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Solution-TVM-1 - Tutorial for chapter the value of money
Fundamental of finance management (International University - VNU-HCM)
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Fundamentals of Financial Management 2019
TUTORIAL – Lecture 2: TIME VALUE OF MONEY
1. Textbook, Chapter 4: 4.1, 4.2, 4.3, 4.4, 4.6, 4.8, 4.13, 4.14. 4.15, 4.18, 4.19, 4.20, 4.23, 4.24,
4.26, 4.25, 4.30, 4.31, 4.33, ST2
Any problem do not mention to use compound interest or simple interest, use
compound interest
2. A wealthy relative give Barney $5,000 to help provide for his new born child’s university
fee. He decides to invest this money at 10% p.a. until his child is ready to go to university.
How much will be in the account 18 years from now?
Present value = $5,000
$5,000
i = 10% ; n = 18
Future value at year 18th?
today 1
…
2
18
Answer: FV = PV(1+i)n  The amount in account in 18 years from now is
FV = 5000*(1+10%)18 = $27,299.59
3. Barney wants to provide $20,000 for his new born child’s education, which will begin 18
years from today. How much should Barney invest now, if interest rate is 10% p.a.?
Future value = $20,000
$20,000
i = 10% ; n = 18
Present value now?
Today
1
2
…
18
𝑭𝑽
Answer: PV = (𝟏+𝒊)𝒏  the amount to invest NOW (at the present) =
𝟐𝟎𝟎𝟎𝟎
(𝟏+𝟏𝟎%)𝟏𝟖
= $3,597.18
4. How long will it take for a $1,000 investment to double in size when invested at the rate of
8% per year?
n = ???
Present value = $1,000
1000
Future value = $2,000 (double PV)
today
i = 8% , n = ?
Answer: PV =
𝑭𝑽
(𝟏+𝒊)𝒏
 PV =
𝟐𝑷𝑽
(𝟏+𝟖%)𝒏
 (1+8%)n = 2  n*ln(1+8%) = ln(2)  n =
Prepared by: Ms. C. Tien and Ms. H. Nhung
2000
𝒍𝒏(𝟐)
𝒍𝒏(𝟏+𝟖%)
= 9 years
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5. Suppose you have $500 to invest and you want to know how long will it take for this
amount to double in size if the interest rate is 10% p.a.
n = ???
Present value = $500
Future value = $1000 (double PV)
i = 10% , n = ?
500
1000
today
Answer: PV =
𝑭𝑽
𝒏  PV =
(𝟏+𝒊)
𝟐𝑷𝑽
(𝟏+𝟏𝟎%)𝒏
 (1+10%)n = 2  n*ln(1+10%) = ln(2)
𝒍𝒏(𝟐)
 n = 𝒍𝒏(𝟏+𝟏𝟎%) = 7.27 years
6. You borrow money on your credit card at 17.5% p.a., compounding quarterly. What is the
effective annual interest rate?
Answer: Effective annual interest rate = (𝟏 +
𝒂𝒏𝒏𝒖𝒂𝒍 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 𝒏
)
𝒎
- 1 = (𝟏 +
𝟏𝟕.𝟓% 𝟒
)
𝟒
- 1 = 18.68%
7. The following interest rates are being offered by three competing banks: 4% compounded
monthly; 4.1% compounded quarterly; 4.15% compounded annually. Which one is the
most attractive?
We can compare these interest rate by using its effective annual interest rate
1st bank: i = 4%, compound monthly
Effective annual interest rate = (1 +
𝑎𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑛
)
𝑚
2nd bank: i = 4.1% compound quarterly
Effective annual interest rate = (1 +
𝑎𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑛
)
𝑚
3rd bank: i = 4.15%, compound annually
- 1 = (1 +
- 1 = (1 +
4% 12
)
12
4.1% 4
)
4
- 1 = 4.07%
- 1 = 4.16%
Effective annual interest rate = 4.15%
 The 2nd bank gives us the highest interest rate  MOST ATTRACTIVE
8. Barney lends $100,000 to John, his cousin in 10 years at 5% p.a.
a. how much John have to pay after 10 years (if use simple interest rate)
b. how much John have to pay after 10 years (if use compounded interest rate)
c. assume that Barney lends $100,000 to John, using simple interest rate, but wants
to receive the payment which is equal to that if using compounded interest. What
should the interest rate be?
