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I
Late
Math
LOAN TABLES
J. B. CHERRIMAN, M.A.,
LaU Fellow of St John's CoUcge, Cambridge; Prof, ofyat. PliU. in Univertity
College, Toronto; Actuary to the Confederation Life Association.
AND
JAMES LOUDON, M.A.,
Uathe^r^aVical Tutor and Dean in Univ. Coll., Toronto; Consulting Actuary to
the Farmers' and Mechanics' Loan and Savings Company.
TORONTO:
COPP, CLARK <k CO., PEINTERS, KINO ITBEET EAST.
3873.
\1
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The following Tables are an extension of a set published in the
Canadian Almanac for 1870, which gave the values from 1 to 12
years for rates of interest per cent, per annum 10, 10^, 11, IH, 13
payable half-yearly. In the present set, the time runs up to 20
years, and the rates are 8, 9, 10, 11, 12, 13 ; with aome additional
Tables which seemed likely to be of use.
t^
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^
EXPLANATION AND USE OF THE TABLES.
^>
"When a loan is contractod to be repaid (principal and interest)
by equal instalments extending over a term of years, the following
tables will give the amount of the instahnent, monthly, quarterly,
half-yearly, or j-early, for various rates of interest, and will also
give the present worth or surrender value of the instalments for
any unexpired portion of the term.
As interest is in this country generally jiayablo half-yearly, tho
half-yearly rate has been taken as the basis, and the calculations
are made on the true yearly rate correspondiii:;- and not tho
nominal rate. Tlius 1(> per cent, per annum, payable half-yearly,
is understood to mean 5 per cent, per half-year, the true yearly
rate in this case being 10^^; for, $100 being put out for one year
at 5 per cent, per half-year, would amount at the end of the year to
$110.25. Both tho true and the nominal rates for the various
periods are given in a foot note to each table.
(1.) Table I. gives tho instahnent payable at tho end of each
raontli, quarter, half-year, or year, to repay a loan of $1,000 in
au}^ period from 1 to 20 years.
These instalments are given exact to tho nearest cent above the
true value when the dlfforence exceeds one-tenth of a cent.
(2.) Table 11. gives the present value of an instalment of $1
payable at the end of each month, quarter, half-year, or year,
during any nund)er of years from 1 to 20.
B}' aid of this table, the present value o. ^n instalment for an
exact number of years is at once found by multiplying the instal.
ment by the corresponding number in the table.
Example. — Interest at 10 per cent, per annum, payable half-
yearly. An instalment of $20 is payable at the end of each month
for 1 years: required, the i)rescnt value.
6
EXPLANATION AND USE OF THE TABLES.
No, corresponding in Tabic II 60.6 1Y4
Instnlrnent 20
Present value $1 2 1 '2.35
^-^
\*'
(3.) Table III. gives the present value of %\ due at the end of
any number of months from 1 to 12.
By aid of this Table and Table II., tlie present value of an
instalment for a broken period of a number of years and some
months can be found — as follows :
From Table II. take the present value of an instalment of *1 for
the named number of years, and discount it for the number of
months by multiplying it by the factor corresponding to that
number of months in Tabic III.; then add to it tiie present values
taken from Table III. of the several instalments of $1, payable
during the broken period of months, and multiply the final sum
by the given instalment.
Example 1. — Interest at 10 per cent, per annum, payable half-
yearly A yearly instalment of |5U0 has 8 years and 9 months to
run : required, its present value.
From Table II., present value of $1 for 3 years 2.47695
Multiply this by the factor for 9 months from Table III. 0.9294284
giving 2.3U1218
Add the present value of an instalment %\ due at the cud
of 9 months, from Table III 0.929 128
3.230G46
Multiply the instalment ^600 by this number, giving
Present value .... fll 615.32
Example 2. — Same rate.
A liulf-vearlv instalment of ^250 has G year.s and 11 months to
run: required, its present value.
From Table 11., present value of half-yearly instalment
$1 for G years 8.8632
Multiply this by the factor for 11 months from Table III. 0.9144350
giving 8.10482
>;►.»
EXPLANATION AND USE OF THE TABLES. 7
Brought forward 8.10482
AJd the present vnhic of nn Instftlmcnt $1 duo at the end
of 5 months, from Table III 0.90018
And the present vahie of nn instahnent $1 due at the end
of 11 months 0.01443
9.97941
Multiply the instalment $260 by this number, giving
Present value |2404.85
ExAMPLR 3. — Required the present value of a quarterly instal-
ment of §100, unexpired period being 7 years and 5 months*
interest at 11 per cent, per annum, payable half-yearly.
From Table 11., present value of instalment of $1 for
7 years, is 19.4396
Multiply this by the factor for 6 mos., from Table III. 0.95fi3C33
giving 18.5912
Add the present value of $1 duo at end of 2 months. . 0.9823
And the present value of .$1 duo at end of 5 months. . 0.9564
giving 20.6299
Multiply the instalment ^100 by this number, giving. .
Present value $2052 . 99
In the same way by aid of these two Tables, the present value
of a monthly instalment for a broken period may bo found, but
this will be more conveniently effected by Table IV.
(4.) Table IV. gives the present values of a monthly instalment
for any number of months not cx'ceeding a year.
