Post on 21-Dec-2015
Exponential Growth
Exponential Growth
Discrete Compounding Suppose that you were going to
invest $5000 in an IRA earning interest at an annual rate of 5.5%. How much interest would you earn during the 1st year? How much is in the account after 1 year?
Exponential Growth
Interest after 1 year:
Account value after 1 year:
What would happen during the 2nd year?
275$055.050001 rP
i
5275$055.15000055.050005000275$
1
1
iP
F
Exponential Growth Interest made during the 2nd year:
Value of account after 2nd year:
What about for the 3rd year?
13.290$055.052751
2 rF
i
5,565.13$055.15275055.05275527513.290$
2
21
iF
F
Exponential Growth
Interest made during 3rd year:
Value of the account after 3rd year:
08.306$055.0$5,565.132
3 rF
i
$5,871.21055.113.5565055.013.556513.556508.306$
3
32
iF
F
Exponential Growth
Summarizing our calculations:
$5,871.21055.15000055.113.5565
$5,565.13055.15000055.15275
275,5$055.15000
5000
3
055.15000
3
2
055.150002
1
2
F
F
F
P
Exponential Growth
From our calculations, a $5,000 investment into an account with an annual interest rate of 5.5% will have a value of F after t years according to the formula:
ttF 055.15000
Exponential Growth In general, P dollars invested at an annual
rate r, has a value of F dollars after t years according to:
Notice that the interest was paid on a yearly basis, while our money remained in the account. This is called compounding annually or one time per year.
tt rPF 1
Exponential Growth
What would happen if the interest was paid more times during the year?
Suppose interest is collected at the end of each quarter, (interest is paid four times each year). What would happen to our investment?
Exponential Growth
Since the annual interest rate is 5.5% this rate needs to be adjusted so that interest is paid on a quarterly basis. The quarterly rate is:
%375.14
5.541 n
rr
Exponential Growth
Interest made during 1st quarter:
Value of account after 1st quarter:
$68.754
055.05000
41
41
r
P
i
$5,068.754
055.015000
4
055.050005000
41
41
i
P
F
Exponential Growth
Interest made during the 2nd quarter:
Value of account after 2nd quarter:
$69.704
055.05068.75
41
41
42
r
F
i
$5,138.454
055.015068.75
4
055.05068.755068.75
$69.7042
41
42
i
F
F
Exponential Growth
Interest made during the 3rd quarter:
Account value after 3rd quarter:
70.65$4
055.05138.45
41
42
43
r
F
i
$5,209.104
055.015,138.45
4
055.05138.455138.45
$70.6543
42
43
i
F
F
Exponential Growth
Interest made during the 4th quarter:
Account value after 4th quarter:
$71.634
055.05209.10
41
43
44
r
F
i
$5,280.724
055.015209.10
4
055.05209.105209.10
$71.6344
43
44
i
F
F
Exponential Growth
Summarizing our results for 1 year:
$5,280.724
055.015000
4
055.015209.10
$5,209.104
055.015000
4
055.015138.45
$5,138.454
055.015000
4
055.0175.5068
$5,068.754
055.015000
5000
4
3
2
43
44
42
43
41
42
41
F
F
F
F
F
F
F
P
Exponential Growth
Notice that the exponent corresponds to the number of quarters in a year: So for 1 year there are 4 quarters So for 2 years there are 8 quarters So for 3 years there are 12 quarters So for 4 years there are 16 quarters So for t years there are 4t quarters
Exponential Growth
So the value of a $5,000 investment with an annual interest rate of 5.5% compounded quarterly after t years is given by:
t
tF4
4
055.015000
Exponential Growth
In general, P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years according to:
nt
n
rPF
1
Exponential Growth
From the last slide, we can also say:
In other words, we can find the present value (P) by knowing the future value (F).
nt
n
rFP
1
Exponential Growth
Notice for each of the 3 years the account that is compounded quarterly is worth more than the one compounded annually
n=1 n=4
t F F
1 $5,275.00 $5,280.72
2 $5,565.13 $5,577.21
3 $5,871.21 $5,890.34
Exponential Growth
It would seem the larger n is the more an investment is worth, but consider:
n=52 n=365
t F F
1 $5,282.55 $5,282.68
2 $5,581.07 $5,581.34
3 $5,896.45 $5,896.89
Exponential Growth
Notice value of the investment is leveling off when P, r, and t are fixed, but n is allowed to get really big.
This suggests that is leveling off to some special number
n
n
r
1
Exponential Growth
There is a clever technique that allows us to find this value. We let m = n/r, so that n = mr. For any value of r, m gets larger as n increases. We rewrite the expression:
rtmmrt
mP
rm
rPF
111
Exponential Growth
As m gets big,
10 2.59374246100 2.7048138291000 2.71692393210000 2.718145927100000 2.7182682371000000 2.71828046910000000 2.718281694100000000 2.718281786
m
m
11m
em
m
59057182818284.2
11
Exponential Growth So as m gets large,
This is for continuous compounding In Excel, use the function EXP(x)
rt
rtmmrt
Pe
mP
rm
rPF
111
Exponential Growth
So P dollars will grow to F dollars after t years compounded continuously at r % by the equation:
We can also find P by knowing F as follows:
rtPeF
rtFeP
Exponential Growth How do we compare investments with
different interest rates and different frequencies of compounding? Look at the values of P dollars at the end
of one year Compute annual rates that would
produce these amounts without compounding.
