Math 373 Chapter 3 Homework Spring 2015jbeckley/q/WD/MA373/S15/S15... · 2015-02-24 · Chapter 3...
Transcript of Math 373 Chapter 3 Homework Spring 2015jbeckley/q/WD/MA373/S15/S15... · 2015-02-24 · Chapter 3...
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
Math 373 Chapter 3 Homework
Spring 2015
1. Trey is receiving an annuity immediate which pays 150 each year for 20 years. Calculate the present value of this annuity at an annual effective interest rate of 5%.
Solution:
1 n
n
va
i
201
11.05
150 1869.3315510.05
Or with the calculator: N=20, I/Y=5, PMT=-150, CPT PV=1869.31551
2. Davis is the beneficiary of an annuity immediate pays 3000 each year for 15 years. Calculate the
accumulated value of this annuity at an annual effective interest rate of 8.2%.
Solution:
(1 ) 1n
n
is
i
15(1.082) 13000 82735.4774
.082
With the calculator: N=15, I/Y=8.2, PMT=-3000, CPT FV=82735.4774
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
3. An annuity due pays 250 each year for 12 years. Calculate the present value of this annuity at an annual effective interest rate of 6%.
Solution:
121
11.06
(1.06)(250) 2221.7186440.06
With the calculator set to BGN: N=12, I/Y=6, PMT=-250, CPT PV=2221.718644
4. An annuity due pays 325 each year for 18 years. Calculate the accumulated value of this annuity
at an annual effective interest rate of 4%.
Solution:
18
1.04 1(325) 8668.149555
0.04
1.04
With the calculator set to BGN: N=18, I/Y=4, PMT=-325, CPT FV=8668.149555
5. Madi is the beneficiary of an annuity that will pay her 60 at the end of each month for 8 years.
Calculate the present value of this annuity at 6% compounded monthly.
Solution:
12 8
11
0.061
1260 4565.713095.06
12
N=96, I/Y=.5, PMT=-60, CPT PV=4565.713095
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
6. Madi is the beneficiary of an annuity that will pay her 60 at the end of each month for 8 years.
Calculate the present value of this annuity at an annual effective interest rate of 6%.
Solution: 12
(12)
(12)
(1.06) 112
0.00486755112
i
i
81
11.06
60 4592.7115540.004867551
N=96, I/Y=0.4867551, PMT=-60, CPT PV=4592.711
7. Colin is buying a house for 113,450. He is going to finance the entire amount with a loan which
he will repay with monthly payments for 15 years. The interest rate on Colin’s loan is 9.6%
compounded monthly.
Calculate the amount of Colin’s monthly payment. Solution:
12 15
11
.0961
12 113,450.096
12
95.1385953 113,450
1191.528214
P
P
P
N=180, I/Y=.8, PV=-113450, CPT PMT=1191.528214
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
8. Cameron is taking a loan of L . He will repay the loan with monthly payments of 97 for 10 years.
Calculate L , the amount of the loan, assuming that the loan has an annual effective interest rate of 6.9%.
Solution:
12(12)
(12)
10
1.069 112
.0055757912
11
1.06997 8469.992766
.00557579
i
i
N=120, I/Y=.557579, PMT=-97, CPT PV=8469.99
9. Mitchell has inherited 250,000. He decides to take his inheritance and buy an annuity with quarterly payments for the next 20 years. The first payment will be made immediately.
Calculate the amount of the quarterly payment if the interest rate is 6% compounded quarterly. Solution:
4 20
11
0.061
.064250,000 10.06 4
4
250,000 47.10343335
5307.468739
P
P
P
Calculator set to BGN
N=80, I/Y=1.5, PV=-250,000, CPT PMT=5307.468739
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
10. Pri wants to accumulate 30,000 to buy a new car at the end of five years. She deposits D into a
bank account at the start of each month over the next five years. The bank account earns an annual effective interest rate of 9%.
Calculate D in order that she will have the 30,000 at the end of 5 years.
Solution: 12
(12)
(12)
5 12
1.09 112
.00720732312
(1.007207323) 1(1.007207323) 30,000
.007207323
398.5572601
i
i
D
D
Calculator set to BGN
N=60, I/Y=.7207323, FV=-30,000, CPT PMT=398.5572601
11. Laura has a loan which requires payments of 1000 semi-annually for n years. The amount of the loan is 13,003.17 and the loan has an interest rate of 12% compounded semi-annually.