By aid of this and the previous tables, the present value of a
monthly instalment for a broken period of a number of jears and
some months can be found, as follows :
From Table II. take the present value of an instalment of $1
for the named number of years, and discount it for the number
of months by multiplying it by the factor corresponding to that
iiumber of months in Table III ; then add the present value of an
instalment of .<;1 from Table IV. for the number of months, and
multiply the final sum by the given instalment.
Example. — Interest 10 per cent, per annum, payable half-yearly.
An instalment of $12.50 per month has 4 years and 8 months
to run: required, its present value.
>.
8
EXPLANATION AND USE OF THE TABLES.
Present vnlno of monthly instnlrnont $1 for 4 j'onrs, from
Tablo II V.0.6105
Multiply this Ly tho factor for 8 months, from TublolII.
o.U.'nunt, n'lvin^' 37.0860
Add present value of monthly instuhnent of §^l for 8
mouths, from Table IV 7.7139
44.8U05
Multiply tho instalment $12.60 by this number, uiving
rresent value. . . . $600.01
(5.) Table V. gives tho amount with interest of $1 after any
number of months from I to 12.
lly aid of thia and Table II., the present value of an instalment
for a broken period of a number of years and some months can
r.lso be found, as follows:
From Tablo II. take tho present value of nn iiistalnipnt of $1 fop
the nuiubcr of years next <jjreator than the broken period, and find
tho amount of it for tho difrcrmcc! between the <ii^ivcn number of
months and one year, by muliiplyiny; it by tho factor correspond-
inij to that diU'orouco in Tablo V. Then subtract from it tho
amounts (with interest^ of tho several instalments of ^\ (if any)
paid durin;^ that dllFerenco as given in Table V., and multiply tho
rcsuli. by the given inatalment.
Example 1. — Interest 10 per cent, per annum, payable half-
yearly. A yearly instalment of $5U0 has 3 years and 9 months to
run : required, its present vnlus.
From Tablo II., present value of yearly instalment of
$1 for 4 years 3 .15279
Multiply this by the factor for 3 mcnths, from Table V.,
1.021695, giving 3.23065
No instalment hus boon paid during the 3 months.
Therefore, multiply tho instalment $500 by the above,
giving Present value )B1G 1 5. 32
f
t'
EXPLANATION AND USE OF THE TABLES. 9
ExAMPLK 2. — A linlf-yoarly instalment of $250 hm yenra and
11 months to run: roquirod, its presi'ut valiio. (Same rftto.)
From Tiiblo II., prosont value of hnlf-yeurly instalment
of 5-1 for 7 years, ia 9 . 8986
Multiply this by the factor for 1 month, from Toble V.,
1.008165, giving 9.919-i
No instalment has been paid during the 1 month.
Therefore, multiply the instalment $2rtO by the above,
giving Present value .... !^2494 . 86
Example 3. — Required the present value of a quarterly instal-
ment of ^100, unexpired period being 7 years and 6 months .
(Same rate.)
From Table II., present value of quarterly instalment of
$1 for 8 years, is 21 .2082
Multiply this by the factor for 7 months, from Table V. 1 .Oi)M56
giving 22.5761
Subtract the amount of the instalment of $1
paid 4 months before 1 .Oa63
And the amount (»f the instalment of 1^1
paid 1 month before 1 .0089 2 .0452
giving 2.05299
Multiply the instalment $100 by this number, giving
Present value. . . . §2052.99
In the same way by the aid of these two Tables the present
value of a monthly instalment for a b"oken period can be found ;
but this can be more conveniently effected by Tal)le VI.
(6). Table YI. gives the amount at the end of any number of
months, not exceeding a year, of a mouthly instalment of ^1 paid
at the end of each month during tliat period.
By aid of this, and Tables II, and V., the present value of a
monthly instalment for a broken period of a number of years and
some months can be found as follows: From Table II. take the
present value of a monthly instalment of $1 for the number of
years next greater than the broken period, and find the amount of
10
EXPLANATION AND USE OF THE TABLES.
^
it for the diflference betvvoen the given number of months and
one year by multiplyinij it by the fuctor corresponding to that
difference in Table V.; then subtract from it the amount of an
instalment of $1 for that difference given in Table VI., and
multiply the result by the given instalment.
Example. — Interest 10 per cent, per annum, payable half-yearly.
An instalment of ^12 50 per month has four years and eight
monihs to run; required, its present value.
Present value of monthly instalment of $1 for 5 years,
from Table II 47 . 2866
Multiply this by the factor for 4 months, from Table V.
1.088062, giving 48.8498
Subtract from this the amount corresponding to 4 months
in Table VI 4.0493
giving 44.8005
Multiply the instalmeut ^12.50 by this number, giving. . $5G0 01
(7). Table VIII. gives the present value of $1 due at the end
of any number of years, from 1 to 20.
With the aid of Table III. this also gives the present value of a
sura due after a broken period of a number of years and some
months, as follows ;
Multiply the sum by the factor corresponding to the number of
years in Table VIII., and then multiply this product by the factor
corresoondinii; to the number of months in Table III.
Example, — Required the present value of $4,000, due 17
years and 8 months hence, interest 10 per cent, paj-able half-yearly.
Multiply ^4,000 by the factor for 17 years, from Table
VIII., 0.1903548, giving 761.4192
Multiply this by the factor for 8 months from Table III. . 9370174
Present value.. $713 46
(8) By the aid of this and previous Tables, the value of an
ordinary mortgage, where tiie priucipal i^ paid at the end of the
period, can be calculated for any of the given rates of interest.
