APPENDIX A
THE TIME VALUE OF MONEY
EXERCISES
EA–1
Time Periods (Years)
Compound Interest
Rates 51015
5%$150 1.27628$150 1.62889$150 2.07893
=$191.44=$244.33=$311.84
10%$150 1.61051$150 2.59374$150 4.17725
=$241.58=$389.06=$626.59
15%$150 2.01136$150 4.04556$150 8.13706
=$301.70=$606.83=$1,220.56
EA–2
Time Periods (Years)
Compound Interest
Rates 51015
5%$10,000=$7,835.26$10,000=$6,139.15$10,000=$4,810.17
1.05^51.05^101.05^15
10%$10,000=$6,209.21$10,000=$3,855.44$10,000=$2,393.92
1.10^51.10^101.10^15
15%$10,000=$4,971.76$10,000=$2,471.85$10,000=$1,228.94
1.15^51.15^101.15^15
The above problem has also been attempted in an alternate way to demonstrate the use of formulas.
EA–3
Time Periods (Years)
Compound
Interest
Rates 51015
5%$150 5.52563$150 12.57789$150 21.57856
=$828.84=$1,886.68=$3,236.78
10%$150 6.10510$150 15.93743$150 31.77248
=$915.77=$2,390.61=$4,765.87
15%$150 6.74238$150 20.30372$150 47.58041
=$1,011.36=$3,045.56=$7,137.06
EA–4
Time Periods (Years)
Compound
Interest
Rates 51015
5%$150 5.80191$150 13.20679$150 22.65749
=$870.29=$1,981.02=$3,398.62
10%$150 6.71561$150 17.53117$150 34.94973
=$1,007.34=$2,629.68=$5,242.46
15%$150 7.75374$150 23.34928$150 54.71747
=$1,163.06=$3,502.39=$8,207.62
EA–5
Time Periods (Years)
Compound
Interest
Rates 51015
5%$10,000 4.32948$10,000 7.72173$10,000 10.37966
=$43,294.80=$77,217.30=$103,796.60
10%$10,000 3.79079$10,000 6.14457$10,000 7.60608
=$37,907.90=$61,445.70=$76,060.80
15%$10,000 3.35216$10,000 5.01877$10,000 5.84737
=$33,521.60=$50,187.70=$58,473.70
EA–6
Time Periods (Years)
Compound
Interest
Rates 51015
5%$10,000 4.54595$10,000 8.10782$10,000 10.89864
=$45,459.50=$81,078.20=$108,986.40
10%$10,000 4.16987$10,000 6.75902$10,000 8.36669
=$41,698.70=$67,590.20=$83,666.90
15%$10,000 3.85498$10,000 5.77158$10,000 6.72448
=$38,549.80=$57,715.80=$67,244.80
EA–7
a.($50 .85734) + ($100 .68058)+ ($80 .54027)
=$42.87 +$68.06 +$43.22
=$154.15
b.($100 3.31213)+($100 .54027)
=$331.21+$54.03
=$385.24
c.($60 .68058)+($60 .63017)+($60 .58349)+($60 .54027)+($100 .46319)
=$40.83 +$37.81+$35.01+$32.42+$46.32
=$192.39
d.($90 .58349)+($90 .54027)+($90 .50025)
=$52.51+$48.62+$45.02
=$146.15
EA–8
a.($50 .85734) + ($100 .68058)+($80 .58349)
=$42.87 +$68.06+$46.68
=$157.61
b.($100 3.57710)+($100 .58349)
=$357.71+$58.35
=$416.06
c.($60 .73503)+($60 .68058)+($60 .63017)+($60 .58349)+($100 .50025)
=$44.10+$40.83+$37.81+$35.01+$50.03
=$207.78
d.($90 .63017)+($90 .58349)+($90 . 54027)
=$56.72+$52.51+$48.62
=$157.85
EA–9
a.Dollar amount=$25,000 Future value factor for i = 10% and n = 4
=$25,000 1.46410 (from Table 1)
=$36,603
Dollar amount=$36,603 Future value factor for i = 12% and n = 3
=$36,603 1.40493 (from Table 1)
=$51,425
Dollar amount=$51,425 Future value factor for i = 15% and n = 5
=$51,425 2.01136 (from Table 1)
=$103,434
b.Ben should not accept $36,000 for $25,000 at the end of 4 years. Why not? Because if he invests the initial $25,000 at 10 percent per annum compounded annually, he will have a total of $36,603, $603 more than the amount the person offered him.