Compute n .
Solution: 2
2
2
11
.121
121000 13,003.170.12
2
11 .7801902
1.06
1ln ln .2198098
1.06
12 ln ln .2198098
1.06
2 26.000017
13
n
n
n
n
n
n
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
12. Antony is receiving an annuity with annual payments of 250 at the end of each year for 20 years. The present value of this annuity is 3323.59.
Calculate the interest rate for this annuity.
Solution: N=20, PMT=-250, PV=3323.59, CPT I/Y=4.25%
13. Sam deposits 1000 into a bank account at the end of each year for 17 years. During this 17 year
period, the bank pays an annual effective interest rate of i . At the end of 17 years, James has
accumulated 30,000.
Determine i . Solution:
N=17, PMT=-1000, FV=30,000, CPT I/Y=6.69116992
14. Jie has 100,000 and uses it to purchase an annuity due which pays 18,000 annually for 7 years. Calculate the annual effective interest rate for this annuity.
Solution:
Calculator set to BGN N=7, PMT=-18,000, PV=100,000, CPT I/Y=8.485631304
15. If 0.1d , calculate 12
a .
Solution:
12
14
0.10.11111
1 0.9
11
1.111116.458134178
0.11111
di
d
a
16. The accumulated value of an n year annuity is four times the present value of the same annuity.
Calculate 2100(1 ) ni .
Solution:
an(1+ i)n = 4a
n® (1+ i)n = 4
(1+ i)2n = 42 = 16
100(1+ i)2n = 100(16) = 1600
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
17. Sandra wants to accumulate a sum of money at age 65 so she can retire. In order to accomplish this goal, she can deposit 80 per month at the beginning of the month or 81 per month at the
end of the month. Calculate the annual effective rate of interest earned by Sandra.
Solution: The number of payments n is not given, but this is true for any n so we just set it up and n will cancel out.
(12)
80 81 but (1 ) 80(1 ) 81
Now divide both sides by 80(1 ) 81 1 81/ 80
Since payments are monthly, the i above is really but we need the annual effective interest r12
n n n n n n
n
a a a i a i a a
a i i
i
12 12(12)
ate
81(1+i)= 1+ 1.160754518 16.075%
12 80
ii
18. Tyler is the beneficiary of a trust fund which pays him 1000 at the end of each month forever.
At an interest rate of 9.6% compounded monthly, calculate the present value of Tyler’s
perpetuity.
Solution:
1000125,000
0.096( )
12
19. Kevin wants to fund a scholarship at Purdue. He wants the scholarship to pay 3000 at the start
of each year with the first payment being made now. These payments are to continue into perpetuity. If the annual effective interest rate is 4%, determine the amount that Kevin must donate to Purdue to fund this scholarship.
Solution:
30003000 78,000
.04
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
20. Drew is the beneficiary of a trust fund that has 100,000 in the fund. At the end of each month he or his decendents will receive 1000 forever.
Calculate the annual effective interest rate earned by the trust fund.
Solution:
(12)
(12)
1000100,000
( )12
1000 100,00012
i
i
(12)
12
1000.01
12 100,000
(1.01) 1 1.126825 1
12.6825%
i
i
i
21. Tyler has inherited $1 million. He has decided to use his inheritance to purchase one of the following:
a. A 30 year annuity immediate with annual payments of 106,079.25; or b. A perpetuity due with quarterly payments of P.
Both options are based on the same interest rate.
Calculate P.
Solution: N=30, PMT=-106,079.25, PV=1,000,000 CPT I/Y=10%
4(4)
(4)
1.1 14
0.02411374
11 1,000,000
0.0241137
23,545,92
i
i
P
P
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
22. The value of a perpetuity immediate where the payment is P is 1000 less than the value of a perpetuity due where the payment if P. Calculate P.
Solution:
1000 (1 )
1000
1000
P P Pi P
i i i
P P P PP
i i i i
P
23. A perpetuity is funded by a donation of 500,000. Payments of P are to be made at the end of
every second year. In other words, P will be paid at time 2, 4, 6, etc. If the fund earns an annual
effective interest rate of 8%, calculate P. Solution:
1.08 = 1+i(.5)
.5
æ
èçö
ø÷
i(.5)
.5= .1664
P
i(.5)
.5
= 500,000
P
.1664= 500,000
P = 83,200
24. A deferred annuity has 18 annual payments of 100 where the first payment is made in 7 years.
Calculate the present value of this deferred annuity using an annual effective interest rate of 5.75%.