Consider the interest payable on the mortgage as an instalment
and calculate its present value by the preceding rules ; then add
%
%
EXPLANATION AND USE OF THE TABLES. H
to it the present value of the principal payable at the end of the
period, which will be found as in the last example.
Ex. 1. — A mortgage of :j;5,00r>, interest at 6 per cent, per annum,
ha3 7 years and 10 montiis to run: find its present value, interest
at 10 per cent, per annum, payable half-yearly.
The interest, $300, may be considered a yearly instalment.
The present value of yearly instalment of $1 for 1 years,
from Table II, is 4 . 82860
Multiply this by the factor for 10 months from Table III. . 9219014
giving 4.451491
Add the present value of the instalment of $1 due at
the end of 10 mouths . 921 901
giving 5. 378^93
Multiply the instalment $300 by this number, giving
Present value $1,G12 02
Again : multiply the principal, .?5,000, by the factor for
7 years from Table VUI, 0.5050679, giving 2525.84
Multiply this by the factor for 10 mos. from Table III. 0.9219014
giving !C.2,328 11
Adding these results, the present value required is $^3,940 13
Ex. 2.— A mortgage of $4,000, interest at 5 per cent, per annum,
payable half-yearly, has 17 years and 8 months to run: fina ita
present value, interest 10 per cent, per annum, payable half-yearly.
The interest, $100, will be considered a half-yearly instalment.
The present value of a half-yearly instalment of |1 for
17 years, from Table II, is 16 . 1 929
Multiply this by the factor for 8 months, from Table III. . 9370174
giving 15.17303
Add the present value of instalment of $1 due at end
of 2 months, from Table III . 98387
And the present value of the stime, due at end of 8 mos. . 93702
givinij 17.09393
12
EXPLANATION AND USE OF THE TABLES,
Multiply the interest, $100, by this number, giving;. . . $1,709 89
The present value of the principal, 84,000, due 17 years
and 8 months hcucc, is found, in the last example
but one, to be $713 46
Adding these two results, we find the present value-
required to bo $2,422 85
(9). Sometimes the borrower may wish to pay off part of his
debt by a lump sum at some time during tiie running of his pay-
ments. In this case, having calculated the present value of his
future payments, deduct from it the sum he pays down, and consi-
der the remainder as a now loan. If the borrower wishes to con-
tinue his payments at a reduced rate for Mie same period, the new
rate of instalment can be obtained at once from Table I, when the
period is an exact number of years; but if he wishes to continue
to pay the same amount of instalment, reducing the period during
which the payments are to continue, obtain from Tables II. and III.
the period which will give for that instalment a present value next
less than the above-mentioned remainder; take the difference
between tliis present value and such remainder; this differ"nce
improved for the period at the given rate of interest will be an
additional sura, which he must pa}- along with the final instalment
to clear the account. To obtain this ditference so improved, if for
a broken period of a number of years and some months, divide the
difference by the factor corresponding to the number of j'cars ia
Table VIII, and then divide this quotient by the factor correspond-
ing to the number of months in Table III.
ExAMPLK 1, — Interest at 10 per cent, per annum, payable half-
yearly. A monthly instalment of ^20 as a loan has 6 years to run,
and the borrower wishes to pay down §500 along with his instal'
ment: to find the reduced instalment for the remaining period.
The present value of the instalments from Table II $945 73
Deduct 500 00
Remainder 445 73
Multiply this by the number for o years from Table I, 21.15,
Rud dividing by 1000, we obtain $9 43 fo^ the reduced instalment,
i
%
1
't
I
EXPLANATION AND USE OF THE TABLES, 13
Example 2. — Interest .'is abovo.
A monthly instalment of $10 has 1 years and 3 months to run,
and the borrower wishes to pay down along with his instalment
$200: to find the reduction in the period.
Present value of instalments, found as in (3) $621 08
Deduct 200 00
Remainder 421 08
The present value of a monthly instalment, $10, ia now to be
found, which is next below |421 08 in the Tables.
From Table 11. it is seen that the presen value of a monthly
instalment of $10 for 4 years is $395 80, and on trial with Table Ilf.
it is found that the present value of an instalment of $10 for a
period of 4 years and 3 months is $415 78, and is the next below
the given remainder $421 08. Hence the period required is 4
years and 3 months, and the remaining difference $5 30, amounting
in this period to $8 03, leaves this additional sum to be paid along
with the final Instalment.
(10). Given the loan, the instalment, and the number of years for
repayment: it is required to determine the rate of interest.
AVhen the instalment for $1,000 lies between two values in the
Tables, an approximate value of the rate of interest can at once be
obtained by interpolation, and this will in general be near enough
for practical purposes ; or from tins approximate value, closer and
closer values may be obtained by the method of " trial and error,"
or " double position."
When the instalment for $1,000 is beyond the Tables, the method
of proceeding will be found in the Appeadij{.
m
U
Interest 8 per Cent, peii Annum, Payable Half-yearly.
TABLE T.
Instalments to repay a Loan of ^1000 in the named number
of years, payable
Years.
Yearly.
Half-yearly.
Quarterly.
Monthly.
Years.
