Question 476268: You are thinking of building a new machine that will save you $1000 in the first year. The machine will then begin to wear out so that the savings declines at a rate of 2% per year forever. What is the present value of the savings? (r=6%)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your have savings of 1000 per year.
the present worth of those savings in perpetuity would be 1000 / .06 assuming that the 1000 per year did not decrease by 2% per year.
that value would be equal to 16,666.67
it assumes end of year payments are being made.
this means that the first savings is at the end of the first year but the present value of all payments is brought back to the beginning of the first year.
this includes the first payment.
the present value of the first payment is 1000 / 1.06 = 943.3962.
after the end of the first year, the savings are being diminished by 2% of the current value of the payments.
this means that the savings at the end of the second year is equal to .98 * 1000 and the savings at the end of the third year is equal to .98^2 * 1000, etc.
the present value of the savings at the end of the second year brought back to the end of the first year would therefore be 1000 * .98 / 1.06 = 924.5283019
since (1000 * .98) / 1.06 is equivalent to 1000 / 1.081632653, we can use the rate of 1.081632653 to simulate a loss of 2% each year divided by 6% each year.
Assuming this to be true, then we'll look at the first 3 years to get you comfortable with what is happening.
out cash flow would be as follows:
time point 0 is the beginning of the first year.
time point 1 is the end of the first year and the beginning of the second year.
gime point 2 is the end of the second year and the beginning of the third year, etc.
here's what our cash flow would look like if we did the calculations by multiplying each succeeding savings by .98 before discounting to the present value.
cumulative
time savings present value present value
point of savings of savings
0
1 1000 1000/1.06=943.3962264 943.3962264
2 1000*.98 = 980 980/1.06^2=872.1965112 1815.592738
3 1000*.98^2=960.4 960.4/1.06^3=806.3703594 2621.963097
here's what our cash flow would look like if we did the calculations by dividing by a composite rate of 1.06/.98 = 1.081632653
we use the composite rate to get the present value in time point 1 and then use the regular rate to get the present value from time point 1 to time point 0.
it works like this:
cumulative
time savings present value present value
point of savings of savings
0
1 1000 1000/1.06=943.3962264 943.3962264
2 1000 1000/1.081632653/1.06
=872.1965112 1815.592738
3 1000 1000/1.081632653^2/1.06
=806.3703594 2621.963097
Both methods get the same present value.
What this allows us to do, however, is extend the logic to perpetuity as follows:
The present value of the payments that are reduced each year brought back to time point 1 would be equal to:
1000/.081632653 = 12250
that includes all payments except the payment in time point 1.
we now add that payment to the total value to get 13250 and then bring that back to time point 0 at 6% to get a present value of 12,500.
that is the present value of savings, assuming that the value of savings is going down by 2% each year after the first year.
this is tricky, but think about what we just did.
we looked at the problem from the perspective of time point 1 when we started losing 2% of our savings each year.
that made the composite rate equal to 1.06/.98 = 1.081632653
this is equal to an interest rate of 8.1632653% per year which is equivalent to an interest rate of .081632653 per year (interest rate percent is equal to interest rate * 100%).
our perpetuity formula gets us a present value of 1000/.081632653 which equals 12,250.
those savings include all savings except the 1000 savings in time point 1.
we need to add the savings in time point 1 and then bring back the new total to time point 0 at 6% interest rate.
that equals 12250 + 1000 = 13250 / 1.06 = 12,500.
that's your answer.
if you looked at this from the perspective of a cash flow, this is what you would have seen.