Solution:
6
18100a v N=18, I/Y=5.75, PMT=-100, CPT PV = 1103.381873
61
1103.381873 788.93929661.0575
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
25. An annuity immediate has 12 annual payments of 600. The annual effective interest rate is 10%.
Calculate the accumulated value of this annuity 5 years after the last payment.
Solution:
600a12
(1.10)5 ®N=12, I/Y=10, PMT=-600, CPT FV=12830.57
12830.57(1.10)5 = 20,663.76
26. An annuity due has 12 annual payments of 600. The annual effective interest rate is 10%.
Calculate the accumulated value of this annuity 5 years after the last payment.
Solution:
27. A deferred annuity has 18 annual payments of 100 where the first payment is made in 7 years. Calculate the current value of this deferred annuity at the end of ten years using an annual
effective interest rate of 5.75%. Solution:
4 14100( )s a
4100s = N=4, I/Y=5.75, PMT=-100, CPT FV=435.8415109
14100a = N=14, I/Y=5.75, PMT=-100, CPT PV=944.0576449
435.8415109+944.0576449=1379.899156
28. John buys a series of payments. The first payment of 50 is in six years. Annual payments of 50 are made thereafter until 14 total payments have been made. Calculate the price John should pay to realize an annual effective return of 7%.
Solution:
50a14v5 ®N=14, I/Y=7, PMT=-50, CPT PV=437.2734
437.27341
1.07
æ
èçö
ø÷
5
= 311.77
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
29. Anji is the beneficiary of a perpetuity which pays 1000 per year with the first payment at the end of ten years.
Calculate the present value of Anji’s perpetuity at an annual effective interest rate of 8%.
Solution:
9 91000 112500( ) 6253.112089
0.08 1.08v
30. For a given interest rate, n
s = 21.4953 and n
a = 7.90378. Calculate n.
Solution:
1 1
1 18%
7.90378 21.4953
1 11 ( ) 1 ( )
1 1.087.90378.08
10.3676976 ( )
1.08
1ln(0.3676976) *ln( ) 13
1.08
n n
n n
n
n
ia S
i i
iai
n n
31. You are given that 10.17847n
a and 19.01987n
s . Calculate 2n
a .
Solution:
28
14*2
1 1
1 14.567%
10.17847 19.01987
1 11 ( ) 1 ( )
1 1.0456710.17847.04567
10.535149275 ( )
1.04567
1ln(0.535149275) *ln( ) 14
1.04567
11 ( )
1.04567 15.625471390.04567
n n
n n
n
n
ia S
i i
iai
n n
a
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
32. You are given 37.3537n
a and 37.7272n
a . Calculate n .
Solution:
(1 )
37.3537(1 ) 37.7272 0.01
11 ( )
1.0137.3537.01
10.626463 ( )
1.01
1ln(0.626463) *ln( ) 47
1.01
n n
n
n
n
a i a
i i
a
n n
33. Yuanzheng has a mortgage loan on his house. The loan was for 100,000 is being repaid with
level monthly payments for 15 years. The interest rate on the loan is 9% compounded monthly.
Calculate the outstanding loan balance right after the 120th payment.
Solution: First find amount of payments.
15 12100,000Pa
N=180, I/Y=9/12=.75, PV=-100,000, CPT PMT=1014.266584
120 180 120 601014.266584OLB Qa a
N=60, I/Y=.75, PMT=-1014.266584, CPT
PV=48860.64301
34. If Yuanzheng (from Problem immediately above) pays an extra 200 per month, what is his
outstanding balance right after the 120th payment? Solution:
120
120 1201014.266584 200 1214.266584 100,000(1.0075) 1214.266584OLB s
1201214.266584s N=120, I/Y=.75, PMT=-1214.266584, CPT FV=234977.9202
120100,000(1.0075) 234977.9202 10,157.78761
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
35. Robert borrowed 30,000 to be repaid with annual payments of 2000 for n years followed by a smaller payment at the end of 1n years.
The annual effective interest rate on this loan is 3.75%.