EA–10
a.Dollar amount=($40,000 Present value factor for an ordinary annuity factor for
i = 10% and n = 10) + ($500,000 Present value factor for
i = 10% and n = 10)
=($40,000 6.14457 from Table 5) + ($500,000 .38554 from Table 4)
=$245,782.80 + $192,770.00
=$438,552.80
b.There are two different ways to calculate the dollar amount. The two ways are shown below.
Dollar amount=($40,000 Present value factor for an annuity due for i = 10% and n= 10)
+ ($500,000 Present value factor for i = 10% and n = 10)
=($40,000 6.75902 from Table 6) + ($500,000 .38554 from Table 4)
=$270,360.80 + $192,770.00
=$463,130.80
Dollar amount=$40,000 + ($40,000 Present value factor for an ordinary annuity factor
for i = 10% and n = 9) + ($500,000 Present value factor for
i = 10% and n = 10)
=$40,000 + ($40,000 5.75902 from Table 5) + ($500,000
.38554 from Table 4)
=$40,000 + $230,360.80 + $192,770.00
=$463,130.80
EA–11
Option 1
Present value=$500,000 Present value factor for an ordinary annuity for i = 10% and n = 20)
=$500,000 8.51356 (from Table 5)
=$4,256,780
Option 2
Present value=$4,500,000
Option 3
Present value=$1,000,000 + [($2,100,000 Present value factor for an ordinary annuity for i = 10% and n = 3) Present value factor for i = 10% and n = 4]
=$1,000,000 + [($2,100,000 2.48685 from Table 5) .68301 from Table 4]
=$1,000,000 + $3,566,941
=$4,566,941
Option 3 should be chosen because it has the highest present value. In other words, if receiving the equivalent amounts for each of the 3 payment patterns, alternative 3 would yield the largest payout today.
EA–12
Ordinary AnnuityAnnuity Due
a.$700 2.48685 (from Table 5) $1,740.80
$700 2.73554 (from Table 6) $1,914.88
b.$700 + ($700 1.73554 from Table 5) 1,914.88
($700 1.10000 from Table 1) + $700
+ ($700 .90909 from Table 4) 2,106.36
c.($700 1.10000 from Table 1) + $700
+ ($700 .90909 from Table 4) 2,106.36
$700 2.31000 (from Table 3) + $700 2,317.00
d.$700 3.31000 (from Table 2) 2,317.00
$700 3.64100 (from Table 3) 2,548.70
e.The present value is the value of future cash flows at the current point in time. Thus, the values in Part (a) represent the present value of the two different annuities.
f.The future value is the value of future cash flows at a future point in time. Since the ends of Periods 1, 2, and 3 are all in the future, the value of the cash flows at those points in time all qualify as future values.
g.Annuity due is most valuable. The present value of annuity due is $174.08 more than the present value of ordinary annuity. In other words, if we were to receive $700 each year for the next 3 years, the payment pattern of the annuity due (payment to be received at the beginning of each year) should be more preferable to us than the payment pattern of the ordinary annuity (payment to be received at the end of each year).
EA–13
a.Option 1
Present value=$240,000
Option 2
Present value=$500,000 Present value factor for i = 12% and n = 8
=$500,000 .40388 (from Table 4)
=$201,940
Option 3
Present value=$600,000 Present value factor for i = 12% and n = 10
=$600,000 .32197 (from Table 4)
=$193,182
Option 4
Present value=$50,000 Present value factor for an annuity due for i = 12% and n = 6
=$50,000 4.60478 (from Table 6)
=$230,239
b.By computing the present value of each option's future cash flows, the cost of each option is comparable. Since Option 3 has the lowest present value, it appears to the best deal for Dunn Drafting Company.
c.Option 1:
Present value=$240,000
Option 2:
Present value=$500,000 Present value factor for i = 8% and n = 8
=$500,000 .54027 (from Table 4)
=$270,135
Option 3:
Present value=$600,000 Present value factor for i = 8% and n = 10
=$600,000 .46319 (from Table 4)
=$277,914
Option 4:
Present value=$50,000 Present value factor for an annuity due for i = 8% and n = 6
=$50,000 4.99271 (from Table 6)
=$249,636
Option 1 now minimizes the present value of future cash flows. Thus, it appears that Option 1 is now the best option for Dunn Drafting Company.
