February 11, 2014 Math 373 Fall 2013 Homework – Chapter 2 Chapter 2, Section 2 1. Brandon borrows 2000 at an interest rate of 7.2% compounded monthly. Brandon repays the loan at the end of 6 years. Determine the amount that Brandon repays. Solution: 2000(1 + .072 12 ) 12i 6 = 3076.70 2. Guanling invests 600 in an account. After 6 years, Guanling has 1000. Calculate the annual effective interest rate that Guanling earn on his investment. Solution: 600(1 + i ) 6 = 1000 (1 + i ) 6 = 1.666666 (1 + i ) 6 6 = 1.666666 6 1 + i = 1.0888669 i = 8.88669% 3. Kathy invests 10,000 in an account earning a nominal rate of interest of 8% compounded quarterly. Determine the amount of time in years that it will take for Kathy’s investment to grow to 25,000. Solution: 10,000(1 + .08 4 ) 4i T = 25, 000 (1 + .02) 4i T = 2.5 ln(1.02) 4i T = ln 2.5 4T ln(1.02) = ln 2.5 T = ln 2.5 4 ln(1.02) = 11.5678
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February 11, 2014
Math 373
Fall 2013
Homework – Chapter 2
Chapter 2, Section 2
1. Brandon borrows 2000 at an interest rate of 7.2% compounded monthly. Brandon repays the
loan at the end of 6 years.
Determine the amount that Brandon repays.
Solution:
2000(1+
.072
12)12i6 = 3076.70
2. Guanling invests 600 in an account. After 6 years, Guanling has 1000.
Calculate the annual effective interest rate that Guanling earn on his investment.
Solution:
600(1+ i)6 = 1000
(1+ i)6 = 1.666666
(1+ i)66 = 1.6666666
1+ i = 1.0888669
i = 8.88669%
3. Kathy invests 10,000 in an account earning a nominal rate of interest of 8% compounded
quarterly.
Determine the amount of time in years that it will take for Kathy’s investment to grow to
25,000.
Solution:
10,000(1+.08
4)4iT = 25,000
(1+ .02)4iT = 2.5
ln(1.02)4iT = ln2.5
4T ln(1.02) = ln2.5
T =ln2.5
4 ln(1.02)= 11.5678
February 11, 2014
4. Avisek deposits 1000 in the bank earning an annual effective interest rate of i . Based on the
Rule of 72, Avisek believes that he will have 4000 at the end of 20 years.
Calculate the amount that Avisek will really have at the end of 20 years.
Solution:
Rule of 72: 0.72
i= amount of time for investment to double
In 20 years the investment would need to double twice in order to reach 4,000.
It would take the investment 10 years to double the first time, so we use this with the rule of 72.
.72
i= 10
i = .072
Now find the amount Chris will really have using i = 0.072.
(Initial Amount)(1 + i)T = (Final Amount)
1000(1.072)20 = 4016.94
5. Colleen invests 5250 in an account earning an annual effective interest rate of 7.2%. Colleen
wants to know when she will have 10,500. Her banker estimates that it will take X years using
the Rule of 72. Colleen calculates it exactly and gets Y years.
Calculate X Y .
Solution:
Banker:
X = 0.72 0.72
0.072i = 10 years
Colleen:
5,250 1.072 10,500
1.072 2
ln 1.072 ln 2
9.9696
Y
Y
Y
Y years
X – Y = 10 – 9.9696 = 0.0304
February 11, 2014
Chapter 2, Section 3
6. Camille borrows 1000 to buy a high definition television. She agrees to repay the loan with a
payment of P at the end of 2 years and an additional payment of P at the end of 3 years.
Camille is paying on the loan is an annual effective interest rate of 7% on the loan.
Determine P .
Solution:
1000(1.07)3 = P(1.07)+ P
1000(1.07)3 = P[(1.07)+1]
1225.04 = 2.07P
P = 591.81
7. Jose borrows 5000 from Yong. Jose repays the loan with a payment of 3000 at the end of 1 year
and 2500 at the end of 2 years.
Calculate the annual effective interest rate that Jose will pay on this loan.
