TVM Sample Problems (ver. 2.1 Fall 13)
More Than One Future Cash Flow?
Yes
No
Single FV
(4 parameters)
Even or Uneven Cash Flows
Even
Uneven
Annuity
(5 parameters)
CF Worksheet
More Than One
Pmt per year? Use
Periodic Rate
More Than One Pmt per year?
Yes
No
Compute n
Draw Time Line
Draw Time Line
Add Cash Flows
Add Cash Flows
Ord. Annuity or An. Due?
Inventory What You Know
To Identify What
You Don’t Know
List Inputs
Note: Usually (but not always),
One of the Five Parameters
Is Irrelevant
Know What You Need to Find?
No
Yes
Compute the Answer
1
2
1. To complete your business school education, you will need $10,000 a year for
the next four years, starting next year (that is, you will need to pay for one year’s
worth of schooling at the beginning of the year, one year from today). Your rich
uncle offers to put you through school and he will deposit a lump sum into a
saving account paying 7% p.a. sufficient to defray these expenses. The deposit
will be made today. How large must the deposit be? (Assume your uncle is rich
because he won the lottery and he doesn’t know much about finance stuff.)
PMT = 10,000
nominal = 7%
0
1
2
3
4
PV =
Soln. Opt 2: P/Y=1, N=4, I/Y=7, PMT=10000; CPT,PV: $33,872.11
2. You are considering financing a new car which cost $51,300 with an
amortized loan. The nominal rate is 2.9% p.a., the term of the loan is 6 years
and you will make monthly payments. How much will each payment be?
PV = $51,300
0
nominal = 2.9%
1
2
m = 12
T=6
n = m x T = 12 x 6 = 72
70
71
72
PMT = ?
rperiodic = nominal / m = 2.9/12 = 0.24167
P/Y=1, N=72, I/Y= 0.24167, PV=51300; CPT,PMT: PMT = $777.14
OR
P/Y=12, N=72, I/Y=2.9, PV=51300’ CPT,PMT: PMT = $777.14
3
3. How many years will it take for your savings account to accumulate $1m if
it pays 4% interest p.a. compounded semiannually and you deposit $10k every
6-months at the end of the 6-month period?
FV = $1m
r = 4%
0
1
2
3
4
5
n-2
n-1
n=?
PMT = (-)10,000
a) Find number of periods
rperiodic = nominal / m = 4%/2 = 2%
P/Y=1, I/Y= 2, PMT=-10000, FV=1000000; CPT,N: N = 55.48 periods
OR
P/Y=2, I/Y= 4, PMT=-10000, FV=1000000; CPT,N: N = 55.48 periods
b) Find number of years: Years = N/m = 55.48/2 = 27.74 years
4. Today you deposited $1,300 in a bank that pays 5% p.a., compounded
quarterly. How much money would you have in this account 20 months from
now?
m = 4, T = 20/12 = 1.666667
FV = ?
n = m x T = 6.666667
nominal = 5%
0
1
2
5
PV = $1,300
6
7
n = 6.666667
rperiodic = rnominal/m = 5%/4 = 1.25%
P/Y=1, N=6.666667, I/Y=1.25, PV=1300; CPT,FV: FV = $1,412.25
OR
P/Y=4, N=6.666667, I/Y=5, PV=1300; CPT,FV: FV = $1,412.25
4
5. What is the future value of an annuity due yielding 9.6400% p.a. that pays
quarterly payments of $1,000 for 9 mos? (Do the math; do not use the
financial functions on your calculator. Draw a cash flow diagram)
nominal = 9.64%
T = 9/12, m = 4, n = T x m = 9/12 x 4 = 3
PMT = 1,000
0
1
2
3
FV = ?
FV = PV(1 + r/m)n + PV(1 + r/m)n-1 + …………..
= 1000(1 + 0.0964/4)3 + 1000(1 + 0.0964/4)2+ 1000(1 + 0.0964/4)1
= 1000(1.0241)3 + 1000(1.0241)2 + 1000(1.0241)1
= 1000(1.0741) + 1000(1.0488)+ 1000(1.0241)
= 1,074.0564 + 1,048.8000 + 1,024.1000
= $3,146.96
5
6. On 1 January 2008 you deposited $700 in a mutual fund that has been
yielding a constant 6% p.a. for the last several years. You plan to deposit $700
every 3 months thereafter for the next 4 years (i.e. the next deposit will be
made 1 April, the subsequent deposit on 1 July, etc.) How much money will
have accumulated after 4 years?