EA–14
a.Since the Croziers plan to invest a lump sum today and then withdraw the money in the form of an annuity, two steps are required to determine how much the Croziers must invest today to pay for Ryan's college education. The first step is to calculate how much money they will need fifteen years from now when Ryan enters college to make the four payments at the beginning of each year Ryan is in college (i.e., the value of the annuity). The second step is to calculate how much they would have to invest now so that it will grow to the value calculated in the first step over the next fifteen years. The calculations are shown below.
Present value of college expenses fifteen years in the future:
Value=$40,000 Present value factor for an annuity due for i = 10% and n = 4
=$40,000 3.48685 (from Table 6)
=$139,474.00
Present value of college expenses today:
Value=$139,474.00 Present value factor for i = 10% and n = 15
=$139,474.00 .23939 (from Table 4)
=$33,389.00
b.The present value of fourteen annual payments must equal the present value of $33,389 calculated in part (a). By using the following formula, the amount of the annual payments can be calculated.
Present value=Annuity payment Present value factor for an ordinary annuity for
i =10% and n = 14
$33,389=Annuity payment 7.36669 (from Table 5)
Annuity payment=$4,532.43
c.Current investment
Present value of college expenses fifteen years in the future:
Value=$40,000 Present value factor for an annuity due for i = 8% and n = 4
=$40,000 3.57710 (from Table 5)
=$143,084
Present value of college expenses today:
Value=$143,084 Present value factor for i = 8% and n = 15
=$143,084 .31524 (from Table 4)
=$45,106
Annuity payment
Present value=Annuity payment Present value factor for an ordinary annuity for i = 8%
and n = 14
$45,106=Annuity payment 8.24424 (from Table 5)
Annuity payment=$5,471.21
EA–15
a.($30,000 .46319) +($30,000 .42888)+($30,000 .39711) +($30,000 .36770)
=$13,895.70 +$12,866.40 + $11,913.30 + $11,031.00
=$49,706.40
b.$49,706.40 6.24689 = $7,956.98
c.($30,000 .55839) + ($30,000 .52679) + ($30,000 .49697)+($30,000 x .46884)
=$16,751.70 +$15,803.70+ $14,909.10+$14,065.20
=$61,529.70
The yearly installment under 6% will be $61,529.70 6.8017 = $9,046.22
PROBLEMS
PA–1
The price that Christie is willing to pay for the stock is comprised of two components: the present value of the dividends she expects to receive from holding the investment and the present value of the proceeds she will receive when she sells the investment. The total present value is calculated as follows.
Present value=Present value of dividends + Present value of proceeds
=[($5 .89286 from Table 4) + ($6 .79719 from Table 4) + ($7 .71178 from Table 4) + ($8 .63552 from Table 4)] + ($100 .63552 from Table 4)
=$4.46 + $4.78 + $4.98 + $5.08 + $63.55
=$82.85
PA–2
a.Investment 1
Future value=($1,000 Future value factor for an ordinary annuity for i = 10% and n = 5)
Future value factor for i = 12% and n = 5
=($1,000 6.10510 from Table 2) 1.76234 from Table 1
=$10,759.26
Investment 2
Future value=$3,000 Future value factor for an ordinary annuity for i = 15% and n = 7
=$3,000 11.06680 from Table 2
=$33,200.40
Therefore, Wharton's total investment at the end of ten years will equal $43,959.66.
b.Current investment=Future value Present value factor for i = 12% and n = 10
=$43,959.66 .32197 from Table 4
=$14,153.69
Therefore, Wharton would have to invest $14,153.69 for ten years earning 12% compounded annually to have an amount equivalent to the two investments.