Solution:
5000(1+ i)2 = 3000(1+ i)+ 2500
5000x2 - 3000x - 2500 = 0
Now use the quadratic formula and solve for x
3000 ± 30002 - 4(5000)(-2500)
2(5000)= 1.068115 = x
x = 1+ i
i = 1.068115 -1 = 6.8115%
February 11, 2014
8. Matt has 10,000 in his bank account today. Five years ago, he deposited 5000 into his account.
Additionally, two years ago, Matt deposited 4000 into his account.
Determine the annual effective interest rate that Matt has earned over the five year period.
Solution:
For this problem, use the financial calculator’s cash flow functionality.
CF0=5000
CO1=0
F01=1
CO2=0
FO2=1
CO3=4000
FO3=1
CO4=0
FO4=1
CO5= -10000
IRR, CPT
IRR=2.8899%
9. Devi deposits 1000 into a bank account on January 1, 2011. Devi also deposits 300 on October
1, 2011. On December 31, 2011, Devi has 1400.
Determine the annual effective interest rate earned by Devi during 2011.
b. Calculate Kylie’s exact annual dollar weighted return using the cash flow functionality in
the BA II Plus.
Solution:
Treat Cash Flows as monthly:
0 5000
01 0; 01 1
02 5000; 02 1
03 0; 03 3
04 4000; 04 1
05 0; 05 9
06 3000; 06 1
07 0; 07 7
08 15,000; 08 1
2.478375127%
Since the periods are months, the IRR is the monthly
effective intere
CF
C F
C F
C F
C F
C F
C F
C F
C F
IRR CPT
(12)
12(12)
12
st rate which is 2.478375127%12
(1 ) 1 (1 0.02478375127) 34.149%12
i
ii i
February 11, 2014
18. Jing has a bank account balance of X on January 1, 2010. During the next two years Jing deposits 8000 into his account and withdraws 3000. On January 1, 2012, Jing has a bank account balance of 28,400. Assuming that all cash flows occur on January 1, 2011, Jing estimates his annual dollar weighted return assuming simple interest to be 4.5383664%. Calculate X.
Solution:
0.5
2
28400
0.5
8000 3000 5000
23400
(1 ) (1 ) 1.045383664
1.045383664 1 0.09287005
234000.092827005
(1 ) 5000(0.5)
0.092827005( 2500) 23400
21200
A X
B
k
C
I X
j i
j
I Xj
A C k X
X X
X
Chapter 2, Section 7
19. Wenxue invested 50,000 in an account three years ago. One year ago, the account had a
balance of 100,000 and Wenxue withdrew $50,000.
Today, Wenxue has 30,000 in the account.
Determine the annual effective time weighted return earned by Wenxue on this account.
Solution:
1+ j1 =B0
B1
=100,000
50,000= 2
1+ j2 =B2
B1 +C1
=30,000
100,000 - 50,000= .6
1+ jTW = (1+ j1)(1+ j2 ) = (2)(.6) = 1.2
1+ iTW = (1+ jTW )1/T
1+ iTW = (1.2)1/3 ® iTW = .06266
February 11, 2014
20. Kylie has an account at Boing Brokerage House. She has made the following deposits and
withdrawals over the last two years.
Date Deposits Withdrawals Balance Before Cash Flow
January 1, 2009 5000 0 0
March 1, 2009 5000 0 4000
July 1, 2009 0 4000 10,000
May 1, 2010 3000 0 9,000
December 31, 2010 0 0 15,000
Calculate Kylie’s annual time weighted return.
Solution:
1+ j1 =4000
5000= .8
1+ j2 =10000
4000 + 5000= 1.111111
1+ j3 =9000
10000 - 4000= 1.5
1+ j4 =15000
9000 + 3000= 1.25
1+ jTW = (.8)(1.1111)(1.5)(1.25) = 1.66667
1+ iTW = (1.66667)1/2 ® iTW = 29.099%
February 11, 2014
21. Evgeny invests in the Marano Mutual Fund. Over the next two years, Evgeny realizes an annual
time weighted yield of 12.5%.
Evgeny initially invests 100,000 with Marano and has 103,551.14 at the end of two years.
During the two year period, Evgeny also withdrew an amount of C to buy a new car. Before
the amount was withdrawn, the fund was worth 110,000.
Calculate C.
Solution:
First, let’s find the time weighted yield for the period of two years.