Dr. Sudhakar Raju
FN 3000
ANSWERS TO CHAPTER 5 – DCF VALUATION
1.)
1 Shift P/Yr
0 CFj
900 CFj
600 CFj
1100 CFj
1480 CFj
10 I/Yr
Shift NPV => $3151.36
18 I/Yr
Shift NPV => $2626.48
24 I/Yr
Shift NPV => $2318.96
2.)
Investment X
1 shift P/yr
4000 PMT
9N
5 I/yr
PV=> 28,431
22 I/yr
PV=> 15,145
Investment Y
1 shift P/yr
6000 PMT
5N
5 I/yr
PV=> 25,977
15 I/yr
PV=> 17,181
3.)
You can figure out the FV of each component cash flow and sum up to get the answer. An more
efficient and quicker way, however, is to do the following:
1 Shift P/Yr
0 CFj
600 CFj
1
800 CFj
1200 CFj
2000 CFj
8 I/Yr
Shift NPV => $3664.09
Now figure out FV of $3664.09 at 8% for 4 years.
Thus:
1 Shift P/Yr
3664.09 +/- PV
4N
8 I/yr
FV=> $4984.95
Similarly, at 11% the FV is $5138.26 and at 24% the FV is $5862.05.
4.)
1 shift P/yr
4500 PMT
15 N
10 I/yr
PV=> 34,227
40N
PV=> 44,006
Forever (Perpetuity) =
FixedCF
InterestRa te
=$4500/.10
PV of Perpetuity => $45,000
5.)
1 Shift P/Yr
15,000 +/- PV
7.50 I/Yr
12 N
PMT=> $1939.17
6.)
1 shift P/yr
60,000 PMT
9N
8.25 I/yr
PV=> $370,948
2
Since the PV of Revenues is $370,948 while PV of Cost is only $325,000 you can afford the
system.
7.)
1 Shift P/Yr
3000 +/- PMT
8.50 I/yr
20 N
FV=> $145,131.04
40 N
FV => $887,047.61
8.)
1 shift P/yr
40,000 FV
7N
5.25 I/yr
PMT=> $4,876
9.)
1 Shift P/Yr
30,000 PV
7N
9 I/yr
PMT => $5960.72
10.) & 11.)
PVPerpetuity =
=
AnnualPMT
InterestRate
20,000
.08
= $250,000
PVperp=
AnnualPMT
InterestRate
$270,000=
$20,000
InterestRa te
Interest Rate = 7.41%
3
12.)
4 Shift P/yr
8 I/yr
Shift Eff% => 8.24%
Similarly, 10.47%, 15.02% and 18.81%
13.)
2 Shift P/Yr
12 Shift Eff%
Shift Nom% => 11.66%
Note than Nominal Percent (Nom %) is another term for the Stated Rate or APR. Similarly, the
other values are: 16.67%, 6.77%, 10.44%.
14.)
12 shift P/yr
13.10 I/yr
Shift Eff%
13.92%
2 shift P/yr
13.40 I/yr
Shift Eff%
13.85%
15.)
365 Shift P/Yr
17 Shift Eff%
Shift Nom% => 15.70%
The law requires lending institutions to report the APR. The APR is 15.70% which is actually an
underestimate of the true, effective rate. Because of the compounding effect, the effective rate is
higher at 17%.
16.)
2 Shift P/yr
1575 +/- PV
13 N
Shift N
4
10 I/yr
FV=> 5600
17.)
The 3.90% is NOT a daily rate but an APR.
365 Shift P/Yr
3.90 I/Yr
6000 +/- PV
5N
Shift N
FV=> $7291.79
10 N
Shift N
FV=> $8861.70
20 N
Shift N
FV => $13,088.29
18.)
365 shift P/yr
70,000 FV
6N
Shift N
10 I/yr
PV => 38,420
19.)
25% per month x 12 months = 300% per year (APR)
12 Shift P/Yr
300 I/Yr
Shift Eff% => 1355.19%
20.)
12 Shift P/yr
62,500 PV
0 FV
(The 0 FV is not really necessary since this is the default setting in the calculator. It simply
indicates that the loan is completely paid off at the end of 5 years.
8.20 I/yr
5N
Shift N
PMT => $1273.27
5
Shift Eff%=>8.52%
21.)
1.30% per month x 12 months => 15.60% per year
12 Shift P/Yr
12,815 PV
15.60 I/Yr
400 +/- PMT
N => 41.71 months
Divide by 12 => About 3.48 years
22.)
$5  $4
= 25% per week
$4
Annual Rate => 25% x 52 weeks = 1300% p.a.
Weekly Rate =>
EAR%?
