Question

The United States purchased Alaska in 1867 for $7,200,000. Assume that federal tax revenue from the state of Alaska (net federal expenditures) is $58,200,000 in 2014 and that tax revenue started in 1868 and has steadily increased by 3% annually since then. Assume that the cost of capital (or interest rate) is 6%. What is the NPV of the Alaska purchase, assuming that you are in 1867 looking forward?

Answer 1

As we know, US purchased Alaska in 1867 for $7,200,000

Therefore, Cash outflow (CO) = $7,200,000

In order to compute NPV of tax revenue net of federal expenditure (CI) we have to compute present value (PV) of CI from 1868 till 2014.

Since we are given CI for 2014 as $58,200,000 we have to compute the CI from 1867 to 2013 by doing at back calculation since the CI increases 3% every year. The calculations will be by theis formula

CI for 2013 = CI of 2014/1.03

CI of 2012 = CI of 2013/1,03 and so on and so forth till CI of 1867 is computed

The calculations will be as per below table

Year	Year No.	Tax Revenue net of expenditure	DF @ 6%	PV @ DF
1868	1	7,77,478	0.943396	7,33,470
1869	2	8,00,802	0.889996	7,12,711
1870	3	8,24,827	0.839619	6,92,540
1871	4	8,49,571	0.792094	6,72,940
1872	5	8,75,058	0.747258	6,53,895
1873	6	9,01,310	0.704961	6,35,388
1874	7	9,28,350	0.665057	6,17,405
1875	8	9,56,200	0.627412	5,99,932
1876	9	9,84,886	0.591898	5,82,953
1877	10	10,14,433	0.558395	5,66,454
1878	11	10,44,866	0.526788	5,50,422
1879	12	10,76,212	0.496969	5,34,844
1880	13	11,08,498	0.468839	5,19,707
1881	14	11,41,753	0.442301	5,04,998
1882	15	11,76,005	0.417265	4,90,706
1883	16	12,11,286	0.393646	4,76,818
1884	17	12,47,624	0.371364	4,63,323
1885	18	12,85,053	0.350344	4,50,210
1886	19	13,23,604	0.330513	4,37,468
1887	20	13,63,313	0.311805	4,25,087
1888	21	14,04,212	0.294155	4,13,057
1889	22	14,46,338	0.277505	4,01,366
1890	23	14,89,728	0.261797	3,90,007
1891	24	15,34,420	0.246979	3,78,969
1892	25	15,80,453	0.232999	3,68,243
1893	26	16,27,867	0.21981	3,57,821
1894	27	16,76,703	0.207368	3,47,694
1895	28	17,27,004	0.19563	3,37,854
1896	29	17,78,814	0.184557	3,28,292
1897	30	18,32,178	0.17411	3,19,001
1898	31	18,87,143	0.164255	3,09,972
1899	32	19,43,758	0.154957	3,01,200
1900	33	20,02,071	0.146186	2,92,675
1901	34	20,62,133	0.137912	2,84,392
1902	35	21,23,997	0.130105	2,76,343
1903	36	21,87,717	0.122741	2,68,522
1904	37	22,53,348	0.115793	2,60,922
1905	38	23,20,948	0.109239	2,53,538
1906	39	23,90,577	0.103056	2,46,362
1907	40	24,62,294	0.097222	2,39,390
1908	41	25,36,163	0.091719	2,32,614
1909	42	26,12,248	0.086527	2,26,031
1910	43	26,90,615	0.08163	2,19,634
1911	44	27,71,334	0.077009	2,13,418
1912	45	28,54,474	0.07265	2,07,378
1913	46	29,40,108	0.068538	2,01,509
1914	47	30,28,311	0.064658	1,95,805
1915	48	31,19,161	0.060998	1,90,264
1916	49	32,12,735	0.057546	1,84,879
1917	50	33,09,117	0.054288	1,79,647
1918	51	34,08,391	0.051215	1,74,562
1919	52	35,10,643	0.048316	1,69,622
1920	53	36,15,962	0.045582	1,64,821
1921	54	37,24,441	0.043001	1,60,156
1922	55	38,36,174	0.040567	1,55,624
1923	56	39,51,259	0.038271	1,51,219
1924	57	40,69,797	0.036105	1,46,939
1925	58	41,91,891	0.034061	1,42,781
1926	59	43,17,648	0.032133	1,38,740
1927	60	44,47,177	0.030314	1,34,813
1928	61	45,80,593	0.028598	1,30,998