1
1081.60
630.20
262.50
86.93
1
2
662.00
275.49
186.40
45.17
2
3
S89.16
190.77
94.45
31 28
3
4
308.00
148.53
73.64
24.36
4
5
251.52
123.29
61.04
20.22
6
6
217.87
106.66
52.76
17.47
7
193.13
94.67
46.87
15.63
1
8
175.08
85.82
42.49
14. ^^7
8
9
161.15
79.00
39.11
12.96
9
10
150.11
73.59
36 43
12.07
10
11
141.17
69.20
34.26
11.85
11
12
133.80
05.59
82.48
10.76
12
13
127.64
62.57
30.98
10.26
13
U
122.43
60.02
29.72
9.84
14
16
117.98
67.83
28.64
9.49
15
16
114.14
65.95
27.70
9.18
16
17
110.81
54.32
26.90
8.91
17
18
107.89
52.89
26.19
8.68
18
19
105.33
61.64
25.51
8.47
19
20
103.07
60.53
25.02
8.29
20
Nominal Rates of Intkrest : 8 per cent, per annum payable half-yearly ;
7.921G per cent, per annum payable ijuarterly; 7.8708 percent, per annum
payable monthly.
Actual IIATE3 : 8.16 per cent, per annum ; 4 per cent, per half-year ; 1.980^
percent, per quarter; 0.6559 per cent, per month,
J
1
15
Interest 8 per Cent, per Annum, Payabt.e Half-ybarlt.
ii
M'
TABLE II.
Present Value of Instalment of %\ payahle
Years.
Yearly.
Half-Yearly.
Quarterly,
Monthly.
Years.
1
0.92466
1.8S61
3.8095
11.5037
1
2
1.77936
3.6299
7.3317
22.1396
2
8
2.56968
5.2421
10.5881
31.9730
3
4
8.30037
6.7327
13.5988
41.0646
4
5
3 97593
8.1109
16.3824
49.4703
6
6
4.60053
9.3851
18.9560
67.2418
6
7
5.17800
10 5631
21.3354
64.4270
7
8
5.71191
11.6523
23.5353
71.0701
8
9
6.20554
12.6593
25.5693
77.2121
9
10
6.66192
13.5903
27.4498
82.8906
10
11
7.08388
14.4511
29.1884
88.1408
11
12
7.47400
15.2470
80.79i'9
92.9948
12
13
7.83469
15.9828
32.2821
97.4827
13
14
8.16S17
16 6631
r3.6561
101.6320
14
15
8.47649
17.2920
34.9265
105.4682
15
16
8.76154
17.8735
36.1011
109.0150
16
17
9.02510
18.4112
37.1870
112.2943
17
18
9.26876
18.9083
33.1910
115.3261
18
19
9.49405
19.3679
39.1193
118.1292
19
20
9 . 70234
19.7928
39.9775
120.7208
20
16
Interest 8 per Cent, per Annum, Payable Haly-tearlt.
TABLE III.
Present value of $1 due
any number of months,
1 to 12.
after
from
Months, i Present Value.
1
0.9934845
2
0.9870115
8
. 0805808
4
0.9741915
5
0.9678443
6
0.9615386
7
0.9552736
8
0.9490496
9
0.9428660
10
0.9367229
11
0.9306197
12
0.9245563
TABLE IV.
The present value of an Instal-
ment of^l, pay able at the end
of each month during the named
nwnbtr of months.
Months.
Present Value.
1
0.993485
2
1.9S0496
3
2.961077
4
8.935268
5
4.903113
6
6.864651
7
^•819925
8
7.768974
9
8.711840
10
9.648563
11
10.579183
12
11.503739
i
17
Interest 8 per Cent, per Annum, Patablb Half-yearly.
TABLE V.
Amount of ^1 in any number of
months, from 1 ^o 12.
Months.
1
2
8
4
5
6
7
8
9
10
11
12
Amounts.
1.006559
1.013160
1.019804
1.026492
1.03.3225
1.040000
1.046820
1.053685
1.060596
1.067552
1.074552
1.081600
TABLE VL
Amonut at the cud of any number
of mon/hx. froin. 1 to 12, of a
Mouthlji 7n.st<jhiu7tt of ^\, paid
at the end (f each month during
that period.
Jlonths. '
1
2
3
4
5
6
7
8
9
10
11
12
Amounts.
1.000000
2.006559
3.019719
4.039523
6.066016
6.099210
7.139240
8.186060
9.239745
10.300341
11.367893
12.442445
18
Interest 8 per Cent, per Annum, Payable Half-yearly.
i
TABLE VII.
Amount at the end
of any number of
quarters, from 1 to
4, of a qnarterhj
Jutttalment o/" $1,
paid at the end of
each quarter dur-
ing that period.
Qrs.
Amounts.
1
1.000000
2
2.019804
8
3.059804
4
4.120400
TABLE VIII.
Present Value of $1 due at the end of any
mirnber of years, from 1 to 20.
i
1
Yr3.
1
Present Values.
Yrs.
11
Present Values.
0.924556
0.421955
2
0.854804
12
0.390121
8
0.790315
13
0.360689
4
0.730690
11
0.333477
5
0.675564
15
0.308319
6
0.624597
16
0.285058
1
0.577475
17
0.263552
8
0.5:J3908
18
0.243669
9
0.493628
19
0.225285
10
0.4563S7
20
0.208289
4-
19
Interest 9 peu Cent, per Annum, Payaqi.e TIat.f-yearlt.
TABLE I.
Instalments to repay a Loan of §1000 in the named number
of years, payable
Years.
Yearly.
-7.3. : — -r~
Ilalf-yearly.