52 Shift P/yr
1300 I/yr
Shift Eff% => 10,947,544%
23.)
PVPerpetuity =
$175,000=
MonthlyFix edPMT
MonthlyInt erestRate
$3000
InterestRa te
Interest Rate = .0171 (1.71%)
APR=.0171 x 12=.2057 or 20.57%
12 Shift P/Yr
20.57 I/Yr
Shift Eff% => 22.62%
24.) & 25.)
12 Shift P/Yr
250 +/- PMT
11 I/yr
6
30 N
Shift N
FV => $701,129.93
Shift Eff% => 11.57%
Note that 11% compounded monthly is equivalent to an EAR of 11.57%. Suppose now that
instead of making monthly deposits of $250 you make one annual deposit of $250 x 12 or $3000
at 11.57%.
1 Shift P/Yr
3000 +/- PMT
11.57 I/yr
30 N
FV=> $666,237.87
26.)
Notice that we have quarterly compounding here.
The APR = .75% per quarter x 4 = 3% p.a.
4 Shift P/yr
2000 +/- PMT
4N
Shift N
3 I/yr
PV=> 30,049
27.)
1 Shift P/Yr
0 CFj
700 CFj
900 CFj
400 CFj
800 CFj
10 I/Yr
Shift NPV => $2,227.10
28.)
1 Shift P/Yr
0 CFj
1500 CFj
3200 CFj
6800 CFj
8100 CFj
7.83 I/Yr
7
Shift NPV => 15,558
29.)
Assume a PV of $100.
FV = PV [1 + (r) (n)]
Simple Interest => FV= $100 [1+ (.09)(10)]
FV=> $190
Compound Interest => FV = PV [1+ x]10
$190 = $100 [1+ x ]10
1 Shift P/Yr
100 +/- PV
190 FV
10 N
I/Yr => 6.63%
Thus, a simple interest of 9% is equivalent to a compound rate of 6.63% over a 10 year horizon.
30.)
Note that this is an Annuity Due rather than an Ordinary Annuity.
Shift Beg/End
12 Shift P/yr
56,000 PV
5N
Shift N
8.15 I/yr
PMT=> 1131.82
31.)
Calculate amount owed at the end of the first six months
12 Shift P/Yr
6000 PV
2.10 I/Yr
.50 N [6 months = .50 year]
Shift N
FV => $6063.28 (Amount owed after six months)
Next six month period:
12 Shift P/Yr
6063.28 PV
21 I/Yr
.50 N
8
Shift N
FV => $6728.44
Thus, the total amount of interest owed over the year on a $6000 loan is $728.44.
32.)
Simple Interest
FV = PV [1 + (r) (n)]
$150,000= $83,000 [1+ (.05)(n)]
1.8072 = 1+.05 n
.8072 =.05 n
N=.8072 /.05 Thus, n = 16.14 years
Compound Interest
12 Shift P/yr
150,000 FV
83,000 +/- PV
5 I/yr
N =>142.33 months
Divide by 12 => 11.86 years
33.)
12 Shift P/Yr
1+/- PV
14.28 I/Yr [i.e. 1.19 % x 12 = 14.28 %]
1N
Shift N
FV => $1.15
2N
Shift N
FV => $1.33
34.)
1 Shift P/Yr
440 +/- PV
60 PMT
31 N
I/Yr => 13.36%
35.)
12 Shift P/Yr
6200 PMT
2N
Shift N
8 I/yr
9
PV=> $137,085.37
OR
12 shift P/yr
4900 PMT
2N
Shift N
8 I/yr
PV=> $108,341.66 + $30,000 (signing bonus) => $138,341.66
The second option is better.
36.)
1 shift P/yr
18,000 PMT
20 N
10 I/yr
PV=> 153,244
Note that “effective annual return” (EAR) and APR are the same in this problem since we are
using annual compounding. EAR and APR will differ only if one uses compounding other than
annual compounding.
37.)
12 Shift P/Yr
140 +/- PMT
12 I/Yr
35,000 FV
N=> 125.90 payments
38.)
12 shift P/yr
60,000 PV
1300 +/- PMT
0 FV
(The 0 FV indicates that the loan reduce to zero( i.e. is completely paid off) at by the end of 5
years)
5N
Shift N
I/yr => 10.85%
39.)
10
Yr 1 => 2.9 m
2=> 3.77 [$2,900,000 + $870,000 = $3.77m]
3=> 4.64
4=> 5.51
5=> 6.38
6=> 7.25
7=> 8.12
8=> 8.99
9=> 9.86
10=> 10.73
1 Shift P/Yr
0 CFj
2.9 CFj
3.77 CFj
.
.
.