1929	62	47,18,010	0.02698	1,27,290
1930	63	48,59,551	0.025453	1,23,688
1931	64	50,05,337	0.024012	1,20,187
1932	65	51,55,497	0.022653	1,16,786
1933	66	53,10,162	0.02137	1,13,480
1934	67	54,69,467	0.020161	1,10,269
1935	68	56,33,551	0.01902	1,07,148
1936	69	58,02,558	0.017943	1,04,115
1937	70	59,76,634	0.016927	1,01,169
1938	71	61,55,933	0.015969	98,305
1939	72	63,40,611	0.015065	95,523
1940	73	65,30,830	0.014213	92,820
1941	74	67,26,755	0.013408	90,193
1942	75	69,28,557	0.012649	87,640
1943	76	71,36,414	0.011933	85,160
1944	77	73,50,506	0.011258	82,750
1945	78	75,71,022	0.01062	80,408
1946	79	77,98,152	0.010019	78,132
1947	80	80,32,097	0.009452	75,921
1948	81	82,73,060	0.008917	73,772
1949	82	85,21,251	0.008412	71,684
1950	83	87,76,889	0.007936	69,655
1951	84	90,40,196	0.007487	67,684
1952	85	93,11,401	0.007063	65,768
1953	86	95,90,744	0.006663	63,907
1954	87	98,78,466	0.006286	62,098
1955	88	1,01,74,820	0.00593	60,341
1956	89	1,04,80,064	0.005595	58,633
1957	90	1,07,94,466	0.005278	56,974
1958	91	1,11,18,300	0.004979	55,361
1959	92	1,14,51,849	0.004697	53,794
1960	93	1,17,95,405	0.004432	52,272
1961	94	1,21,49,267	0.004181	50,792
1962	95	1,25,13,745	0.003944	49,355
1963	96	1,28,89,157	0.003721	47,958
1964	97	1,32,75,832	0.00351	46,601
1965	98	1,36,74,107	0.003312	45,282
1966	99	1,40,84,330	0.003124	44,000
1967	100	1,45,06,860	0.002947	42,755
1968	101	1,49,42,066	0.00278	41,545
1969	102	1,53,90,328	0.002623	40,369
1970	103	1,58,52,038	0.002475	39,227
1971	104	1,63,27,599	0.002334	38,116
1972	105	1,68,17,427	0.002202	37,038
1973	106	1,73,21,950	0.002078	35,989
1974	107	1,78,41,608	0.00196	34,971
1975	108	1,83,76,856	0.001849	33,981
1976	109	1,89,28,162	0.001744	33,019
1977	110	1,94,96,007	0.001646	32,085
1978	111	2,00,80,887	0.001553	31,177
1979	112	2,06,83,314	0.001465	30,294
1980	113	2,13,03,813	0.001382	29,437
1981	114	2,19,42,928	0.001304	28,604
1982	115	2,26,01,215	0.00123	27,794
1983	116	2,32,79,252	0.00116	27,008
1984	117	2,39,77,629	0.001094	26,243
1985	118	2,46,96,958	0.001033	25,501
1986	119	2,54,37,867	0.000974	24,779
1987	120	2,62,01,003	0.000919	24,078
1988	121	2,69,87,033	0.000867	23,396
1989	122	2,77,96,644	0.000818	22,734
1990	123	2,86,30,543	0.000772	22,091
1991	124	2,94,89,460	0.000728	21,465
1992	125	3,03,74,144	0.000687	20,858
1993	126	3,12,85,368	0.000648	20,268
1994	127	3,22,23,929	0.000611	19,694
1995	128	3,31,90,647	0.000577	19,137
1996	129	3,41,86,366	0.000544	18,595
1997	130	3,52,11,957	0.000513	18,069
1998	131	3,62,68,316	0.000484	17,557
1999	132	3,73,56,365	0.000457	17,060
2000	133	3,84,77,056	0.000431	16,578
2001	134	3,96,31,368	0.000406	16,108
2002	135	4,08,20,309	0.000383	15,653
2003	136	4,20,44,918	0.000362	15,210
2004	137	4,33,06,266	0.000341	14,779
2005	138	4,46,05,454	0.000322	14,361
2006	139	4,59,43,617	0.000304	13,954
2007	140	4,73,21,926	0.000287	13,559
2008	141	4,87,41,584	0.00027	13,176
2009	142	5,02,03,831	0.000255	12,803
2010	143	5,17,09,946	0.000241	12,440
2011	144	5,32,61,245	0.000227	12,088
2012	145	5,48,59,082	0.000214	11,746
2013	146	5,65,04,854	0.000202	11,414
2014	147	5,82,00,000	0.000191	11,091
			Total	2,55,35,155

Therefore, PV of all CIs will be $25,535,155

Therefore NPV = $25,535,155-$7,200,000 = $18,335,155