Solutions to TVM Practice Set II
1. In this problem you are solving for the monthly rate of return and then the yearly rate
of return. There are 228 months in 19 years (19 * 12 = 228).
1
$13,438 228
)  1 = 1.104%
$1,100
N = 228, PV = -$1,100, FV = $13,438; compute I
a) To determine the monthly rate of return: (
1
$13,438 19
b) To determine the yearly rate of return : (
)  1 = 14.079%
$1,100
N = 19, PV = -$1,100, FV = $13,438; compute I
2. You should disagree with Jim. You do not owe $1,900 for the entire year since you
are making monthly payments that would reduce the outstanding loan balance. If
your total interest payments are $380, the true rate of interest is much higher than
20%. You would need to set up the problem and solve for r (a monthly rate) in the
PVIFA equation. You could convert the monthly rate to an A.P.R. by multiplying it
by 12. The true A.P.R. is 35.07%, not 20% as Jim is suggesting.
1
(1  r )12
=> $1,900 = $190 * [
]
r
Now, use a financial calculator to solve for the monthly rate of interest (r).
1
On a financial calculator: N = 12, PV = 1,900, PMT = -190; compute I
=> r = 2.923% per month
=>
2.923% * 12 = 35.07% APR
=> The E.A.R on the loan would be found as follows: (1.02923)12 –1 = 41.30%
3. Here you must solve for the E.A.R of an annuity due. The easiest way to solve this
problem is to set your calculator to BEGIN mode, directly solve for the monthly rate,
and then convert the monthly rate to an effective rate.
ON BEGIN MODE: N = 24, PV = $1150, PMT = -$60; compute I
I = 2.0632% per month
Convert to an E.A.R.:
(1.020632)12 –1 = 27.77%
F301 TVM practice set II solutions
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As an alternative, if you recognize that the first payment is at time zero, you can solve
the problem as follows:
1
1
(1  r ) 23
=> $1150 - $60 = $60 * [
] , which is equivalent to solving the problem as
r
if the present value were $1090 and there are 23 months of $60 payments.
1
1
(1  r ) 23
=> $1090 = $60 * [
] ; solve for r on a financial calculator as follows:
r
N = 23, PV = $1090, PMT = -$60; compute I; I = 2.0632% per month,
Convert to an E.A.R.:
(1.020632)12 –1 = 27.77%
4. There are several ways to solve this problem. First, you can use brute force and
discount all the cash flows back individually (YUCK!). Alternatively, you can see
that there are effectively two annuity streams here; a five period ordinary annuity of
$253 with the first cash flow one period away and then a four period ordinary annuity
of $253 with the first cash flow starting 7 periods from now. Recalling that the
annuity present value formula assumes that payments start at the end of the period
(i.e., one period away), you can solve the for present value of these cash flows as
follows:
$253 * PVIFA(7.9%, 4)
(1.079) 6
1
1
1
1
5
1
(1.079)
(1.079) 4
=> = $253 * [
+
$253 * [
]
]
]*[
(1.079) 6
.079
.079
=> =
$1,012.82 + $532.19 = $1,545.01
=> PV = $253 * PVIFA(7.9%,5) +
On a financial calculator:
Step 1:
Determine the present value of the first 5 payments:
PMT = $253, N = 5, I = 7.9; compute PV = $1,012.82
Step 2:
Determine value of the last 4 payments as of t = 6:
PMT = $253, N = 4, I = 7.9; compute PV = $839.84
This is the value at the end of 6th year, so you must discount it 6 more years.
FV = $839.84, N = 6, I = 7.9; compute PV = $532.19
Step 3:
Add the two values together
$1,012.82 + $532.19 = $1,545.01
F301 TVM practice set II solutions
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Now, a simpler approach. Calculate the value of a 10-year, $253 annuity and then
subtract off the present value of the missing payment at t = 6.
=> PV = $253 * PVIFA(7.9%,10) - $253 * PVIF(7.9%,6)
1
1
$253
(1.079)10
=> $253 * [
](1.079) 6
.079
=> $1,705.33 - $160.32 = $1,545.01
On a financial calculator:
Step 1:
Determine the present value of a 10 year annuity:
PMT = $253, N = 10, I = 7.9; compute PV = $1,705.33
Step 2:
Determine the present value of the 6th payment (single CF) that
was missed:
FV = $253, N = 6, I = 7.9; compute PV = $160.32
Step 3:
Take the difference between these two values:
$1,705.33 - $160.32 = $1,545.01
5. Fifteen-percent compounded monthly implies a monthly interest rate of .15 / 12 =
.0125. Because the perpetuity formula assumes that the first cash flow is one period
away, you can directly find the perpetuity’s value at t = 0 as follows:
=> PV = C/r = $257/.0125 = $20,560
6. First, find the value as of t = 4 (recall that the perpetuity formula assumes that the first
cash flow occurs one period away), then discount 4 periods (months) to get a value as
of t = 0.