Present value = $100,000 ; n = 10; i = 5% p.a
100,000
0
1
FV
2
…
10
a. FV = PV(1 + i*n) = 100,000(1 + 5%*10) = $150,000
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b. FV = PV(1 + i)n = 100,000(1 + 5%)10 = $162,889.46
c. Suppose i’ is what we are looking for
PV(1 + i’* n) = PV(1 + i)n  (1 + 10i’) = (1 + 5%)10  i’ =
(1+5%)10 − 1
10
= 6.2%
9. You have just received a letter from your aunt, which advises that she has written
you into her will. On her passing, you will receive an inheritance of $50,000.
Assuming r = 8% p.a., what’s the inheritance worth in today’s dollars if your aunt
lives another 5 years? 15 years?
FV = $50,000 ; i = 8%
PV
1
2
For n = 5
PV =
4
5
50,000
𝐹𝑉
(1+𝑖
For n = 15
PV =
3
)𝑛
𝐹𝑉
(1+𝑖
)𝑛
=
=
50,000
(1+8% )5
50,000
(1+8% )15
= $34,029
= $15,762
10. On a contract you have a choice of receiving $25,000 six years from now or
$50,000 twelve years from now. At what implied compound annual interest rate
should you be indifferent between the two contracts?
I propose 3 ways to solve this problem, I hope you guys would have a comprehensive way in understanding
the time value of money.
The contract would be indifferent if they have the same present value now or the same future value at
year 6th or the same future value at year 12th
1st way: Discount both amount back to present value now
PV =
𝐹𝑉
(1+𝑖)𝑛

25000
(1+𝑖)6
=
50000
(1+𝑖)12
now
50000
 (1 + i)6 = 25000  i = 12.24% PV
6
25,000
12
50,000
2nd way: Discount 50,000 at year 12th back to year 6th  n = 6
now
PV =
𝐹𝑉
(1+𝑖)𝑛
50000
 25000 = (1+𝑖)6  i = 12.24%
6
25,000
12
50,000
3rd way: Compound 25,000 at year 6th to year 12th  n = 6
now
FV = PV(1 + 𝑖)𝑛  50000 = 25000(1 + i)6  i = 12.24%
Prepared by: Ms. C. Tien and Ms. H. Nhung
6
25,000
12
50,000
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Fundamentals of Financial Management 2019
11. You have just won the prize in the State lottery. A recent innovation is to offer prize
winners a choice of payoffs. You must choose one of the following prizes:
a. $1,000,000 paid immediately
b. $ 600,000 paid exactly one year from today, and another $600,000 paid exactly 3
years from today
c. $ 70,000 payment at the end of each year forever (first payment occurs exactly 1 year
from today)
d. an immediate payment of $600,000, then beginning exactly 5 years from today, an
annual payment of $50,000 forever
e. an annual payment of $200,000 for the next 7 years (first payment occurs exactly 1
year from today
f. Require: You believe that 8% p.a. compounded annually is an appropriate discount rate.
Assuming you wish to maximize your current wealth, which is the best prize?
Calculate current wealth  Present value
n = 8%
a.
PV = $1,000,000
b.
PV =
c.
600,000
1 +
(1+8%)
600,000
(1+8%)3
= $1,031,855
PV
1
2
Now
600,000
3
600,000
Forever  perpetuity.
Note for perpetuity: the present value of perpetuity is the value before the time the first
payment occur (1 year - compound yearly, 1 month – compound monthly, 3 months –
compound quarterly or 6 month – compound semi-annually before the 1st payment occur)
𝑐
PV = 𝑖 =
d.
70,000
8%
= $875,000
4
PV
PV = 600,000 +
(𝟏+𝟖%)𝟒
= 600,000 +
𝟓𝟎,𝟎𝟎𝟎
𝟖%
(𝟏+𝟖%)𝟒
𝐶
𝑖
(1 −
1
(1+𝑖 )𝑛
) =
200000
8%
(1 −
1
(1+8% )7
7
… … ..
= $1,059,393
e. This is an ordinary annuity – 1st payment occur 1 year from now
Now 1
PV =
6
PV4 50K 50K 50K
600,000
𝐏𝐕𝟒
5
) = $1,041,274
2
3
PV 200K 200K 200K
Prepared by: Ms. C. Tien and Ms. H. Nhung
…… 7
200K
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12. Ann buys a new computer. The stated priced is $2,000, but the retailer has a special offer
whereby she have to pay $400 immediately and then another $400 each year of the next 7
years. If the interest rate is 8% p.a., what is the effective cash price?
We will discount the annuity of $400 each year of the next 7 years back to the present to compare with
the payment of $2,000 immediately.
