 5 years) • CPT I/Y = 3.714%
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Discount Rate – Example 2
qSuppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?
vN = 6
vPV = -10,000
vFV = 20,000 vCPT I/Y = 12.25%
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Discount Rate – Example 3
qSuppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?
vN = 17; PV = -5,000; FV = 75,000 vCPT I/Y = 17.27%
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Finding the Number of Periods
qStart with basic equation and solve for t (remember your logs)
v FV = PV(1 + r)t
v t = ln(FV / PV) / ln(1 + r)
qYou can use the financial keys on the calculator as well; just remember the sign convention.
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Number of Periods – Example 1
qYou want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?
v I/Y = 10; PV = -15,000; FV = 20,000 v CPT N = 3.02 years
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Number of Periods – Example 2
qSuppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year. How long will it be before you have enough money for the down payment and closing costs?
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Number of Periods – Example 2 Continued
qHow much do you need to have in the future? v Down payment = .1(150,000) = 15,000
v Closing costs = .05(150,000 – 15,000) = 6,750
v Total needed = 15,000 + 6,750 = 21,750
qCompute the number of periods v PV = -15,000; FV = 21,750; I/Y = 7.5 v CPT N = 5.14 years
qUsing the formula
v t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years
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Examples of Discounting…
$100 discounted at 4%, 10%, and 20% for 2, 5, and 10 years
Principle
As the discount rate rises, and/or the length of time that must elapse before a cash flow is received increases, the present value of that payment shrinks.
2 yrs
5 yrs
10 yrs
4%
$92.46
$82.19
$67.56
10%
$82.64
$62.09
$38.55
20%
$69.44
$40.19
$16.15
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Spreadsheet Example
qUse the following formulas for TVM calculations
v FV(rate,nper,pmt,pv) v PV(rate,nper,pmt,fv) v RATE(nper,pmt,pv,fv) v NPER(rate,pmt,pv,fv)
qThe formula icon is very useful when you can’t remember the exact formula
qClick on the Excel icon to open a spreadsheet containing four different examples.
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A summary
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