Note: 12% compounded monthly implies a monthly rate of 1% (.12/12).
$845
= $84,500
.01
$84,500
=> Value at t = 0:
= $81,202.84
(1.01) 4
=> Value at t = 4:
7. a) Solve for the monthly payment in each case and then take the difference:
30 year loan:
1
1
(1.006)360
$132,000 = $C * [
];
.006
$C = $896.00
On a financial calculator: N = 360, I = .6, PV = 132,000; compute PMT
1
1
(1.006)180
15 year loan:
$132,000 = $C * [
$C = $1,201.26
];
.006
F301 TVM practice set II solutions
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On a financial calculator: N = 180, I = .6, PV = 132,000; compute PMT
The payments are $305.26 ($1,201.26 - $896.00) higher per month for the 15 year
loan.
b) ($896.00 * 360) - ($1,201.26 * 180) = $106,333.20.
c and d)
The E.A.R.s are both the same. Both loans have the same A.P.R. and both
.072 12
] 1 =
are compounded on a monthly basis: E.A.R. = [1 
7.442%
12
8. Note that you will be financing 80% of the $140,000 purchase price, or a total of
$112,000. Also, the monthly rate is .066/12 = 0.0055.
=> $C = $112,000
;
$C = 841.65
1
1
(1.0055) 240
[
]
.0055
On a financial calculator:
N = 240, I = 6.6/12 = .55, PV = $112,000; compute PMT = |$841.65|
You will need to construct an amortization schedule for the first 12 months of the
loan in order to answer these questions. A sample is provided below
Amortization Schedule.
Period
Loan Balance
Payment
Interest
Reduction of Principal
End Balance
1
$112,000.00
$841.65
$616.00
$225.65
$111,774.35
2
$111,774.35
$841.65
$614.76
$226.89
$111,547.46
3
$111,547.46
$841.65
$613.51
$228.14
$111,319.32
4
$111,319.32
$841.65
$612.26
$229.39
$111,089.93
5
$111,089.93
$841.65
$610.99
$230.66
$110,859.27
6
$110,859.27
$841.65
$609.73
$231.92
$110,627.35
7
$110,627.35
$841.65
$608.45
$233.20
$110,394.15
8
$110,394.15
$841.65
$607.17
$234.48
$110,159.67
9
$110,159.67
$841.65
$605.88
$235.77
$109,923.89
10
$109,923.89
$841.65
$604.58
$237.07
$109,686.82
11
$109,686.82
$841.65
$603.28
$238.37
$109,448.45
12
$109,448.45
$841.65
$601.97
$239.68
$109,208.77
$7,308.57
a) As you can see from the amortization schedule, the total amount of interest that
will be paid over the first 12 months of the loan is $7.308.57.
b) The loan balance after one year of monthly payments would be $109,208.77.
F301 TVM practice set II solutions
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9. MAKE A TIME LINE!
=> In 8 years (96 months), Betsy will have $100,000(1.005)96 =
$161,414.27
=> In 8 years (96 months), Betsy will need $132,000 +$558,323.09 = $690,323.09
1
1
(1.005) 240
($4,000 * [
= $558,323.09)
]
.005
=> Betsy will have to make up the difference ($528,908.82) with monthly payments
over the 8 year (96 month) period.
(1.005)96  1
=> $528,908.82 = $C * [
] $C = $4,306.07
.005
Betsy must save $4,306.07 per month for the next 8 years (96 months). Wish
Betsy luck!
10. Jonathan’s contract does not include a signing bonus and can be valued as follows:
1
1
(1.005)12
The value of the first year’s cash flows (as of t = 0) is $180,000 * [
]=
.005
$2,091,407.77.
The value of the 2nd year’s cash flows (as of t = 1) is $2,091,407.77 * 1.03 =
$2,154,150.01.
The value of the 3rd year’s cash flows (as of t = 2) = $2,091,407.77 * (1.03)2 =
$2,218,774.51.
The present value of Jonathan’s contract is therefore:
$2,154,150.01 $2,218,774.51
=> $2,091,407.77 
= $6,088,878.11

(1.005)12
(1.005) 24
The value of Jamal’s contract (before the signing bonus) can be found as follows:
1
1
(1.005)12
The value of the first year’s cash flows (as of t = 0) is $120,000 * [
]=
.005
$1,394,271.85.
The value of the 2nd year’s cash flows (as of t = 1) is $1,394,271.85 * 1.05 =
$1,463,985.44.
The value of the 3rd year’s cash flows (as of t = 2) = $1,394,271.85 * (1.05)2 =
$1,537,184.71.
The present value of Jamal’s contract before the signing bonus is therefore:
$1,463,985.44 $1,537,184.71

=> $1,394,271.85 
= $4,136,975.80
(1.005)12
(1.005) 24
In order for the two deals to have the same present value, Jamal’s signing bonus must
have been: $6,088,878.11 - $4,136,975.80 = $1,951,902.31.
F301 TVM practice set II solutions
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