Now 1
400
2
3
4
5
6
400 400 …
7
400
Because this is an annuity of $400 each year and the first payment is now (pay immediately) so we
use the formula of annuity due
We have: C = $400 ; i = 8% ; n = 7
Answer: PV =
𝑪
𝒊
𝟏
[𝟏 − (𝟏+𝒊)𝒏 ](𝟏 + 𝒊) =
𝟒𝟎𝟎
𝟖%
𝟏
[𝟏 − (𝟏+𝟖%)𝟕 ] (𝟏 + 𝟖%) = $𝟐, 𝟐𝟒𝟗
(this is the effective cash price)
This is present value of the amount we have to pay if we choose the annuity due of $400, so
we should choose to pay $2,000 now instead of the “special offer” of the retailer
13. Mr. and Mrs. Haiku have two offers for their apartment in Tokyo. The first calls for
payment of ¥50 million now and ¥50 million in one year. The second would pay ¥90
million immediately. The appropriate interest rate is 4%. Which offer has the higher PV?
1st call:
PV = 50,000,000 +
2nd call:
50,000,000
(1+4%)
= ¥98,076,923
PV = ¥90,000,000
 The 1st call has higher PV
14. Muffin Megabucks is considering two different savings plans in 10 years. The first plan
would have her deposit $500 every six months, and she would receive interest at 7% p.a.
(compounding semi-annually). Under the second plan she could deposit $1,000 every year
with the rate of interest of 7.5% p.a. (compounding annually).
Which plan should Muffin use? (Assuming that the initial deposit of Plan 1 would be made
6 months from now, and with Plan 2, one year hence)
Since this is the saving plan which will be used after the next 10 years, so we should consider the future
value of each plan to compare which one results a higher future value
1st plan:
C = $500 per six month, i = 7%, compound semi-annually , n = 10
FV = C [
(1+𝑖)𝑛 −1)
𝑖
] = 500 [
(1+
7% 10∗2
)
−1
2
7%
2
] ] = $14,139
now 1
Prepared by: Ms. C. Tien and Ms. H. Nhung
2
500 500 …..
20
500
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2nd plan
C = $1,000 per year, i = 7.5%, compound annually, n = 10
FV = C [
(1+𝑖)𝑛 −1)
] = 1000 [
𝑖
(1+7.5%)10 − 1
] = $14,147
7.5%
now 1
2
1000 1000
10
…
1000
So we should choose the 2nd plan because it has the greater future value
15. Kate’s financial advisor tells her that she will need $2 million to fund her retirement. She
plans to work for another 30 years before retiring. She will make 30 contributions to a
pension plan. How much will each contribution be, if the interest rate is 9% p.a.?
n = 30 ; i = 9% ; FV = $2,000,000 ; C = ?
This is an ordinary annuity  FV = C
(1+𝑖 )𝑛 −1
𝑖
 2,000,000 = C
Each equal contribution is $16,472.2 per year
(1+9% )30 −1
9%
= = $16,472.7
16. Mary has just retired and has $1 million in her retirement account. Her bank offers an
arrangement whereby the bank takes her $1 million now and pays her $110,000 at the end
of each year for the next 20 years. Is it a fair deal, if the offered rate is 10% p.a.?
We will compare this deal by using its present value now or future value in the next 20 years
1st way: using present value
C = $110,000 ; n = 20 ; i = 10% ; PV = ?
PV =
𝑪
𝒊
𝟏
[𝟏 − (𝟏+𝒊)𝒏] = =
𝟏𝟏𝟎,𝟎𝟎𝟎
𝟏𝟎%
𝟏
[𝟏 − (𝟏+𝟏𝟎%)𝟐𝟎] = $936,492 < $1,000,000  not a FAIR deal
2nd way: using future value in year 20th
C = $110,000 ; n = 20 ; i = 10% ; FV = ?
FV1 = C [
(1+𝑖)𝑛 −1)
𝑖
] = 110000 [
(1+10%)20 − 1
10%
] = $6,300,250
PV = $1,000,000 ; n = 20 ; i =10% ; FV = ?
FV2 = PV(1+i)n = 1000000(1 + 10%)20 = $6,727,450
FV1 < FV2  still not a FAIR deal
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17. Joe Hernandez has inherited $25,000 and wishes to purchase an annuity that will provide
him with a steady income over the next 12 years. He has heard that the local bank is
currently paying 6% p.a. (compounding annually). If he were to deposit his fund, what is
the equal amount would he be able to withdraw at the end of each year for 12 years?
PV = $25,000 ; n = 12 ; i = 6% ; C = ???
PV =
𝑪
𝒊
𝟏
[𝟏 − (𝟏+𝒊)𝒏 ]  25000 =
𝑪
𝟔%
𝟏
[𝟏 − (𝟏+𝟔%)𝟏𝟐]  C = $2,982
18. Your company anticipates the introduction of environmental protection laws 3 years from
now. Under these laws you will have to pay an environment tax of $5,000 at the end of
each year. If the rate is 6% p.a., what is the present value of your company’s obligation
under this law?