Quarterly.
Monthly.
Years.
1
1
1092.03
534.00
264.06
87.88
2
5'70.04
278.75
137.84
45.61
2
3
396.48
193 88
95 . 88
31.73
3
4
310.05
151.61
74.97
24.81
4
6
258.45
126.38
62.50
20.68
6
6
224.27
109.67
54.23
17.95
6
n
200.05
97.82
48.38
16.01
7
8
182.04
89.02
44.02
14.57
8
9
168.18
82.24
40 . 67
13.46
9
10
157.22
76.88
88.02
12.68
10
11
148.36
72.55
35.83
11.87
11
12
Ml. 08
68.99
34.12
11.29
12
13
135.02
66.03
32.65
10.81
13
14
129.90
63.52
31.41
10.40
14
15
125.55
61.40
30.36
10.05
15
16
121.81
69.57
29.46
9.75
16
17
118.58
57.99
28.68
9.49
17
18
115.76
56.61
28.00
9.27
18
19
113.30
55.41
27.40
9.07
19
20
111.14
54.35
26.88
8.90
20
NoJiiNAL Rate of Tntkrest : 9 per cent, per annum payable half-yearly ;
8.9008 per cent, per annum payable quarterly; 8.S350 per cent, per annum
payable monthly.
Actual Rate: 9.2025 per cent, per amir.m ; 4 J per cent, per half-year;
2.2252 per cent, per quarter ; 0.7363 per cent, per month.
20
Interest 9 per Cent, per Annum, Payable Half-yearly.
TABLE II,
Present Value of hmtahncnt of $1 payable
Years
Yearly.
Ilalf-Yeaily.
Quarterly.
Montlily.
Years.
1
0.91573
1.8727
3.7870
11.4449
1
2
1.75429
3.5875
7.2549
21.9253
2
4
2.52219
5.1579
10.4305
31.5225
• 3
3.22537
0.5959
13.3385
40.8110
4
5
3.86930
7.9127
16.0015
48.3589
5
6
4.45896
9.1180
18.4401
55. '7285
6
1
4.99894
10.2228
20.6731
62.4772
7
8
6.49341
11.2340
22.7180
68.6571
8
9
5.94621
12.16'.'
24.5906
74.3162
9
10
6.3G085
13.0079
26.3053
79.4985
10
11
6 . 74055
13.7844
27.8756
84.2440
11
12
7.08825
14.4955
29.3135
88.5896
12
13
7.40666
15.1466
80.6303
92.5690
13
14
7.69823
15.7429
31.8361
96.2131
14
15
7.96523
16.2889
82 9402
99.5501
15
16
8.20973
16.7889
33.9514
102.6('59
16
17
8.43362
17.2468
34.8773
105.4042
17
18
8.C3865
17.GC.60
3.-). 7252
107.9(;66
18
19
8.82640
18.0500
36.5016
110.3131
19
20
8.99833
18.4016
37.2126
112.4619
20
21
Interest 9 per Cent, per Annum, Payable Half-yearly.
V
TABLE III.
Present value of $1 due after
any number of months, from
1 to 12.
TABLE IV.
The present value of an Instal-
vicnt of ^\, payable at the end
of each month during the named
number of months.
Months.
Present Values.
Months.
Present Values.
1
0.9926908
1
0.992691
2
0.9854349
2
1.978126
S
0.9782321
3
2.956358
4
0.9710818
4
3.927440
5
0.9639839
6
4.891423
6
gsegsvs
6
6.848361
1
0.9499483
7
6.798305
8
0.9429998
8
7.741304
9
. 0.9361072
9
8.677412
10
0.9292650
10
9.606677
11
0.9224722
11
10.529149
12
0.9157301
12
11.444879
it
Interest 9 per Cent, per Annum, Patable Half-yearly.
TABLE V.
Amount of $1 iti any number of
monl/isfrom 1 to 12.
Months.
Amounts.
1
1.007368
2
1.014780
3
1.022252
4
1.029779
6
1.037362
6
1.045000
1
1.052695
8
1.060446
9
1.068254
10
1.076119
11
1.084044
12
1.092025
1
TABLE VL
Amount at the end of any number
of months, from 1 to 12, of a
monthly Instalment of$,l, paid
at the end of each month during
that period.
Months.
Amounts.
1
1.000000
2
2.007363
3
3.022143
4
4.044395
6
5.074174
6
6.111536 1
7
7.156536
8
8.209231
9
9.269677
10
10.337931
11
11.414050
12
12.498094
23
Interest D per Cent, per Annum, Payible IIalf-tearlt.
TABLE VII.
Amount at the end
of any number of
quarters, from 1 to
4, of a quarterly
Instalment of %\,
paid at the end of
each quarter dur-
ing that period.
Qrs.
Amounts.
1
2
8
4
1.000000
2.022262
3.067252
4.135506
TABLE VIII.
Present value o/ $1 due at the end of any
number of years from 1 to 20.
Yrs.
1
2
8
4
6
1
8
9
10
Present Values.'' Yrs.
0.9157300 I 11
0.8885613 I 12
0.7678957 ' 13
0.7081851 14
0.6439277 16
0.6896689 16
0.6399729 17
0.4944698 18
0.4528004 19
0.41''-6429 20
Present Values.