Calculate the outstanding loan balance right after the 10th payment.
Solution: If we use the retrospective approach, we do not need to know n or the amount of the smaller
payment.
10
10 10(30,000)(1.0375) 2000(s )
(30,000)(1.445043943) 2000(11.86783847) 19615.64135
OLB
36. Aaron is repaying a loan with monthly payments of 200. The interest rate on the loan is 6.9% compounded monthly.
Xiao has 14 payments remaining with the next payment due in one month.
Calculate the outstanding balance on this loan. Solution:
14200OLB a N=14, I/Y=6.9/12=.575, PMT=-200, CPT PV=2682.864348
37. Tracy borrows 30,000 to buy a new car. Her loan carries a monthly effective interest rate of 1%.
She will repay the loan by making monthly payments of 721.70.
Tracy makes k payments of 721.70. Immediately after the kth payment, she pays off the
outstanding balance of her loan by making a payment of 13,608.94. Determine k.
Solution:
OLBk =13,608.94
First find N
I/Y=1, PMT=-721.70, PV=30,000, CPT N=54
54721.70 13,608.94k N k k
OLB Qa a
I/Y=1, PMT=-721.70, PV=13,608.94, CPT N=21
54-k = 21
k=33
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
38. A loan of 10,000 is being repaid with 20 non-level annual payments. The interest rate on the loan is an annual effective rate of 6%. The loan was originated 4 years ago. Payments of 500 at
the end of the first year, 750 at the end of the second year, 1000 at the end of the third year and 1250 at the end of the fourth year have been paid. Calculate the outstanding balance
immediately after the fourth payment. Solution:
410000(1.06) accumulated value of payments made
3 2
4
500(1.06) 750(1.06) 1000(1.06) 1250 3748.208
10000(1.06) 3748.208 8876.56
39. Calculate the outstanding balance to the loan in Number 38 one year after the fourth payment
immediately before the fifth payment.
Solution:
5 4 3 210,000(1.06) 500(1.06) 750(1.06) 1000(1.06) 1250(1.06) 9409.16
40. Linhan bought a new high definition television for 2000. Linhan paid for the television using a
15 month loan with an interest rate of 9% compounded monthly.
LInhan forgot to make the 8th payment on the loan.
Determine Linhan’s outstanding loan balance at the end of the 12th month.
Solution:
Pa15
= 2000®N=15, I/Y=9/12, PV=-2000, CPT PMT=141.47
OLB12 =141.47a3®N=3, I/Y=.75, PMT=-141.47, CPT PV=418.123
4418.123 141.47(1.0075) 563.90
41. An annuity immediate has annual payments for 30 years. The payments for the first five years are 5000 each. The payments during the second five years are 4000 each. The payments during the last 20 years are 1000 each.
Calculate the present value at an annual effective interest rate of 6.5%.
Solution:
30 10 51000 3000 1000a a a
N=30, I/Y= 6.5, PMT = -1000, CPT PV = 13,058.67591
N=10, I/Y= 6.5, PMT = -3000, CPT PV = 21,566.49067
N=5, I/Y= 6.5, PMT = -1000, CPT PV = 4,155.679438
13,058.67591 + 21,566.49067 + 4,155.679438 = 38,780.84602
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
42. An annuity due has annual payments for 30 years. The payments for the first five years are 1000 each. The payments during the second five years are 3000 each. The payments during the last
20 years are 4000 each.
Calculate the present value at an annual effective interest rate of 4%. Solution:
30 10 54000 1000 2000a a a
Calculator set to BGN
N=30, I/Y= 4, PMT = -4000, CPT PV = 71,934.85853
N=10, I/Y= 4, PMT = -1000, CPT PV = 8,435.331611
N=5, I/Y= 4, PMT = -2000, CPT PV = 9,259.790449
71,934.85853 - 8,435.311611 – 9,259.790449 = 54,239.75647
43. Joseph is receiving an annuity with monthly payments for 10 years. The payments at the end of the first 60 months are 200 and the payments at the end of the second 60 months are 100. Joseph takes each payment and invests them in a fund earning 6% compounded monthly.
How much will Joseph have in the fund at the end of 10 years?
