Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.
-
Upload
adrianna-kinslow -
Category
Documents
-
view
212 -
download
0
Transcript of Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.
![Page 1: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/1.jpg)
Lecture 6
Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.
![Page 2: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/2.jpg)
Review
Review Problems 2.28, 2.30(b) (reviewed other problems, took the entire class)
![Page 3: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/3.jpg)
2.28 Compute P for:
![Page 4: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/4.jpg)
![Page 5: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/5.jpg)
Problem 2.30
![Page 6: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/6.jpg)
Lecture 7
Due TuesdayRead Chapter 3 115-136Problems 3.1, 3.2, 3.5, 3.6, 3.7
![Page 7: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/7.jpg)
Chapter 3
![Page 8: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/8.jpg)
Nominal interest rate or annual percentage rate (APR)
r = the nominal interest rate per year M = the compounding frequency or the
number of interest periods per year r/M = interest rate per compounding
period Effective interest rate = the rate that truly
represents the amount of interest earned in a year or some other time period
![Page 9: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/9.jpg)
ia = (1 + r/M)M – 1
ia = effective annual interest rate
![Page 10: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/10.jpg)
Example If a savings bank pays 1 ½% interest
every three months, what are the nominal and effective interest rates per year,
Nominal %/year, r = 1 1/2% x 4 = 6% Effective interest rate per year, ia = ( 1 + 0.06/4)4 –1 = 0.061 = 6.1%
Notice that when M=1, ia = r
![Page 11: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/11.jpg)
Example
A loan shark lends money on the following conditions,
Gives you $50 on Monday, you owe $60 the following Monday
Calculate nominal interest rate , r, ? Calculate effective interest rate, ia? If the loan shark started with $50, and
stayed in business for one year, how much money would he have in one year?
![Page 12: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/12.jpg)
Example
F=P(F/P,i,n) 60=50(F/P,i,1) (F/P,i,1)= 1.2, Therefore, i = 20% per
week Nominal interest rate per year = 52
weeks x 0.20 = 10.40, 1040% = r Effective interest rate per year ia = ( 1+
10.40/52)52 –1 = 13,104 = 1,310,400% F = P(1+i)n = 50(1+0.2)52 = $655,200
![Page 13: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/13.jpg)
Effective interest rate
Who said crime doesn’t pay? To calculate the effective interest rate
for any time duration we have the equation,
ia = (1 + r/M)C – 1
ia = (1 + r/CK)C – 1
![Page 14: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/14.jpg)
where M = number of interest periods per year (ie quarterly compounding, M = 4; monthly
compounding, M = 12) C = number of interest periods per
payment period K = number of payment periods per year
(ie weekly payments, K = 52, monthly payments K = 12)
![Page 15: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/15.jpg)
Effective Interest
Notice that M = CK or M/K = C Simple case – compounding and
payment are the same
![Page 16: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/16.jpg)
Example Borrow $10,000 at yearly nominal rate of 9%.
Compounding monthly, payment monthly. You pay on the loan for 6 years. What is your monthly payment?
M = 12 (monthly payments),r/M = 0.09/12 = 0.0075 per month, n = 12 months * 6 years = 72 A = P(A/P, i, N) = 10,000 (A/P,
0.0075, 72) = $180/ month
![Page 17: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/17.jpg)
Example Just using equivalence here.
Note that you are really paying.
(1.0075)12 - 1 = 9.38% and not really 9% as stated.
![Page 18: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/18.jpg)
Harder - cases when compounding and payment occur at different time periods.
Must convert one to the same time
period.
![Page 19: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/19.jpg)
Example Invest at yearly nominal of 9%. Compounding monthly, payment
quarterly. You will invest for 8 years. If you want to have a fund of $100,000
at the end of the 8 years, how much do you have to invest in each quarter?
![Page 20: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/20.jpg)
Solution
M = 12 (monthly compound), K = 4 (quarterly payments). Since we compound more frequently
than we pay, we use the CK method. C = number of compound periods per
payment period = 3. iper = [1 + r / (CK)]C - 1 = [1 + .09/12]3 - 1 = .022
![Page 21: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/21.jpg)
Solution N = 4 * 8 years = 32 payments.