FV = ?
m = 4, T = 4
n = m x T = 16
nominal = 6%
0
1
2
14
15
16
PMT = 700
Note: There is no payment at t = 16
Set to “BGN” mode: 2nd, BGN, 2nd, SET, CE/C
rperiodic = rnominal/m = 6%/4 = 1.5%
P/Y=1, N=16, I/Y=1.5, PMT=700; CPT,FV: FV = $12,740.95
OR
P/Y=4, N=16, I/Y=6, PMT=700; CPT,FV: FV = $12,740.95
7. On 1 January 2008 you deposited $700 in a mutual fund that has been
yielding a constant 6% p.a. for the last several years. You plan to deposit $700
every 3 months thereafter for the next 4 years (i.e. the next deposit will be
made 1 April, the subsequent deposit on 1 July, etc.) How much money will
have accumulated after 4 years to include a payment to be made at the
beginning of the first quarter of the fifth year.
Option 1: (Treat as an Annuity Due)
FV = ?
nominal = 6%
m = 4, T = 4
n = m x T = 16
0
1
2
PMT = 700
14
15
16
700
Set to “BGN” mode: 2nd, BGN, 2nd, SET, CE/C
rperiodic = rnominal/m = 6%/4 = 1.5%
[P/Y=1, N=16, I/Y=1.5, PMT=700; CPT,FV: FV = $12,740.95] + $700
= $13,440.95
OR
P/Y=4, N=16, I/Y=6, PMT=700; CPT,FV: FV = $12,740.95] + $700 = 13,440.95
6
7. (continued)
Option 2: (Treat as an Annuity in Arrears)
FV = ?
nominal = 6%
m = 4, T = 4
n = m x T = 16
0
1
2
700
14
15
16
PMT = 700
rperiodic = rnominal/m = 6%/4 = 1.5%
P/Y=1, N=16, I/Y=1.5, PV=700, PMT=700; CPT,FV: FV = $13,440.95
OR
P/Y=4, N=16, I/Y=6, PV=700, PMT=700; CPT,FV: FV = 13,440.95
8. You are considering leasing a car that cost $46,000. The lease will be for 5
years and requires monthly payments. Somewhere on the lease paperwork you
notice a statement to the affect that you will be charged a nominal rate of
6.2850% p.a. Assume the car will be worth $20,000 at the end of the lease.
What kind of annuity is this?___________________ How much will your
payments be?
PV = $46,000
0
m=12, T=5; n = m x T = 60
nominal = 6.2850%
1
58
2
59
5 yrs
60
Turn-in = $20,000
PMT = ?
rperiodic = rnominal/m = 6.2850%/12 = 0.5238%
Set BGN, P/Y=1, N=60, I/Y=0.5238, PV=46000, FV= -20000; CPT PMT:
PMT = $607.67
OR
Set BGN, P/Y=12, N=60, I/Y=6.2850, PV=46000, FV= -20000; CPT PMT:
PMT = $607.67
7
9. You are tasked with estimating the fair market value of a security that
promises uneven future payments. The table below shows the monthly
payment schedule (each cash flow occurs at the end of the month). You
consider 6% p.a. to be the appropriate opportunity cost. What is the theoretical
value of this security?
1st month
2nd month
3rd month
4th month
5th month
6th month
$350
$390
$480
$660
$820
$940
$820
$350
0
1
$390
2
$480
3
$660
4
rperiodic = rnominal/m = 6%/12 = 0.5%
CF, 2nd, CLR WORK (Clears Cash Flow Registers)
0, ENTER,
↓, 350, ENTER
↓, ↓, 390, ENTER
↓, ↓, 480, ENTER
↓, ↓, 660, ENTER
↓, ↓, 820, ENTER
↓, ↓, 940, ENTER
NPV, 0.5, ENTER
↓, CPT = $3,566.31
8
$940
5
6
10. Today you open a new investment account for your company with a
$30,000 deposit. The account has had an average yield of 7.5600% p.a. over
the last three years and compounds every month. You plan to deposit $30,000
into this account every quarter, at the beginning of the quarter. Your next
deposit will be three months from now. How much will you have in this
account 3 years from now?.
m=4, T=3; n = m x T = 4 x 3 = 12
nominal = 7.56% compounded monthly
0
1
2
FV = ?
10
11
12 qtrs
PMT = $30,000
Set BGN, P/Y=4, C/Y=12, N=12, I/Y=7.56, PMT=30000; CPT,FV:
FV = $407,761.65
OR
Convert the monthly rate into a quarterly rate then solve for FV
2nd , ICONV, 7.56, ENTER, ↓, 12, ENTER, ↓, ↓, CPT: EFF% = 7.8275%
↓, 4, ENTER. ↓, CPT: NOM% = 7.6077%
Set BGN, P/Y=4, N=12, I/Y=7.6077, PMT=30000; CPT,FV: FV = $407,761.65
9
11. You are considering buying one of the two securities described below. What
is the fair market value of each security and which one is priced closer to its fair
market value?