Solution:
120 60200 100s s
N=120, I/Y=6/12=.5, PMT=-200, CPT FV=32,775.87
N=60, I/Y=.5, PMT=-100, CPT FV=6977.00
32,775.87 – 6977.00 = 25,798.87
44. A 30 year annuity immediate with annual payments pays 2000 in years 1, 3, 5, …, 29 and pays
6000 in years 2, 4, 6, …, 30.
Calculate the present value of this annuity at an annual effective interest rate of 8%.
Solution: Convert i to biannual rate.
1+i(.5)
.5
æ
èçö
ø÷
.5
= 1.08 ®i(.5)
.5= .1664
2000a30 .08
+ 4000a15 .1664
N=30, I/Y=8, PMT=-2000, CPT PV=22, 515.57
N=15, I/Y=16.64, PMT=-4000, CPT PV=21,649.53
22,515.57 + 21, 649.53 = 44,165.15
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
45. Reid wants to buy an annuity immediate for his parents. The annuity will have annual payments
for 30 years. The payments will increase each year. The payment in the first year will be 8,000.
The payment in the second year will be 105% of the payments in the first year. The payment in
the third year will be 105% of the payments in the second year. This same pattern will continue
for all 30 years.
At an interest rate of 7% calculate the price that Reid should pay for this annuity. (The price is
the same as the present value.)
Solution:
1 30 318000(1.07) 8000(1.05) (1.07)172,895.56
1 1.05 /1.07
46. Zhe has won the lottery and will receive annual annuity payments at the beginning of each year
for 20 years. The first payment is 50,000. Each subsequent payment is 10% larger than the prior
payment. In other words, the first payment is 50,000. The second payment is 50,000(1.1). The
third payment is 50,000(1.1)2, etc.
Calculate the present value of Zhe’s winnings at an annual effective interest rate of 6%.
Solution:
201.10
11.06
50,000 1,454,407.831.10
11.06
47. If Zhe took each payment in the problem immediately above and invested it in a fund earning
5% interest, how much would he have at the end of 20 years.
Solution:
20
20
1.101
1.0550,000 (1.05) 4,277,912.36
1.101
1.05
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
48. Joanna is receiving an annuity due with 28 payments. The first payment is 13,000. Each
subsequent payment is 95% of the previous payment.
Calculate the present value of Joanna’s annuity at an annual effective rate of 4.5%.
Solution: 28
0.951
1.04513,000 133,083.90
0.951
1.045
49. An annuity immediate has geometrically increasing payments made annually for 24 years. The
first payment is 1000. The second payment is 11000(1.08) . The third payment is 21000(1.08)
and payments continue to increase at a rate of 8% each year.
Calculate the present value of this annuity at an annual interest rate of 8%.
Solution:
1 2 23
2 3 24
1 1 1 1
1000 1000(1.08) 1000(1.08) 1000(1.08)...
1.08 (1.08) (1.08) (1.08)
1000 1000 1000 1000...
(1.08) (1.08) (1.08) (1.08)
1000(24)22,222.22
1.08
PV
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
50. An annuity due has monthly payments for 7 years. The first payment is 800 with each successive payment being 1% larger than the previous payment. Tony invests each payment in
an account that earns 12% compounded monthly.
Calculate the amount that Tony will have in the account at the end of 7 years. Solution:
(12)(12)
1 2 8384 84
1 2 83
84 84
0.12 0.0112
800(1.01) 800(1.01) 800(1.01)(1.01) 800 ... (1.01)
(1.01) (1.01) (1.01)
800 800 800 ... 800 (1.01) 800(84)(1.01) 155,011.77
ii
AV PV
51. A perpetuity makes payments at the end of each year. The first payment is 2000. Each payment
thereafter is 103% of the previous payment. Calculate the present value of this perpetuity at an
interest rate of 8%.
Solution:
12000 0
1.0840,000
1.031
1.08
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
Chapter 3, Section 3.9
52. Brian earned a scholarship which pays him a monthly payment at the end of each month for 48
months. The first payment is 50. The second payment is 60. The third payment is 70. The
same pattern continues with each payment being 10 greater than the previous payment.
Calculate the present value of the scholarship using an interest rate of 9% compounded
monthly.