(Note: the first payment will be three years from now)
PV
2
3
4
5
… … ..
PV2 5K 5K 5K
This is a perpetuity with C = 5,000 ; i = 6%
PV =
𝑷𝑽𝟐
(𝟏+𝟔%)𝟐
=
𝟓,𝟎𝟎𝟎
𝟔%
(𝟏+𝟔%)𝟐
= $74,166.37
19. Vernal Equinox wishes to borrow $10,000 for three years. A group of individuals agrees to
lend him this amount if he contracts to pay them $16,000 at the end of the three years.
What is the implicit compound annual interest rate implied by this contract?
PV = $10,000 ; n = 3 ; FV = $16,000 ; i = ??
FV = PV(1+i)n  16000 = 10000(1 + i)3  i =
3
√
16000
10000
– 1 = 16.96%
20. The Happy Hang Glide Company is purchasing a building and has obtained a $190,000
mortgage loan for 20 years. The loan bears a compound interest rate of 17% p.a. and calls
for equal annual installment payments at the end of each of 20 years. What is the amount
of annual payment?
PV = $190,000 ; n = 20 ; i = 17% ; C = ?
PV =
𝐶
1
[1 − (1+𝑖)𝑛 ]  190000 =
𝑖
𝑪
𝟏
[𝟏 − (𝟏+𝟏𝟕%)𝟐𝟎 ]  C = $33,761
𝟏𝟕%
21. Mary enters into a loan agreement to borrow $90,000 to help finance the purchase of her
new home.
a. The agreement specifies the term of 20 years with monthly repayment at the fixed rate
of 9% p.a. (compounded monthly). What is her monthly payment?
b. Five years has passed. A rival lender offers to refinance Mary her loan at fixed rate of 8%
p.a. (compounded monthly). Cost associated with this refinancing is $1,500. Should
her refinance?
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c. Suppose 9 years have passed since Mary enters the original loan. She’s considering
making an extra payment of $10,000 off her loan. If she plans to keep the term of the
loan the same, how much will her monthly repayment reduce?
PV = $90,000 ; n = 20 ; i = 9% ; compound monthly
a. PV =
b.
𝐶
𝑖/12
i’ = 8%
1
[1 − (1+𝑖/12)𝑛∗12 ]  90000 =
𝐶
9%/12
1
[1 − (1+9%/12)20∗12]  C = $809.75
The refinance decision is based on the monthly amount Mary has to pay, so we will compare C of 2
options. Determine the new C’ of the refinancing by the following steps:
1. Calculate the amount Mary still owned after 5 years with monthly payment C; Suppose we are in
year 5th and there are still 15 years left to finish the mortgage  n = 15
PV5
=
𝐶
𝑖/12
1
[1 − (1+𝑖/12)𝑛∗12 ] =
809.75
9%/12
1
[1 − (1+9%/12)𝟏𝟓∗12 ] = $79,836
2. Calculate the amount we have to pay if we refinance
Amount have to pay from now (in the 5th year) = $79,836 + $1,500 = $81,336
3. Calculate the new monthly payment C’ with the new debt of $81,336 and new interest i’ = 8%
PV5 =
𝐶′
𝑖/12
1
[1 − (1+𝑖/12)𝑛∗12 ]  81336 =
 C’ = $777.28 < C = $809.75
𝐶′
8%/12
1
[1 − (1+8%/12)15∗12 ]
 Refinance
c. Suppose her new monthly payment is C1 ; We calculate the new C1 and compare it
with C
1. Calculate the amount still owned after 9 years. Suppose now we are in year 9th, so there are 11
years left until we finish the mortgage  n = 11
PV9
=
𝐶
𝑖/12
1
[1 − (1+𝑖/12)𝑛∗12 ] =
809.75
9%/12
1
[1 − (1+9%/12)𝟏𝟏∗12 ] = $67,700
2. Calculate the amount we have to pay after additional payment
After we add extra payment of $10,000, we still owned PV9’ = $67,700 - $10,000 = $57,700
3. Calculate the new monthly payment C1 with the new PV9’
PV9’ =
𝐶1
𝑖/12
1
[1 − (1+𝑖/12)𝑛∗12 ]  57700 =
𝐶1
9%/12
 C1 = $690.14
1
[1 − (1+9%/12)11∗12 ]
Her old monthly payment is C = $809.75, so she has reduced $809.75 - $690.14 = $119.61 of her
monthly payment
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