I went out 300 years which looks like its enough to get a value that is very close to perpetuity if not right on.
the first column is the time points
the second column is the savings each year. this is reduced by 2% each year after time point 1
the third column is the present value of the savings each year. those savings are being brought back to time point 0 by dividing by 1.06 for each year that they are removed from time point 0. the value in time point 1 is being divided by 1.06. the value in time point 2 is being divided by 1.06^2, etc.
the fourth column is the cumulative present value of the savings each year.
the sum of all the present value of savings for each year is the cumulative present value of savings which is the number you are looking for.
if you go down to time point 300 you will see that the cumulative present value of those savings is equal to 12,500 which is exactly the value we calculated using the formulas.
time point
savings
present value of savings at 6% discount rate per year
cumulative present value of savings
0
1 1000 943.3962264 943.3962264
2 980 872.1965112 1815.592738
3 960.4 806.3703594 2621.963097
4 941.192 745.5122191 3367.475316
5 922.36816 689.247146 4056.722462
6 903.9207968 637.2284934 4693.950956
7 885.8423809 589.1357769 5283.086732
8 868.1255332 544.6726994 5827.759432
9 850.7630226 503.5653259 6331.324758
10 833.7477621 465.5603956 6796.885153
11 817.0728069 430.423762 7227.308915
12 800.7313507 397.9389498 7625.247865
13 784.7167237 367.9058215 7993.153687
14 769.0223893 340.1393444 8333.293031
15 753.6419415 314.4684505 8647.761482
16 738.5691026 290.7349825 8938.496464
17 723.7977206 268.7927197 9207.289184
18 709.3217662 248.5064767 9455.79566
19 695.1353309 229.7512709 9685.546931
20 681.2326242 212.4115523 9897.958484
21 667.6079718 196.3804918 10094.33898
22 654.2558123 181.5593226 10275.8983
23 641.1706961 167.8567322 10443.75503
24 628.3472822 155.1882996 10598.94333
25 615.7803365 143.4759751 10742.41931
26 603.4647298 132.6475996 10875.0669
27 591.3954352 122.63646 10997.70336
28 579.5675265 113.3808781 11111.08424
29 567.976176 104.8238307 11215.90807
30 556.6166524 96.91259822 11312.82067
31 545.4843194 89.59843987 11402.41911
32 534.574633 82.83629346 11485.25541
33 523.8831403 76.58449773 11561.8399
34 513.4054775 70.80453563 11632.64444
35 503.137368 65.4607971 11698.10524
36 493.0746206 60.52035958 11758.6256
37 483.2131282 55.95278527 11814.57838
38 473.5488656 51.72993355 11866.30831
39 464.0778883 47.82578762 11914.1341
40 454.7963306 44.21629422 11958.3504
41 445.700404 40.87921541 11999.22961
42 436.7863959 37.79399161 12037.0236
43 428.050668 34.94161488 12071.96522
44 419.4896546 32.30451187 12104.26973
45 411.0998615 29.8664355 12134.13617
46 402.8778643 27.6123649 12161.74853
47 394.820307 25.52841283 12187.27694
48 386.9239008 23.60174016 12210.87868
49 379.1854228 21.82047676 12232.69916
50 371.6017144 20.17364832 12252.87281
51 364.1696801 18.65110883 12271.52392
52 356.8862865 17.24347797 12288.76739
53 349.7485608 15.94208341 12304.70948
54 342.7535895 14.7389073 12319.44839
55 335.8985177 13.62653694 12333.07492
56 329.1805474 12.59811906 12345.67304