0.3797009
0.3477036
0.3184026
0.2915707
0.2670000
0.2444999
0.2238959
0.2050282
0.1877504
0.1719287
24
Interest 10 per Cent, per Annum, Payable Half-yearly.
TABLE I.
Instalments to repay a Loan of $1000, in the named number
of years, payable
Years.
Yearly.
Half-yearly.
Quarterly.
Monthly.
Years.
1
1102.50
537.81
265.63
87.83
1
2
578.18
282.02
139.29
46.06
2
8
403.89
197.02
97.31
32.18
8
4
317.18
154.78
76.42
25.27
4
5
265.49
129.51
63.97
21.15
5
6
231.30
112.88
55.73
18.43
6
7
207.10
101.08
49.90
16.50
7
8
189.16
92.27
45.58
15.07
8
9
176.87
85.55
42.26
13.97
9
10
164.50
80.25
39.64
13.11
10
11
155.74
75.98
37.58
12.41
11
12
148.57
72.48
85.80
11,84
12
18
142.61
69.57
84.36
11.36
18
14
137.61
67.13
33.16
10.96
14
15
133.86
65.06
32.13
10.63
15
16
129.78
63.28
31.26
10.34
16
11
126.60
61.76
80.51
10.09
17
18
123.89
60.44
29.85
9.87
18
19
121.54
59.29
29.28
9.68
19
20
119.47
58.28
28.79
9.52
20
Nominal Rates of Interest : 10 per cent, per annum, payable half-yearly ;
9.86304 per cent, per annum, payablo quarterly ; 9.71788 por cent, per annum,
payable monthly.
Actual Rates: 10.25 per cent, per annum; 5 per cent, per half-year
2.46591 per cent, per quarter ; 0.81649 per cent. p«r month.
I
L>5
Interest 10 teu Cent, peh Annum, Payable IIalf-Yeakly .
TABLE II.
Present Value of Instalment of %\ payable
V'
Years.
Yearly.
Half-Yearly.
Quarterly.
Monthly.
Years.
,1 1
0.90703
1.8594
1
3.7647
11.3S67
1
2
1.72973
3.5460
7.1795
21.7147
2
3
2.47595
5.0757
10.2767
31.0826
3
4
3. 15279
6.4632
13.0860
39.5795
4
5
3.70G70
7.7217
15.6342
47.2865
5
6
4.32354
8.8632
17.9454
54.2769
6
7
4.82800
9.8986
20 0417
60.6174
7
8
5.28672
10.8378
21.9432
66.3685
8
9
5.70224
11.6896
23.6679
71.5848
9
10
6.07913
12.4622
25.28-22
76.3163
10
11
6.42098
13.1630
26.6511
80.6078
11
12
6.73104
13.7986
27 9381
84 . 5003
12
13
7.01229
14.3752
29.1054
88.0310
13
14
7.20738
14.8981
30.1642
91.2334
14
15
7.49876
15 3725
31.1245
94.1380
15
16
7.70862
15.8027
31.9956
96.7726
16
17
7.89898
16.1929
32.7857
99.1623
17
18
8.07163
16.6469
33.5023
101.3298
18
19
8.22824
16.8679
34.1523
103.2958
19
20
8 37029
17.1591
34.7419
105.0790
20
•i
26
Interest 10 per Cent.
PER Annum. Payaptt^ tt ,
TABLE III.
Preseni value of $i am after
Months. Present Vulii
es.
1
2
3
4
5
6
7
8
9
10
11
12
0.901 'JO 13
0.9838680
0.9759000
0.9679964
0.9601569
0.9623S09
0.9446679
0.9370174
0.9294284
0.9219014
0.9144350
0.9070294
TABLE IV.
The prmmt value of an InsiaJ
y^^\n^,paj,a/leahtend
^^^^niber of mouilis.
.Months. Present Val
lies.
3
4
5
6
7
8
9
10
11
12
0.991901
1.975769
2.951669
3.919666
4.879823
5.832203
6.776871
7.713889
8.643317
0.665219
^0.479654
11.386683
-^T^T-TBAELT,
27
Interest 10 peu Cent, peu Anxhm, P.vablb HALP-.EABLr.
IV.
of an Instal
'^^G at the end
'^^'[f the named
s.
'' Values.
51901
^5769
'1669
9666
9823
i203
i87l
8S9
317
219
554
183
TABLE Y,
Amount of $1 hi any number of
niouth!^ from \ to \ 2.
Aronths,
Amounts.
1
1.00S165
2
1 016397
3
1.024695
4
1.033062
5
1.041496
6
1.050000
7
1.058573
8
1.067216
9
1.075930
10
1.084714
11
1.093571
12
1.102500
TABLE VL
Amount at tJieendofanynumhtr
of months from 1 to' 12, of a
monthl,/ instalment of i^l paid
at the end of each month during
that period.
Months.
Amounts.
1
1.000000
2
2.008165
3
3.024562
4
4.049257
5
5.082319
•6
6.123S15
7
'7.173815
8
8.232388
9
9 299604
10
10.375534
11
11.460248
12
12.553819
28
Interest 10 per Cent, per Annum, Payable Half-yearly.
TABLE VIT.
Amount at the end
of any number of
quarters from 1 to
4 of a qnarterhi
instalment of .$1,
paid at the end of
each quarter dur-
ing that period.
Qrs.
Amounts.
1
2
3
4
1.000000
2.024695
3.074695
4.150624
TABLE VIIL
Present value of %\ due at the end of any
number of years, from 1 to 20.