Annual rates represent the effective annual yield
Exponential Growth
In our current example when we compounded quarterly, after one year we had:
Notice we gained $280.72 on interest after a year. That interest represents a gain of 5.61% on $5000:
72.5280$4
055.015000
14
1
F
0561.05000
72.280y
Effective Annual Yield (y)
Exponential Growth
Effective annual yield (Discrete): find the difference between our money
after one year and our initial investment and divide by the initial investment.
Therefore, interest at an annual rate r, compounded n times per year has yield y:
111
n
n
n
r
P
Pnr
Py
Exponential Growth
You may need to find the annual rate that would produce a given yield.
Need to solve for r :
ryn
n
ry
n
ry
n
ry
n
ry
n
n
n
n
n
11
11
11
11
11
/1
/1
/1
This tells you the annual interest rate r that will produce a given yield when compounding n times a year. Note: This is only for Discrete Compounding
Exponential Growth
Effective Annual Yield (Continuous):
Annual interest rate:
1
rr
eP
PePy
1ln yr
Exponential Growth
Ex. Find the final amount if $10,000 is invested with interest calculated monthly at 4.7% for 6 years.
Soln. 49.250,13$
1000,10
1612
12047.0
nt
nrPF
Exponential Growth Ex. Find the annual yield on an
investment that computes interest at 4.7% compounded monthly.
Soln.
About 4.80%
0480.0
1048025794.1
11
1112
12047.0
n
nry
Exponential Growth
• Ex. Find the rate, compounded weekly, that has a yield of 9.1%
• Soln.
About 8.72%
087167685.0
1091.0152
1152/1
/1
nynr
Exponential Growth Examples that use the word
continuous to describe compounding period mean you use:
Ex. How much would you have after 3 years if an investment of $15,000 was placed into an account that earned 10.3% interest compounded continuously?
rtPeF
Exponential Growth
Soln.
94.430,20$
000,15 3103.0
e
PeF rt
Exponential Growth Ex. Find the annual rate of an investment
that has an annual yield of 9% when compounded continuously.
Soln.
Approx 8.62%
0862.0
109.0ln
1ln
yr
Exponential Growth
Where else can compound interest be used? Financing a home Financing a car Anything where you make monthly
payments (with interest) on money borrowed
Exponential Growth
The average cost of a home in Tucson is roughly around $225,000. Suppose you were planning to put down $25,000 now and finance the rest on a 30 year mortgage at 7% compounded monthly. How much would your monthly payments be?
Exponential Growth For a 30 year mortgage, you’ll be making
360 monthly payments.
At the end of the 360 months we want the present value (P) of all the monthly payments to add up to the amount you plan to finance, e.g. $200,000
The $200,000 is called the principal
Exponential Growth
Let’s say that Pk represents the present monthly value k months ago.
Then after 360 months, we want:
360321000,200 PPPP
Exponential Growth Since we’re borrowing money here,
each Pk can be expressed as
But where F represents the future value for Pk. In other words, F is your monthly payment.
k
k n
rFP
1
Exponential Growth
Remember we want:
So if we insert:
We have instead:
360321000,200 PPPP k
k n
rFP
1
36021
111000,200
n
rF
n
rF
n
rF
Exponential Growth
Now for a little algebra (factor out F):
Divide both sides by the stuff in [ ]
36021
111000,200n
r
n
r
n
rF
F
nr
nr
nr
36021
111
000,200
Exponential Growth The last result will tell us our monthly
payment F:
Notice that all we need to is figure out how to add up the numbers in the bottom. This is where we use Excel.
F
nr
nr
nr
36021
111
000,200
Exponential Growth
Since we’re compounding monthly at 7%, r = 0.07 and n = 12
So:
F
36021
1207.0
11207.0
11207.0
1
000,200
Exponential Growth
We’ll do the rest of our calculation in Excel
So our monthly payments F:
60.1330150.3076
000,200
1207.0
11207.0
11207.0
1
000,20036021
F
Exponential Growth
Now that we know what F is we can figure out what each Pk is.
Again, each Pk will tell us what F dollars was worth k months ago
We’ll again use Excel to answer this question.
Exponential Growth
In Excel:This number tells us that our monthly payment of $1330.60 was worth $1322.89 one month ago. Notice that as we descend down the table the values get smaller because we’re going farther back in time.
This number tells us how much of the monthly payment is for interest. Notice that as we descend the table the interest goes up. This tells us that in the beginning of a payment plan a lot of the monthly payment is toward interest and only a small portion is going toward principal while the reverse is true at the end.
End
Start
Exponential Growth
What your outstanding balance looks like with each monthly payment?
Balance
-500000
50000100000150000200000250000
0 100 200 300 400
Months
Bal
ance
$
Balance
Exponential Growth Things to notice:
After 360 months of payments of 1330.61, you’re really paying $479,019.60 on $200,000 borrowed.
The mortgage company has made 139% profit on your borrowing $200,000.