57 322.5969364 11.64731762 12357.32036
58 316.1449977 10.76827478 12368.08863
59 309.8220978 9.955574796 12378.04421
60 303.6256558 9.20421066 12387.24842
61 297.5531427 8.509553252 12395.75797
62 291.6020798 7.867322818 12403.6253
63 285.7700382 7.273562605 12410.89886
64 280.0546375 6.724614484 12417.62347
65 274.4535447 6.21709641 12423.84057
66 268.9644738 5.747881586 12429.58845
67 263.5851844 5.314079202 12434.90253
68 258.3134807 4.913016621 12439.81555
69 253.1472111 4.542222914 12444.35777
70 248.0842668 4.199413637 12448.55718
71 243.1225815 3.882476759 12452.43966
72 238.2601299 3.589459645 12456.02912
73 233.4949273 3.31855703 12459.34768
74 228.8250287 3.068099896 12462.41578
75 224.2485282 2.836545187 12465.25232
76 219.7635576 2.622466305 12467.87479
77 215.3682864 2.42454432 12470.29933
78 211.0609207 2.241559843 12472.54089
79 206.8397023 2.072385515 12474.61328
80 202.7029082 1.915979061 12476.52926
81 198.6488501 1.771376868 12478.30063
82 194.6758731 1.637688047 12479.93832
83 190.7823556 1.51408895 12481.45241
84 186.9667085 1.399818085 12482.85223
85 183.2273743 1.294171437 12484.1464
86 179.5628268 1.196498121 12485.3429
87 175.9715703 1.106196376 12486.44909
88 172.4521389 1.022709857 12487.4718
89 169.0030961 0.945524208 12488.41733
90 165.6230342 0.87416389 12489.29149
91 162.3105735 0.808189257 12490.09968
92 159.0643621 0.747193841 12490.84688
93 155.8830748 0.690801853 12491.53768
94 152.7654133 0.638665864 12492.17634
95 149.710105 0.590464667 12492.76681
96 146.7159029 0.545901296 12493.31271
97 143.7815849 0.504701198 12493.81741
98 140.9059532 0.466610542 12494.28402
99 138.0878341 0.431394652 12494.71542
100 135.3260774 0.398836565 12495.11425
101 132.6195559 0.368735692 12495.48299
102 129.9671648 0.340906583 12495.82389
103 127.3678215 0.315177784 12496.13907
104 124.8204651 0.291390782 12496.43046
105 122.3240558 0.269399025 12496.69986
106 119.8775746 0.249067023 12496.94893
107 117.4800231 0.230269512 12497.1792
108 115.1304227 0.212890681 12497.39209
109 112.8278142 0.196823459 12497.58891
110 110.5712579 0.181968859 12497.77088
111 108.3598328 0.16823536 12497.93912
112 106.1926361 0.155538352 12498.09466
113 104.0687834 0.143799608 12498.23845
114 101.9874077 0.132946808 12498.3714
115 99.94765958 0.122913086 12498.49431
116 97.94870639 0.113636627 12498.60795
117 95.98973226 0.105060278 12498.71301
118 94.06993762 0.0971312 12498.81014
119 92.18853887 0.089800544 12498.89994
120 90.34476809 0.083023144 12498.98297
121 88.53787273 0.076757246 12499.05972
122 86.76711527 0.070964247 12499.13069
123 85.03177297 0.065608454 12499.1963
124 83.33113751 0.060656873 12499.25695
125 81.66451476 0.056078996 12499.31303
126 80.03122446 0.051846619 12499.36488
127 78.43059997 0.047933666 12499.41281
128 76.86198797 0.044316031 12499.45713
129 75.32474821 0.040971425 12499.4981
130 73.81825325 0.037879242 12499.53598
131 72.34188818 0.035020431 12499.571
132 70.89505042 0.03237738 12499.60338
133 69.47714941 0.029933804 12499.63331
134 68.08760642 0.027674649 12499.66099
135 66.7258543 0.025585996 12499.68657
136 65.39133721 0.023654978 12499.71023
137 64.08351047 0.021869696 12499.7321