1
Yrs.
Present Values.
Yrs.
Present Values.
1
0.9070295
11
0.3418499
2
0.8227025
12
0.3100679
3
0.7462154
13
0.2812407
4
0.6768394
14
2550936
5
0.6139132
1
15
0.2313774
• 6
0.5568374
16
0.2098662
7
0.5050679
17
0.1903548
8
0.4581115
18
0.172i;574
9
0.4155206
19
0.1566054
10
0.3768895
20
0.1420457
29
4
Interest 11 per Cent, per Annum, Payable Half-yearly.
TABLE I.
Instalmentu to repaij a Loan o/'^lOOD in the named number
of years, payable
Years.
Yearly.
Half-yearly.
Quarterly.
Monthly.
Years.
1 '
1
1113.03
541.62
267.19
88.27
1
2
586.29
285.30
140.74
46.50
o
3
411.37
200 18
98.75
32.63
8
4
324.42
157.87
77. 88
25.73
4
5
272.64
132.67
65.45
21.63
6
6
238.44
116.03
57.24
18.91
6
n
214.30
104.28
51.45
17.00
7
8
196.43
95.59
47.16
15.58
8
9
182.74
88.92
43.87
14.50
9
10
171.97
83.68
41.28
13.64
10
11
163.32
79.48
39.21
12.96
11
12
156.26
76.04
87.51
12.40
12
13
150.42
73.20
36.11
11.93
13
U
145.53
70.82
34.94
11.54
14
15
141.40
68 81
33 . 95
11.22
15
16
137.88
67.10
33.10
10.94
16
17
134.87
65.63
32.38
10.70
17
18
132.28
64.37
81.76
10.49
18
19
130.03
63.28
81.22
10.32
19
20
128.07
62.32
30.75
10.16
20
Nominal Rate op Interest : 11 per cent, per annum payable half-yearly ;
10.85'27G per cent, per auuuin payable quarterly ; 10.75008 per cent, per annum
payable montlily.
Actual Rate: 11.30-25 per cent, per annum ; 5i per cent, per half-year;
2.71319 per cent, per quarter ; 0.89634 per cent, per month.
I
B
30
Interest 11 per Cent, per Annum, Payable Half-yearly.
TABLE II.
Present Value of Inatahacnt of §1 ■paijahle
Years
Yearly.
llLilf-Yearly.
Quarterly.
Monthly.
Years.
1
0.89845
1.8463
3.7427
11.3291
1
2
1.Y056Y
3.5052
7.1054
21.5078
2
8
2.43092
4.9955
10.1266
30.6529
3
^ 4
3.08-252
. 33 15
12.8410
38.8693
1
4
5
3.G6795
7.5376
15.2798
46.2514
5
6
4.19393
8.6185
17.4709
52.8838
6
Y
4.66650
9.5897
19.4395
58.8427
7
8
5.09108
10.4622
21.2082
64.1965
8
9
5.4Y254
11.2461
22.7973
69.0067
9
10
5.81527
11.9504
24.2250
73.3284
10
11
6.12320
12.5832
25.6077
77.2112
11
12
6.39986
18.1517
26.6602
80.6997
12
13
6.64841
13.6625
27.6957
83.8340
13
U
6.87174
14.1214
28.6260
86.6500
14
15
7.07238
14.5337
29 4618
89.1801
15
16
7.252G5
14.9042
30.2128
91.4532
16
lY
7.41461
15.2370
30.8875
93.4955
17
18
7.56013
15.5361
31.4937
95.3304
18
19
7.69087
15.8047
32.0S83
96.9790
19
20
7.80833
16.0461
32.5276
98.4601
20
"
o
31
Interest 11 pek Cent, per Annum, Payable IIaly-ye
AULY.
TABLE III.
Prexput value of §!I due after
ani) number of monlh^ from
1 to 12.
tablp: IV.
IVie prese7it value of an In.ttal-
nient of$l, jxiyahle at the end
of each month during the named
nwnbi r of vionthn.
Montlis
1
Present Value.
'
Months
1
' Prt'seiit Value.
1
0.99111H2
i 1
1
0.991116
2
0.98231U
1
i
2
1.973-128
3
0.9785848
3
2.947012
4
9649350
4
3.911948
5
0.9 5 CSC),'] 3
5
4.868311
6
0. 9478673
6
5.816179 t
n
0.9394467
7
0.755625
8
0.9311009
8
7.086726
9
0.9228292
9
8.609555
10
0.9146310
10
9.524186
11
0.9065056
11
10.430692
12
0.S9S4524
i_
12
11.329144 1
i
,ifr-
32
ISTEUEST 11 PEK CeNT. VFAi AnNU.M, PaYABLE HALF-YEARLY.
TABLF. V.
Amount of $1 in any wimher of
months from 1 to 12.
1
Mvoutlis.
Amounts.
1
1.008963
2
1.018008
8
1.027132
4
1.03H338
5
1.045628
6
1.055000
1
1.064456
8
1.073997
9
1.083624
10
1.093387
11
1.103137
12
1.113025
TABLE VI.
Amount at the end of any nnmher
of vioiitliK, from 1 ^o 12, of a
monthly InHtnlment of :«sl, puid
at the Old of each month during
that period.
Months,
Amounts.