Security A: A 2-year investment period yielding 6.35% p.a.; monthly payments
of $75; priced at $1950.
Security B: A 2-year investment period yielding 6.25% p.a.; 3 payments of
$62.50 at the end of each of each successive 6-month period and a final lump
sum payment of $1,062.50; priced at $1300.
PMT = 75
Security A:
nominal = 6.35%
m = 12, T = 2; n = m x T = 24
0
1
2
22
23
24
PV = ?
rperiodic = rnominal/m = 6.35%/12 = 0.529%
P/Y=1, N=24, I/Y=0.529, PMT=75; CPT,PV: PV = $1,686.21
OR
P/Y=12, N=24, I/Y=6.35, PMT=75; CPT,PV: PV = $1,686.21
Percent Above FMV: (1,950 - 1,686.21) / 1,686.21 = 15.6440%
Security B: m = 2, T = 2; n = m x T = 4
Soln Opt 1
PMT = 1062.50
PMT = 62.50
nominal = 6.25%
0
1
2
3
4
PV = ?
Soln. Opt 1: rperiodic = rnominal/m = 6.25%/2 = 3.125%
CF0=0, CF1=62.5, CF2=62.5, CF3=62.5, CF4=1062.5, I/Y=3.125; NPV =
$1,115.82
10
PMT = 1000
Soln Opt 2
PMT = 62.50
nominal = 6.25%
0
1
2
3
4
PV = ?
Soln. Opt 2a: PVA(4, 3.125, 62.50) + PV(4, 3.125, 1000)
[N=4, I/Y=3.125, PMT=62.5] + [N=4, I/Y=3.125, FV=1000]
$231.63 + $884.19 = $1,115.82
Soln. Opt 2b: P/Y=1, N=4, I/Y=3.125, PMT=62.5, FV=1000; CPT,PV: PV =
$1,115.82
Percent Above FMV: (1,300 – 1,115.82) / 1,115.82 = 16.5062%
Security A is more fairly priced
11
12. Retirement Plan You plan to retire on your 65th birthday. You want to
withdraw $100,000.00 each year at the beginning of each year. You plan to
live until your 88th birthday and you want your account balance to be $0 on
that day (this is for planning purposes only). How much money you must
deposit annually into a retirement account starting on your 25th birthday to
fund the above described retirement plan. You estimate the average rate of
return over the rest of your life will be 7% p.a.
Step 1: Find the amount of money the account needs to have on your 65th
birthday.
PMT = $100k
65
PV = ?
66
67
86
87
88
nominal = 7%
Set calculator to “Begin” mode
P/Y=1, N=23, I/Y=7, PMT= 100000; CPT, PV: $1,206,124.05
Step 2: Find the amount of the annual deposit.
nominal = 7%
25
FV = $1,206,124.05
26
27
63
64
65
PMT = ?
You make your deposit at the beginning of each year:
Set calculator to “Begin” mode
P/Y=1,N=40, I/Y=7, FV=1,206,124.05; Compute PMT: $5,646.395
12
13. Martha Mills, manager of the Plaza Gold Emporium, wants to allow her
customers to buy on credit, giving them 3 months in which to pay. However,
Martha will have to borrow from her bank to establish the credit reserve. The
bank will charge 7% p.a. interest compounded monthly. Martha wants to quote
a simple rate to her customers that will exactly cover her financing cost. What
simple (quoted, nominal) should Martha quote to her customers. Assume that
all customers will take the full 3 months to pay. (Hint: the credit account
compounds quarterly.)
Approach: In order to break even, Martha wants credit sales to yield exactly
equal to her cost of debt. Thus both arrangements must have the same EAR.
For the bank loan, m = 12. For credit sales, m = 4.
1) Find EAR on the bank loan:
Solution Opt 1: EAR = (1 + 0.07/12)12 – 1 = 7.229%
Solution Opt 2:
2nd, ICONV
7, ENTER
↓, ↓, 12, ENTER
↓, ↓, CPT : EFF% = 7.229%
2) Find the quoted (simple) rate to apply to credit sales
Solution Opt 1:
EFF%bank = (1 + rnominal/m)m - 1
0.07229 = (1 + rnominal/4)4 - 1
1.07229 = (1 + rnominal/4)4
1.01760 = 1 + rnominal/4
0.01760 = rnominal/4
nominal = 7.041%
Solution Opt 2:
2nd, ICONV
↓, 7.229, ENTER
↓, 4, ENTER
↓, CPT: NOM = 7.041%
13
14. A friend of yours is working as an unpaid intern at a local brokerage firm
and her boss is selling some securities which pay $50 at the end of each of the
next 3 years and a final payment of $1050 at the end of the 4 th year. Your friend
says she can get you some of these securities at a cost of $900 each. Your
money is currently invested in a bank that pays 8% p.a. with quarterly
compounding. You consider the securities as being just as safe and as liquid as
your bank. What is the fair market value (the theoretical value) of these
securities?