Solution:
48
48 48
10 150 48
0.09 /12 1 0.09 /12
n
n n
QPa a nv
i
a a
N=48, I/Y=.75, PMT=-1, CPT PV=40.18478
50(40.18478) + 1333.33(40.18478 - 33.533478)= 10,877.64
53. As a retirement benefit, Jeff receives an annuity due with makes annual payments for 20 years.
The first payment is 10,000. Each payment thereafter, the payment is 1500 greater than the
prior payment. In other words, the first payment is 10,000. The second payment is 11,500. The
third payment is 13,000, etc.
Calculate the present value of this retirement benefit at an annual effective rate of 5%.
Solution:
20
20 20
150010000 20 (1.05)
.05a a v
N=20, I/Y=5, PMT=-1, CPT PV=12.4622
10000(12.4622)+1500
.0512.4622 - 20v20( )é
ëêù
ûú(1.05) = 285,972.46
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
54. An annuity pays 100 at the end of the first quarter, 200 at the end of the second quarter, 300 at
the end of the third quarter, etc. The payments continue for 5 years.
Calculate the accumulated value of this annuity using an interest rate of 8% compounded
quarterly.
Solution:
20
20
20 20
100 1 .08100 20 1
.08.08 / 4 41
4
a a
N=20, I/Y=2, PMT=-1 CPT PV=16.3514
20
20100 1100(16.3514) 16.3514 20 1.02
.02 1.02
16095.18(1.02)20 = 23,916.59
55. An annuity due makes annual payments for 15 years. The first payment is 15,000. The second
payment is 14,000. The payments continue in the same pattern until the last payment of 1000
is made.
Calculate the present value of this annuity at an annual effective interest rate of 7%.
Solution:
15
15 15
100015000 15 (1.07)
0.07a a v
N=15, I/Y=7, PMT=-1, CPT PV=9.107914
15100015000(9.107914) 9.107914 15 (1.07) 90,064.74
.07v
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
56. Spenser is receiving annuity payments at the end of each quarter for 20 years. The first payment is P . The second payment is 2P . The third payment is 3P and payments continue to
increase in the same pattern.
The accumulated value of Spenser’s annuity is 75,000 using a quarterly effective interest rate of 2%.
Determine P . Solution:
Pa80
+P
.02a
80- 80v80( )é
ëêù
ûú(1.08)80 = 75,000
N=80, I/Y=2, PMT=-1, CPT PV=39.7445
(39.7445P +1166.78677P)(1.02)80 = 75,000
5882.37P = 75,000
57. Queenie is the beneficiary of a trust fund that will make a payment on each of her birthdays with the final payment on her 60th birthday. Today is Queenie’s 20th birthday and she will
receive the first payment of 50,000. Each subsequent payment will be 1000 less than the prior payment. In other words, she will receive 49,000 on her 21s t birthday, 48,000 on her 22nd birthday, etc.
Calculate the present value of Queenie’s payments at an annual effective interest rate of 5%.
Solution:
41
41 41
100050,000 41 (1.05)
.05a a v
N=41, I/Y=5, PMT=-1, CPT PV=17.29437
41100050,000(17.29437) 17.29437 41 (1.05) 661,250.05
0.05v
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
58. Sarah is receiving a perpetuity of 1000 payable at the beginning of each year. John is receiving a perpetuity immediate that pays 200 at the end of year one, 400 at the end of year two, 600 at
the end of year three, etc. The present value of Sarah’s perpetuity is equal to the present value of John’s perpetuity if the present values are calculated at i. Calculate i.
Solution:
2
2
2
2
2
2
1000 200 200(1 )
1000 1000 200 200
1000 1000 200 200
1000 800 200 0
.8 .2 0
( 1)( .2) .2
ii i i
ii
i i i
i i i
i i
i i
i i i
59. An annuity pays 10 at the end of year 2, and 9 at the end of year 4. The payments continue decreasing by 1 each two year period until 1 is paid at the end of year 20. Calculate the present value of the annuity at an annual effective interest rate of 5%.
Solution:
.5(.5) (.5)
10
10 10
(1.05) 1 0.10250.5 0.5
110 10
0.1025
i i
a a v
N=10, I/Y=10.25, PMT=-1 CPT PV=6.07913
10110(6.07913) 6.07913 10 38.2524
0.1025v
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
60. A perpetuity immediate makes annual payments. The first payment is 500. Each subsequent payment is 25 greater than the prior payment until the payments reach 750. Thereafter, all
payments will be 750.