PA–3
a.Contract 1
Present value=$8,000 Present value factor for an annuity due for i = 6% and n = 10
=$8,000 7.80169 (from Table 6)
=$62,413.52
Contract 2
Present value=$8,000 + ($20,000 Present value factor for i = 12% and n = 10)
=$8,000 + ($20,000 .32197 from Table 4)
=$14,439.40
Contract 3
Present value=($8,000 Present value factor for an ordinary annuity for i = 10% and n = 3)
Present value factor for i = 10% and n = 3
=($8,000 2.48685 from Table 5) .75131 from Table 4
=$14,947.16
PA–3Concluded
b.(1)Equivalent values at the end of Year 5:
Contract 1
Present value=($8,000 Future value factor for an annuity due for i = 6% and n = 5)+
($8,000 Present value factor for an annuity due for i = 6% and n = 5
=($8,000 5.97532 from Table 3) + ($8,000 4.46511 from Table 6)
=$47,802.56 + $35,720.88
=$83,523.44
Proof:
$83,523.44 .74726 = $62,413 = Present value of Contract 1 in Part (a)
Contract 2
Present value=($8,000 Future value factor for i = 12% and n = 5) + ($20,000 Present value factor for i = 12% and n = 5)
=($8,000 1.76234 from Table 1) + ($20,000 .56743 from Table 4)
=$14,098.72 + $11,348.60
=$25,447.32
Proof:
$25,447.32 .56743 = $14,439 = Present value of Contract 2 in Part (a)
Contract 3
Present value=($8,000 Future value factor for i = 10% and n = 1) + $8,000 + ($8,000 Present value factor for i = 10% and n = 1)
=($8,000 1.10000 from Table 1) + $8,000 + ($8,000 .90909 from Table 4)
=$24,072.72
Proof:
$24,072.72 .62092 = $14,947 = Present value of Contract 3 in Part (a)
(2)Equivalent values at the end of Year 10:
Contract 1
Present value=$8,000 Future value factor for an annuity due for i = 6% and n = 10
=$8,000 13.97164 from Table 3
=$111,773.12
Proof:
$111,773.12 .55839 = $62,413 = Present value of Contract 1 in Part (a)
Contract 2
Present value=($8,000 Future value factor for i = 12% and n = 10) + $20,000
=($8,000 3.10585 from Table 1) + $20,000
=$24,846.80 + $20,000.00
=$44,846.80
Proof:
$44,846.80 .32197 = $14,439 = Present value of Contract 2 in Part (a)
Contract 3
Present value=($8,000 Future value factor for an ordinary annuity for i = 10% and n = 3)
Future value factor for i = 10% and n = 4
=($8,000 3.31000 from Table 2) 1.46410 from Table 1
=$38,769.37
Proof:
$38,769.37 .38554 = $14,947 = Present value of Contract 3 in Part (a)
PA–4
Option 1
Present value=$25,000
Option 2
Present value=$60,000 Present value factor for i = 9% and n = 8
=$60,000 .50187 (from Table 4)
=$30,112.20
Option 3
Present value=$5,000 + ($27,000 Present value factor for i = 9% and n = 3) +
($20,000 Present value factor for i = 9% and n = 20)
=$5,000 + ($27,000 .77218 from Table 4) + ($20,000 .17843 from Table 4)
=$5,000 + $20,848.86 + $3,568.60
=$29,417.46
Hartney should accept bonus option 2 because it has the highest present value. In other words, in terms of today’s dollars, bonus option #2 gives Hartney the most amount of money.