138 62.80184026 0.020219153 12499.75232
139 61.54580345 0.018693179 12499.77101
140 60.31488738 0.017282373 12499.78829
141 59.10858963 0.015978043 12499.80427
142 57.92641784 0.014772153 12499.81904
143 56.76788948 0.013657274 12499.8327
144 55.63253169 0.012626536 12499.84532
145 54.51988106 0.01167359 12499.857
146 53.42948344 0.010792564 12499.86779
147 52.36089377 0.009978031 12499.87777
148 51.3136759 0.009224972 12499.88699
149 50.28740238 0.008528748 12499.89552
150 49.28165433 0.007885069 12499.90341
151 48.29602124 0.007289969 12499.9107
152 47.33010082 0.006739783 12499.91744
153 46.3834988 0.00623112 12499.92367
154 45.45582883 0.005760847 12499.92943
155 44.54671225 0.005326066 12499.93476
156 43.655778 0.004924099 12499.93968
157 42.78266244 0.004552469 12499.94423
158 41.9270092 0.004208886 12499.94844
159 41.08846901 0.003891234 12499.95233
160 40.26669963 0.003597556 12499.95593
161 39.46136564 0.003326043 12499.95926
162 38.67213833 0.00307502 12499.96233
163 37.89869556 0.002842943 12499.96517
164 37.14072165 0.002628382 12499.9678
165 36.39790722 0.002430013 12499.97023
166 35.66994907 0.002246616 12499.97248
167 34.95655009 0.00207706 12499.97456
168 34.25741909 0.001920301 12499.97648
169 33.57227071 0.001775372 12499.97825
170 32.90082529 0.001641382 12499.97989
171 32.24280879 0.001517504 12499.98141
172 31.59795261 0.001402976 12499.98281
173 30.96599356 0.001297091 12499.98411
174 30.34667369 0.001199197 12499.98531
175 29.73974021 0.001108692 12499.98642
176 29.14494541 0.001025017 12499.98744
177 28.5620465 0.000947657 12499.98839
178 27.99080557 0.000876136 12499.98927
179 27.43098946 0.000810012 12499.99008
180 26.88236967 0.000748879 12499.99083
181 26.34472228 0.00069236 12499.99152
182 25.81782783 0.000640106 12499.99216
183 25.30147127 0.000591797 12499.99275
184 24.79544185 0.000547133 12499.9933
185 24.29953301 0.00050584 12499.9938
186 23.81354235 0.000467663 12499.99427
187 23.3372715 0.000432368 12499.9947
188 22.87052607 0.000399736 12499.9951
189 22.41311555 0.000369567 12499.99547
190 21.96485324 0.000341676 12499.99581
191 21.52555618 0.000315889 12499.99613
192 21.09504505 0.000292048 12499.99642
193 20.67314415 0.000270007 12499.99669
194 20.25968127 0.000249629 12499.99694
195 19.85448764 0.000230789 12499.99717
196 19.45739789 0.000213371 12499.99739
197 19.06824993 0.000197267 12499.99758
198 18.68688493 0.000182379 12499.99777
199 18.31314724 0.000168615 12499.99793
200 17.94688429 0.000155889 12499.99809
201 17.58794661 0.000144124 12499.99823
202 17.23618767 0.000133247 12499.99837
203 16.89146392 0.00012319 12499.99849
204 16.55363464 0.000113893 12499.9986
205 16.22256195 0.000105297 12499.99871
206 15.89811071 9.73503E-05 12499.99881
207 15.5801485 9.00031E-05 12499.9989
208 15.26854553 8.32104E-05 12499.99898
209 14.96317462 7.69304E-05 12499.99906
210 14.66391112 7.11243E-05 12499.99913
211 14.3706329 6.57564E-05 12499.99919
212 14.08322024 6.07937E-05 12499.99926
213 13.80155584 5.62055E-05 12499.99931
214 13.52552472 5.19636E-05 12499.99936
215 13.25501423 4.80418E-05 12499.99941
216 12.98991394 4.4416E-05 12499.99946
217 12.73011566 4.10638E-05 12499.9995