A = F (A/F, i, N) = 100,000 (A/F, .0227, 32) = 2160
![Page 22: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/22.jpg)
Example
Invest at yearly nominal 12%. Compounding semi annually,
payment quarterly and you will invest for 10 years.
If you invest $12,000 per quarter, how much will you have at the end of the 10th year?
![Page 23: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/23.jpg)
Solution M = 2 (semi-annual), K = 4 (quarterly
payments). Two alternate approaches for
compounding less frequently that payment.
(1) Bank gives us interest on the dollars invested from the point of investment, we use the CK method.
This transforms the compound period to the payment period!
![Page 24: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/24.jpg)
Solution Here C = number of compound periods
per payment period = ½ iper = [1 + r / (CK)]C - 1 = [1 + 0.12/2]1/2 -
1 = .0296 compute N = 10 years * 4 payments per
year = 40 payments. F = A (F/A, i, N) = 12,000 (F/A, 0.0296,
40) = 896,654
![Page 25: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/25.jpg)
Solution (2)
(2) In the case where the bank does not give interest on middle of period deposits we use the lumping method.
Lump all payments in an interest period at the end of the interest period.
2 payments in each semi-annual interest period.
Payment is now $24,000 semi-annually. This transforms the payment period to
the compound period!
![Page 26: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/26.jpg)
Solution (2) Now, use the r/M formula. r/M = 0.12/2
= .06. N = 10 years * 2 = 20 payments. F = A (F | A, i, N) = 24,000 (F/A, .06, 20) =
882,854 Note that the bank's strategy in the
second case has cost you about $14,000!!
![Page 27: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/27.jpg)
Continuous Compounding As an incentive in investment, some
institutions offer frequent compounding.Continuous Compounding – as M approaches infinity and r/M approaches zero
![Page 28: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/28.jpg)
Continuous Compounding
1
71828.2)1(lim
limit theCalculus from recall
11lim
11lim
/
/1
x
Kr
x
CK
CK
C
CK
ei
ex
CKri
CKri
![Page 29: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/29.jpg)
Continuous Compounding
When K = 1, to find the effective annual interest of continuous compounding
ia = er – 1
![Page 30: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/30.jpg)
Example $2000 deposited in a bank that pays
5% nominal interest, compounded continuously, how much in two years?
ia = e0.05 – 1 = 5.127% F = 2000(1 +0.05127)2 = 2210
![Page 31: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/31.jpg)
Now when compounding and payment periods coincide
1. Identify number of compounding periods (M) per year
2. Compute effective interest rate per payment period, i = r/M
3. Determine number of compounding periods, N = M x (number of years)
![Page 32: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/32.jpg)
When compounding and payment periods don’t coincide, they must be made uniform before equivalent analysis can continue.
1. Identify M, K, and C.2. Compute effective interest rate
per payment periodFor discrete compounding,
i = (1 + r/M)C – 1
For continuous compounding, i = er/K - 1
![Page 33: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/33.jpg)
Equivalence
3. Find total number of payment periods, N = K x (number of years)
4. Use i and N with the appropriate interest formula
![Page 34: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/34.jpg)
Example Equal quarterly deposits of $1000, with
r = 12% compounded weekly, find the balance after five years
M = 52 compounding periods/year K = 4 payment periods per year C = 13 interest periods/payment period
![Page 35: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/35.jpg)
Example
i = (1 + .12/52)13 – 1 =3.042% per quarter
N = K x (5) = 4 x 5 = 20
F = A(F/A, 3.042%,20) = $26,985
![Page 36: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/36.jpg)
Example You are deciding whether to invest
$20,000 into your home at 6.5% continuously compounding, or the same amount into a CD compounded semi-annually at 7%, which is the wiser investment, assume 10 years?
![Page 37: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/37.jpg)
Home Investment
r = 6.5% K = 1 ia = er/K – 1 = e0.065 –1 = 6.7%
F = 20,000(1+0.067)10 = $38,254
![Page 38: Lecture 6 Exam in One week, will cover Chapters 1 and 2. Do Chapter 2 Self test.](https://reader031.fdocuments.us/reader031/viewer/2022013100/5516667d550346b2068b60be/html5/thumbnails/38.jpg)
CD Investment r = 7% M = 2 ia = (1 + r/M)M – 1 = (1 + 7%/2)2 – 1 = 7.12% F = 20,000(1+0.0712)10 = $39,787