1st Step: Find the EAR on the bank account
Why?
Answer: This is the Opportunity Cost of Capital and it must be converted to
the same compounding frequency as the securities
Solution Opt. 1: EAR = (1 + r/m)m - 1 = (1 + 0.08/4)4 – 1 = 8.24%
Solution Opt. 2: 2nd, ICONV
8, ENTER
↓, ↓, 4, ENTER
↓, ↓, CPT: EFF% = 8.24%
2nd Step: Find PV of the securities
PMT = 50
PMT = 1050
nominal = 8.24%
0
1
2
3
4
PV = ?
Why use 8.24% for nominal?
Find PV using uneven CF approach
CF, 2nd, CLR WORK (clears CF registers)
0, ENTER
↓, 50, ENTER
↓, ↓, 50, ENTER
↓, ↓, 50, ENTER
↓, ↓, 1050, ENTER
NPV, 8.24, ENTER
↓, CPT: PV = $893.26
14
Another way: Recognize the cash flows as such (this looks like a bond or a
PMT = 1000
non-amortized loan)
PMT = 50
nominal = 8.24%
0
1
2
3
4
PV = ?
Solution Opt 1:
PV = PVA(4, 8.24, 50) + PV(4, 8.24, 1000)
= [N=4, I/Y=8.24, PMT=50] + [N=4, I/Y=8.24, FV=1000]
= $164.73 + $728.53 = $893.26
Solution Opt 2: N=4, I/Y=8.24, PMT=50, FV=1000; CPT,PV: PV = $893.26
Compare quoted price to fair market value
$900 - $893.26 = $6.74 Your friend is trying to make a profit of $6.74
15
15. On 1 January you deposited $500 in a savings account that pays 3% p.a..
You plan to deposit $500 every three months thereafter (i.e. the next deposit
will be on 1 April, the subsequent deposit on 1 July, etc.) How much money
will have accumulated after 14 months? Wrong Ans: $2,383.37
PMT = $500
0
1
2
n = 4.666667 m = 4
T = 14/12 =1.1666667
n = m x T = 4.666667
3
5
4
FV = ?
Solution Option 1: Compound each CF forward and sum them:
Cash Flow
0
1
2
3
4
PV
$500.00
$500.00
$500.00
$500.00
$500.00
n
4.6667
3.6667
2.6667
1.6667
0.6667
rperiodic
0.7500%
0.7500%
0.7500%
0.7500%
0.7500%
FV
$517.74
$513.89
$510.06
$506.27
$502.50
Sum of FVs: $2,550.46
Solution Option 2:
1) Find the FV at t=4:
PMT = $500
PV = $500
0
1
2
3
5
4
FV = ?
Set “END”, P/Y=4, N=4, I/Y=3, PV=500, PMT=500; CPT, FV: $2,537.78
2) Compound $2,537.78 forward 0.666667 periods
n = 0.666667
PV = $2,537.78
3
4
P/Y=4, N=0.666667, I/Y=3, PV= 2537.78; CPT, FV: $2,550.46
16
5
FV = ?
16. If you save $500 each year for 2 years and then $1,000 each for two years,
how much must you save in the 5th and 6th years to have $10,000 at the end of
10 years if the interest rate is 5% p. a.?
FV = $10,000
0
1
2
3
$500
4
$1,000
5
6
7
8
9
10
PMT = ?
$500[(1.05)9 +(1.05)8] + $1,000[(1.05)7 + (1.05)6] + x[(1.05)5 + (1.05)4] =
$10,000
$500(3.0288) + $1,000(2.7472) + x(2.4918) = $10,000
$1,514.3918 + $2,747.2000 + x(2.4918) = $10,000
x(2.4918) = $10,000 - $1,514.3918 - $2,747.2000
x = $5,738.4082 / 2.4918
x = $2,302.92
17. A football coach is leaving his current school. In doing so, he is giving up
an annuity of $100,000 per year for 10 years that would begin when he turns
60. The coach is 45. His new school has offered to make up the loss of the
annuity with a lump sum payment when he moves. How much should the new
school pay if the interest rate is 7% p.a.?
Step 1: Compute the value of the annuity at age 60
P/Y=1, SET BGN, N=10, I/Y=7, PMT=100000; CPT, PV: PV = $751,523.22
PV = ?
0
PV = ?
1
2
8
9
10
45
46
47
58
59
60
FV = $751,523.22
PMT = $100,000
Step 2: Discount the value computed in Step 1 to age 45
P/Y=1, N=15, I/Y=7, FV=751523.22; CPT, PV: PV = $272,386.60
17