PA–5
a.Value=$5,000 + ($10,000 Present value factor for an ordinary annuity for i = 10%
and n = 5) + ($15,000 Present value factor for i = 10% and n = 5)
=$5,000 + ($10,000 3.79079 from Table 5) + ($15,000 .62092 from Table 4)
=$5,000 + $37,908 + $9,314
=$52,222
b.Value=($5,000 Future value factor for i = 10% and n = 2) + ($10,000 Future value
factor for i = 10% and n = 1) + $10,000 + ($10,000 Present value factor for
an ordinary annuity for i = 10% and n = 3) + ($15,000 Present value factor for
i = 10% and n = 3)
=($5,000 1.21000 from Table 1) + ($10,000 1.10000 from Table 1) + $10,000 +
($10,000 2.48685 from Table 5) + ($15,000 .75131 from Table 4)
=$6,050 + $11,000 + $10,000 + $24,869 + $11,270
=$63,189
c.Value=($5,000 Future value factor for i = 10% and n = 4) + ($10,000 Future value
factor for an ordinary annuity for i = 10% and n = 4) + [($10,000 + $15,000)
Present value factor for i = 10% and n = 1)]
=($5,000 1.46410 from Table 1) + ($10,000 4.64100 from Table 2) +
($25,000 .90909 from Table 4)
=$7,321 + $46,410 + $22,727
=$76,458
d.Value=($5,000 Future value factor for i = 10% and n = 5) + ($10,000 Future value
factor for an ordinary annuity for i = 10% and n = 5) + $15,000
=($5,000 1.61051 from Table 1) + ($10,000 6.10510 from Table 2) + $15,000
=$8,053 + $61,051 + $15,000
=$84,104
PA–5Concluded
Proof:
Value of each equivalent value today
Option 1Option 2Option 3Option 4
1.$52,222 1.00000 $52,222
2.$63,189 0.82645 $52,222
3.$76,458 0.68301 $52,222
4.$84,104 0.62092 $52,222
PA–6
Present values
a.Value=$10,000
b.Value=$2,000 Present value factor for an ordinary annuity for i = 8% and n = 8
=$2,000 5.74664 from Table 5
=$11,493.28
c.Value=$5,000 Present value factor for an annuity due for i = 8% and n = 3
=$5,000 2.78326 from Table 6
=$13,916.30
d.Value=$3,000 Present value factor for an ordinary annuity for i = 8% and n = 5
=$3,000 3.99271 from Table 5
=$11,978.13
e.Value=$25,000 Present value factor for i = 8% and n = 7
=$25,000 .58349 from Table 4
=$14,587.25
f.Value=$3,000 Present value factor for an ordinary annuity for i = 8% and n = 2
=$3,000 1.78326 from Table 5
=$5,349.78
g.Value=$4,000 Present value factor for an annuity due for i = 8% and n = 3
=$4,000 2.78326 from Table 6
=$11,133.04
Future values
a.Value=$10,000 Future value factor for i = 8% and n = 4
=$10,000 1.36049 from Table 1
=$13,604.90
b.Value=$2,000 Future value factor for an ordinary annuity for i = 8% and n = 8
=$2,000 10.63663 from Table 2
=$21,273.26
c.Value=$5,000 Future value factor for an annuity due for i = 8% and n = 3
=$5,000 3.50611 from Table 3
=$17,530.55
PA–6Concluded
d.Value=($3,000 Future value factor for an ordinary annuity for i = 8% and n = 5) Future
value factor for i = 8% and n = 5
=($3,000 5.86660 from Table 2) 1.46933 from Table 1
=$25,859.91
e.Value=$25,000
f.Value=$3,000 Future value factor for an ordinary annuity for i = 8% and n = 2
=$3,000 2.08000 from Table 2
=$6,240.00
g.Value=$4,000 Future value factor for an annuity due for i = 8% and n = 3
=$4,000 3.50611 from Table 3
=$14,024.44
PA–7
a.To determine whether the offer of $110,000 today is a good deal, the future cash flows must be converted into equivalent values in present dollars (i.e., present values). The contract specifies two types of future cash flows: $2,000 at the beginning of each year for ten years and a lump-sum receipt of $300,000 in ten years. The present value of the two types of cash flows are calculated below.
(1)Present value of annual receipts:
Value=$2,000 Present value factor for an annuity due for i = 10% and n = 10
=$2,000 6.75902 (from Table 6)
=$13,518.04
(2)Present value of lump-sum receipt:
Value=$300,000 Present value factor for i =10% and n = 10
=$300,000 .38554 (from Table 4)
=$115,662.00
(3)Total present value:
Value=$13,518.04 + $115,662.00
=$129,180.04
Since the present value of the future cash flows exceeds $110,000, it would not be wise for Joy Don Corp. to accept $110,000 in cash today in place of the note. By accepting the cash of $110,000 now, Joy would be worse off by more than $19,000.
b.As the discount rate increases, the present value of future cash flows decreases. Since the present value of the future cash flows discounted at 10% exceeds $110,000, the discount rate at which Joy Don would be wise to accept $110,000 in cash instead of the note must be greater than 10%. Try i = 12%.
Present value=($2,000 6.32825 from Table 6) + ($300,000 .32197 from Table 4)
=$12,656.50 + $96,591.00
=$109,247.50
With a discount rate of 12%, the present value of the future cash flows is slightly less than $110,000, which implies that Joy Don would be better off accepting $110,000 in cash today rather than accepting the note.
1