Calculate the present value of this perpetuity at an annual interest rate of 6%. Solution:
Break this into two parts: 11 year P & Q annuity and a perpetuity discounted by 11 years.
11
11 11
25500 11
0.06a a v
N=11, I/Y=6, PMT=-1, CPT PV=7.88687
11
11
25500(7.88687) 7.88687 11 4815.1922
0.06
7506584.84
0.06
4815.1922 6584.84 11400.04
v
v
You could also break it up into a 10 year P&Q formula and perpetuity discounted 10 years.
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
61. Suyi invests 1000 at the end of each year for 20 years into Fund A. Fund A earns an annual effective interest rate of 5%. At the end of end of each year, the interest is removed from Fund
A and invested in Fund B. Fund B earns an annual effective interest rate of 8%. What is the total amount that Suyi will have at the end of 20 years.
Solution:
19 19
19 19
1000(0.05) 50
5050 19 (1.08)
0.08a a v
N=19, I/Y=8, PMT=-1, CPT PV=9.6036
19 1950 A= 50(9.6036) (9.6036 19 ) (1.08) 16,101.23
.08
Fund B=1000(20) 20,000 20,000 16,101.23 36,101.23
Fund v
Total
62. Ryan invests 100,000 in Fund A today which earns an annual effective interest rate of 8%
interest. The interest on Fund A is paid at the end of each year into Fund B which earns an
annual effective interest rate of 9%. The interest on Fund B is also paid out at the end of each year into Fund C which earns an annual effective interest rate of 10%. At the end of 12 years,
Ryan liquidates all three Funds. How much does Ryan have? Solution:
11 11
11 11
100,000(0.08) 8000
8000(0.09) 720
720720 11 (1.10)
0.10a a v
N=11, I/Y=10, PMT=-1, CPT PV=6.4951
11 11720720(6.4951) 6.4951 11 (1.10) 67,566.84
.1
Fund A + Fund B+Fund C=100,000 12(8000) 67,566.84 263,566.84
v
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
63. Colleen invests P into Fund A at the start of each year for 25 years. Fund A earns an annual
effective interest rate of 10%.
At the end of each year, Colleen withdraws the interest from Fund A and deposits it into Fund B
which earns an annual effective interest rate of 6%.
At the end of 25 years, Colleen has a total of 135,000 in both accounts.
Determine P .
Solution:
25 25
25 0.06 25 0.06
25 25
25 0.06 25 0.06
After 25 years, Fund A will have 25P.
After 25 years, Fund B will have:
0.10.1 25(1.06) (1.06)
0.06
0.125 0.1 25(1.06) (1.06) 135,000
0.06
PPa a
PP Pa a
P
25 0.06
25 25
1; / 6; 25; 12.78335616
135,0001682.02
0.125 0.1(12.78335616) (12.78335616 25(1.06) (1.06)
0.06
MT I Y N CPT PV a
P
February 24, 2015 Copyright Jeffrey Beckley 2014 2015
64. Will has a loan of 50,000 which is being repaid with annual payments of 6000 followed by a smaller drop payment. The loan has an interest rate of 7%. Calculate the amount of the drop
payment.
Solution: PV=50,000, PMT=-6000, I/Y=7, CPT N=12.9395
So use N=12
12
1250,000(1.07) 6000 5,278.87
5278.87(1.07) 5648.39
s
65. Daniel borrowed 30,000 to be repaid with monthly payments of 1000 with the final payment being a balloon payment. The interest rate on the loan is 14.4% compounded monthly.
Calculate the amount of the balloon payment.
(12) 0.144.012
12 12
i
PV=30,000, PMT=-1000, I/Y=1.2, CPT N=37.413 So use N=36
36
3630,000(1.012) 1000 1393.10
1393.10(1.012) 1409.82
s
66. Mingjing has a loan of 50,000 which is being repaid with monthly payments of 1500. The final
payment will be a Balloon payment of B. The interest rate on the loan is 18% compounded monthly.
Aiman has an identical loan except the last payment is a Drop payment of D.
Calculate B-D. Solution:
PV=50,000, PMT=-1500, I/Y=18/12, CPT N=46.555
For B use N=45, for D use N=46
45
45
46
46
: 50,000(1.015) 1500 2289.35(1.015) 2323.69
: 50,000(1.015) 1500 823.69(1.015) 836.04
2323.69 836.04 1487.65
B s
D s
B D