10.73 CFj
11 I/Yr
Shift NPV=> $35.802 million
40.)
YR 0 => $0
1=> $3m
2=> 3.9m
3=> 4.8m
4=> 5.7m
5=> 6.6m
6=> 7. 5m
7=> 8.4m
1 shift p/yr
0 CFi
3 CFi
3.9 CFi
4.8 CFi
;
;
8.4 CFi
11 I/yr
Shift NPV=> $25.105 million
Robinson’s contract was better.
11
41.)
$1,500,000 x 80% = $1,200,000 (loan amount)
12 Shift P/Yr
1,200,000 PV
8400 +/- PMT
30 N
Shift N
I/Yr => 7.51%
Shift Eff% => 7.78%
42.)
The amount you receive upfront is $10,680 [$12,000 - $1320]. You need to repay $12,000. The
true embedded interest rate on this transaction can be determined thus:
1 Shift P/Yr
10,680 PV
12,000 +/- FV
1N
I/Yr => 12.36%
Shift Eff% => 12.36%
Thus, the true interest rate is higher than the 11% quoted by the lender. Note that the EAR and
APR is the same here. This should not be surprising given that EAR and APR will be different
only if the compounding period is other than annual compounding.
43.)
1 Shift P/Yr
6000 PMT
30 N
8 I/Yr
PV => $67,547
PVPerpetuity =
FixedPMT
InterestRa te
PVPerpetuity =
6000
.08
PVPerpetuity = $75,000
12
Difference = $75,000 - $67,547 = $7453
44.)
To solve this problem do the following:
PV of $890 from Yr 5- Yr 20 =
[PV of $890 from Yr 1 to Yr 20] – [PV of $890 from Yr 1 to Yr 4]
= $8124.41 – $2883.35 => $5241.06
(see calculations below)
1 Shift P/Yr
890 PMT
20 N
9 I/Yr
PV => $8124.41
1 Shift P/Yr
890 PMT
4N
9 I/Yr
PV => $2883.35
45.)
Investment A
12 Shift P/Yr
1600 PMT
10 N
Shift N
10 I/Yr
FV => $327,752
Investment B
1 Shift P/Yr
327,752 FV
10 N
8 I/Yr
PV => $ 151,813
13
46.)
You borrow $20,000 today. This is to be repaid in monthly installments of $1883.33 over the
next 12 months.
12 Shift P/Yr
1883.33 +/- PMT
20,000 PV
0 FV (This indicates that the loan is paid off at the end of one year)
1N
Shift N
I/Yr=> 23.19%!
Shift Eff% => 25.82%
The rate to be legally quoted is the APR of 23.19%. The APR is 23.19% and the EAR is 25.82%,
very different from the quoted rate of 13%.
47.)
Compute the fixed annual payments.
1 Shift P/Yr
45,000 PV
0 FV
11 I/Yr
3N
PMT => $18,414.59
Year
1
2
3
AMORTIZATION SCHEDULE
Beg Balance Total
Interest
Principal
Annual
Paid
Paid
Payment
45,000
18,414.59
4950
13,464.59
[45,000 x
[18,414.5911%]
4950]
31,535.41
18,414.59
3468.90
14,945.69
[31,535.41 x [18,414.5911%]
3468.90]
16,589.72
18,414.59
1824.87
16,589.72
[16,589.72 x [18,414.59 11%]
1824.87]
$10,243.77
Total
Interest Paid
14
Ending
Balance
31,535.41
[45,000 13,464.59
16,589.72
[31,535.4114,945.69]
0
[16,589.72 –
16,589.72]
In the third year, interest of $1824.87 is paid. Total interest over the life of the loan is
$10,243.77.
48.) It is important to remember that in an ordinary annuity all cash flows occur at the end of the
period. This is an implicit assumption of the annuity structure though it is not always explicitly
spelled out.
Today End of Yr 1
End of Yr 7
End of Yr 8
↓
$1000
End of Yr 20
↓
$1000
Compute the PV of cash flows from Yr 8 to Yr 20.The answer you get represents the PV at the
beginning of year 8 (or end of year 7). Now discount this PV value back to year 0. Thus:
1 Shift P/YR
1000 PMT
13 N [From year 8 to year 20]
12 I/YR
PV=> U$ 6,423.55. THIS IS THE PV AT THE END OF YR 7. Now discount this value
back to year zero. Thus:
1 Shift P/Yr
6423.55 FV
7N
12 I/YR
PV => U$ 2906
Another way of answering this question is to recognize that the correct answer must be equal to:
(PV of $1000 from YR 1 to Yr 20) – (PV of $1000 from YR 1 to Yr 7)
= $7469.44 - $4563.76
= $2906 (same answer as above).
15