218 12.47551335 3.79647E-05 12499.99953
219 12.22600308 3.50994E-05 12499.99957
220 11.98148302 3.24504E-05 12499.9996
221 11.74185336 3.00013E-05 12499.99963
222 11.50701629 2.77371E-05 12499.99966
223 11.27687597 2.56437E-05 12499.99969
224 11.05133845 2.37083E-05 12499.99971
225 10.83031168 2.1919E-05 12499.99973
226 10.61370545 2.02648E-05 12499.99975
227 10.40143134 1.87353E-05 12499.99977
228 10.19340271 1.73214E-05 12499.99979
229 9.989534656 1.60141E-05 12499.9998
230 9.789743963 1.48055E-05 12499.99982
231 9.593949084 1.36881E-05 12499.99983
232 9.402070102 1.2655E-05 12499.99984
233 9.2140287 1.16999E-05 12499.99986
234 9.029748126 1.08169E-05 12499.99987
235 8.849153163 1.00005E-05 12499.99988
236 8.6721701 9.24578E-06 12499.99989
237 8.498726698 8.54799E-06 12499.9999
238 8.328752164 7.90285E-06 12499.9999
239 8.162177121 7.30641E-06 12499.99991
240 7.998933578 6.75499E-06 12499.99992
241 7.838954907 6.24518E-06 12499.99992
242 7.682175809 5.77384E-06 12499.99993
243 7.528532293 5.33808E-06 12499.99993
244 7.377961647 4.93521E-06 12499.99994
245 7.230402414 4.56274E-06 12499.99994
246 7.085794365 4.21838E-06 12499.99995
247 6.944078478 3.90001E-06 12499.99995
248 6.805196909 3.60567E-06 12499.99996
249 6.66909297 3.33354E-06 12499.99996
250 6.535711111 3.08196E-06 12499.99996
251 6.404996889 2.84936E-06 12499.99997
252 6.276896951 2.63431E-06 12499.99997
253 6.151359012 2.43549E-06 12499.99997
254 6.028331832 2.25168E-06 12499.99997
255 5.907765195 2.08175E-06 12499.99997
256 5.789609891 1.92463E-06 12499.99998
257 5.673817693 1.77938E-06 12499.99998
258 5.56034134 1.64508E-06 12499.99998
259 5.449134513 1.52093E-06 12499.99998
260 5.340151822 1.40614E-06 12499.99998
261 5.233348786 1.30002E-06 12499.99998
262 5.12868181 1.2019E-06 12499.99999
263 5.026108174 1.11119E-06 12499.99999
264 4.925586011 1.02733E-06 12499.99999
265 4.82707429 9.49795E-07 12499.99999
266 4.730532805 8.78112E-07 12499.99999
267 4.635922149 8.11839E-07 12499.99999
268 4.543203706 7.50568E-07 12499.99999
269 4.452339631 6.93922E-07 12499.99999
270 4.363292839 6.4155E-07 12499.99999
271 4.276026982 5.93131E-07 12499.99999
272 4.190506442 5.48367E-07 12499.99999
273 4.106696314 5.06981E-07 12499.99999
274 4.024562387 4.68718E-07 12499.99999
275 3.94407114 4.33343E-07 12499.99999
276 3.865189717 4.00638E-07 12500
277 3.787885922 3.70401E-07 12500
278 3.712128204 3.42446E-07 12500
279 3.63788564 3.16601E-07 12500
280 3.565127927 2.92707E-07 12500
281 3.493825369 2.70616E-07 12500
282 3.423948861 2.50192E-07 12500
283 3.355469884 2.3131E-07 12500
284 3.288360486 2.13852E-07 12500
285 3.222593277 1.97712E-07 12500
286 3.158141411 1.82791E-07 12500
287 3.094978583 1.68995E-07 12500
288 3.033079011 1.56241E-07 12500
289 2.972417431 1.44449E-07 12500
290 2.912969082 1.33547E-07 12500
291 2.854709701 1.23468E-07 12500
292 2.797615507 1.1415E-07 12500
293 2.741663196 1.05535E-07 12500
294 2.686829933 9.75699E-08 12500
295 2.633093334 9.02061E-08 12500
296 2.580431467 8.33981E-08 12500
297 2.528822838 7.71039E-08 12500
298 2.478246381 7.12848E-08 12500
299 2.428681454 6.59048E-08 12500
300 2.380107824 6.09308E-08 12500
|
|
|
