2 LECTURE TIME VALUE OF MONEY: COMPOUND INTEREST, PRESENT & FUTURE VALUES
Reading: Chapter 3, sections 3.1-3.4, Chapter 4 Sections 4.1-4.2 Chapter 5, Section 5.1
Practice Problems: Lecture 2 online, due next time
Objectives 1. Explain and understand compound interest 2. Compute the present or future value of any cash flow using any compounding interval 3. Understand the concept of Net Present Value and how it relates to the Law of One Price 4. Solve for present value, future value, interest rate or number of periods, given the values of the other thre 5. Derive an effective annual compound rate of interest (also called APY) from an annual percentage rate of interest (APR) 12
MOVE-IN SPECIALS
When I moved into my apartment three years ago, my landloard had a move-in special. Rent on my apartment is $1350/month However, because of the move-in special, the landlord offered to lower my rent to $1125 per month for six months (which was my lease term). Alternatively, I could live rent-free for the first month, and then pay $1350 per month for the remaining five months of my lease. Which option would you take? How can we think of this as a saving/investment problem?
INTRODUCTION TO THE TIME VALUE OF MONEY:
Easy: Choice between $100 and $110 right now.
Not so easy:
Choice between $100 today and $110 in 1 year.
Lets apply the Law of One Price: How can we determine the current market price of the right to receive $110 in 1 year? Many business and personal decisions involve a trade-off of cash flow between points in time. What are some examples? We need market interest rates and the tools of interest rate mathematics to compare bundles or streams of cash flows over time. Present Value Present Value Compound interest Discounting Future Value Future Value
CALCULATING FUTURE VALUE
Suppose you place $100 in a CD that earns 2% interest compounded annually.
FVn = PV (1 + r ) n
n is the number of years r is the annual rate of interest FV is value in n years (Future Value) PV is value today (Present Value)
TOOL
Compound Interest Example: Compounded Annually
Time 0 (Today) 1 year 2 years 3 years 4 years 5 years 6 years ... 100 years 100= PV 102 = PV(1 + .02) 104.04 = PV(1 + .02)(1 + .02) = PV(1+.02)2 106.12 = PV (1 + .02) (1 + .02) (1 + .02) = PV (1+.02)3 108.24 = PV (1 + .02)4 110.41 = PV (1 + .02)5 112.62 = PV (1 + .02) 6 ... 724.46 = PV (1 + .02)100 Amount
COMPOUND INTEREST EXAMPLE, CONTINUED
Interest from this Year Interest from Previous Years Original Investment
Figure 3.1 Compound Interest
200
180
160
140 Value of Account
120
100
80
60
40
20
0 0 1 2 3 4 5 Years 6 7 8 9 10
Compositon of Interest
12
10
8 Total Interest Earned
6
4
2
0 1 2 3 4 5 Years Interest on Original Investment Interest on Interest 6 7 8 9 10
RETURN TO THE EXAMPLE
Easy: Not so easy: Choice between $100 and $110 right now. Choice between $100 today and $110 in 1 year.
Suppose the best 1 year CD you can buy in the market offers you an interest rate of 2%, compounded annually Choice is Between: 100 today which you can invest @ 2% or $110 in 1 year. Calculate how much you would receive in one year if you invested $100 in the CD today (Future Value) or Calculate the market price of the right to receive $110 in one year (present value)
Always organize your analysis with a time line or a table!!
NET PRESENT VALUE
You are an artist and a rich dowager offers to pay you $1,100 in one year if you deliver a sculpture to her It will take you one year to make it You are sure you friend will pay you if you deliver it You need to invest $1,000 in materials today to make the sculpture (the dowager wont give you the money) Assume you have no job skills, so you are not employable, and you enjoy sculpting, so the opportunity cost of your time is zero. Instead of making the sculpture, you can invest your $1,000 in a CD that yields 2% annually. What is the present value of the sculpture to you? What is the net present value? Should you agree to build the sculpture?
MORE FREQUENT COMPOUNDING
r FVn = PV (1 + ) mn m
r is the annual rate of interest n is the number of years m is the number of compounding intervals per year r/m is the interest rate per compounding interval m n is the number of compounding intervals Make sure that r/m and m n reflect the same time interval, such as months. Our original formula is just a special case where the compounding interval equals once per year (so m = 1).
SEMI-ANNUALLY COMPOUNDED FVn INTEREST
r = PV (1 + ) 2n 2
Example: a CD that pays 6%
Time Amount (Compounded Semi-Annually) 0 or Today year 1 years 1.5 yars 2 years 2.5 years 3 years 100 years 100.00 = PV 103.00 = PV (1 + .06/2) = PV (1 + .03) 106.09 = PV (1 + .03) (1 + .03) 109.27 = PV (1 + .03) (1 + .03) (1 + .03) 112.55 = PV (1 + .03)4 115.92 = PV (1 + .03)5 119.40 = PV (1 + .03)6 36,935.58 = PV (1 + .03)200 119.10 = PV (1 + .06) (1 + .06) (1 + .06) 33,930.20 = PV (1 + .06)100 112.36 = PV (1 + .06)2 106.00 = PV (1 + .06) Amount (Compounded Annually) 100.00 = PV
Daily Compounded Interest
FVn = PV (1 +
r 365n ) 365
CONTINUOUSLY COMPOUNDED INTEREST
r mn FVn = limm PV *(1+ ) = PV *ern m
where
TOOL
e = 2 . 7182818284
= 100 x e(.06 x 100) = 100 x e(6) = 40,342.88
FV100
(Rule of 72: # yrs to double is approximately 72/rate. 72/6 = 12 yrs)
TOOL
COMPOUNDING: A FAMOUS EXAMPLE
Peter Minuit bought Manhattan for $24 in 1624: (382 years ago) In 2004 @ 5% FV382 = $24 (1.05)382 = $ 2,982,108,815
In 2004 @ 8% FV382 = $24 (1.08)382 = $ 140,632,545,501,736 In 2004 @ 15% FV382 = $24 (1.15)382 = $ 3.687960465x 1024
The Miracle of Compound Interest
7000 6000 5000 4000
Value
5%
8%
15%
3000 2000 1000 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Years
THE RELATION BETWEEN PV AND FV
FVn = PV (1 + r ) n PV = FVn 1 (1 + r ) n
TOOL
PV is the amount of money that will make you just as happy (or wealthy) today as FV at some date (say n years) in the future.
Today: PV (Present Value) Going from PV to FV Going from FV to PV $100 $110 x
Tomorrow: FV (Future Value) $100 x (1+ r) $110
1 (1 + r )
(r is interest rate appropriate for moving money.)
Example 1: What is the PV of $1000 to be received 10 years from now discounted at an annually compounded rate of 10%?
PV = FVn
1 (1 + r ) n 1 = 1000 (1+.10)10 = 1000(.38554) = 38554 .
Example 2: What is the PV of $1000 to be received 5 years from now discounted at a semi-annually compounded 10% .
PV = FVn
1 r (1 + ) nm m 1 1 = 1000 = 1000 .10 52 1.6289 (1 + ) 2 = 1000(.61391) = 613.91
SINGLE CASH FLOWS: SOLVING FOR n AND r
Thus far we have solved for FV and PV.
What if we want to know the interest rate or number of years?
Just solve the present value equation for n or r!
Solving for r
You have $1000.00. You want it to grow to $1790.85 in 10 years. What is the rate of interest compounded annually? What do we know? FVn = $1790.85 PV = $1000 r = ?% n = 10
Solving for n
How many years will it take an initial investment of $300 to grow to $774 if it is invested at 9% compounded annually? What do we know? FVn = $774 PV = $300 r = 9% n=?
ANNUAL PERCENTAGE YIELD (APY) (ALSO CALLED EFFECTIVE ANNUAL RATE) VERSUS ANNUAL PERCENTAGE RATE (APR) OF INTEREST
A discrepancy arises when the compounding interval is not annual.
Suppose you buy $100 CD at an annual percentage rate of 10% with annual compounding. This means you will pay 10% interest and principal in one year.
In this case, the effective annual rate, or annual percentage yield (APY) is the same as the annual percentage rate (APR). But what if the rate at which the APR compounds is different?
The effective annual rate (or annual percentage yield, APY) is just the annually compounded APR that gives you the same balance after one year:
APR 1+ = (1 + APY ) m
The left hand side is the amount you would have at the end of the year if you invested one dollar in a CD with an APR compounds m times per year.
m
The right hand side is the amount you would have at an APR that compounds annually. This APR is also the APY.
So solving for APY, we get:
APR APY = 1+ 1 m
Example:
m
TOOL
Suppose you buy a CD at $100 at an annual stated percentage rate (APR) of 10% with semi-annual compounding. That is, 5% interest is paid every six months. The APY = (1.05)2 1 = .1025, or 10.25%
APR ignores the effect of compounding when interest is paid more frequently than annually.
What if the CD were compounded monthly? Daily?
Note: some people use Effective Annual Rate, or EAR, instead of APY. The two terms have exactly the same meaning. Up until a few years ago, EAR was the most commonly used term and APY was very rarely used. Today, however, the situation is reversed. Nevertheless, lots of people still refer to the APY as the effective annual rate, so its important that you know this term so that you know what people are talking about.
For next time 1) Do the online homework 2) Read chapter 4, sections 4.3 4.6