1
1.000000
2
2.0089G3
3
1
3.026971
4
4.054103
5
5.090441
6
6.136069
7
7.191069
8
8.255525
9
9.329522
10
10.413146
11
11.506483
12
12.609620
1
V
33
Interest 11 per Cent, per Annum, Payable IIalf-
YEARLY.
TABLE Vir.
Amount at the end
of any number of
quartern, from 1 to
4, of a quarterly
Imlalment of $1,
paid at the end of
each quarter dur-
ing that period.
Q s.
1
2
3
4
Amounts.
1.000000
2.027132
3.082132
4.165756
TABLE Vin.
Present value of $1 due at the end of any
number of years from I to 20.
1
2
3
4
5
6
7
8
9
10
0.8072162
0.7252458
0.6515982
0.5854305
0.5259816
0.4725693
0.4245811
0.3814659
0.3427290
12
13
14
15
16
17
18
19
20
0.3079256
0.2766566
0.2485627
0.2233218
0.2006440
0.1802691
0.1G19632
0.14551G2
0.1307394
0.117^i631
V
I
34
Interest 12 per Cest. per Annum, Payable Half-yearly.
TABLE I.
Instalments to repay a Lodn of f 1000, in the named unmbcr
of i/ears, payable
Years.
Yoarly.
Half-yearly.
Qunrterly.
Monthly.
Years.
1
1123.60
545.44
268.75
88.72
1
2
594.50
288.60
142.20
46.94
2
8
418.93
203 . 37
100.21
33.08
8
4
831.74
101.04
79.35
26.20
4
5
279.89
135.87
06.95
22.10
5
6
215.72
119.28
58.77
19.40
6
7
221.03
107.59
53.01
17.50
7
8
203.85
98.90
48.70
16.10
8
. 9
190.20
92 . 30
45.51
15.03
9
10
170.61
87.19
42.90
14.19
10
11
171. OS
83.05
40.92
13.51
11
12
161.14
79.08
39.26
12.96
12
13
158.43
76.91
37 90
12.51
13
U
153.06
74.00
30.70
'.2.14
14
15
140.06
72.05
35 . 80
11.82
15
IG
140.27
71.01
34 . 99
11.55
10
17
143.38
09 GO
34.30
11.32
17
18
140.90
08.40
33.70
11.13
18
19
138.76
07.38
83.19
10.90
19
20
136.91
66.47
32.75
10.81
20
Nominal Rates of Interest : 12 per cent per annum, payable half-yearly ;
11.8252 per cent, per anuuin, payable quarterly ; 11.71006 per cent, per annum,
payable monthly.
Actual Rate.s : 12.36 per cent, per annum; G per cent, per half-year;
2.9663 per cent, per quarter ; 0.975SS per cent, per niontli.
36
Interest 12 per Cent, per Annum, Payable Halp-ykarlt.
TABLE II.
Present Value of Instalment of %\ payable
Years.
Yearly.
Half-Yearly.
Quarterly.
3Ionthly.
Years,
1
0.89000
1.8334
3.7210
11.2722
1
••
1.G8209
3.4651
7.0326
21.3045
2
8
2.38706
4.9173
9 . 9800
30.2332
8
4
3.01446
6.2098
12.6032
38.1797
4
5
8.57286
7.3601
14.9378
45.2520
5
G
4.06983
8.3838
17.0155
51.5464
6
1
4. 01213
9.2950
18.8ti48
57.1484
7
8
4.90577
10.1059
20.5106
62.1341
8
9
5.25G12
10.8276
21.9753
66.5714
9
10
5.56792
11.4699
23.2789
70.5205
10
11
5.84543
12.0416
24.4392
74.0353
11
12
6.09241
12.5504
25.4717
77.1634
12
13
6.31222
13.0032
26.3908
79.9474
13
14
6.50785
13 4062
27.2087
82.4251
14
15
6.68196
13.7648
27.9806
84.6303
15
1(3
6.83091
14.0840
28.5845
86 . 5929
16
17
6.97483
14.3681
29.1611
88.3397
17
18
7.09757
14.0210
29.6742
89.8942
18
19
7.20681
14.84G0
30.1309
91.2778
19
20
7.30403
15.0463
30.5o74
92.5092
20
3G
IlfTERHST 12 PER CeNT. PER AnXCM, PaYABLE HaL7-TEARLT.
TABLE III.
Present value of %\ due after
any number of montJia, from
1 to 12.
TABLE IV.
The present value of an Instal-
ment of ■$,1, nm/afile at the end
of each month during the named
number of months.
Months
Present Values.
Months.
Present Values,
1
0.9903356
1
0.990836
2
0.9807645
2
1.971100
3
0.9712868
3
2.942386
4
0.9618989
4
8.904285
5
0.9526026
5
4.856887
6
9433960
6
6.800283
1
0.9342786
7
6.734562
8
0.9252495
8
7.659812
9
0.9163074
9
8.576119
10
0.9074517
10
9.483571
11
0,8986816
11
10.382252
12
1
0.8S99965
_ _.
.2
11.272249
'*
t
%i
37
I.T.
Interest 12 per Cent, per Annum, Payable Ha
LP-YEARLY.
led
'»
■
TABLE V.
Amount of $1 in any number of
months, from 1 to 12.
Months.
1
2
3
4
6'
6
7
8
9
10
11
12
Amounts.
1.009759
1.019612
1.029662
1.039610
1.049756
1.060000
1.070344
1.080789
1.091338
1.101987
1.112741
1.123600
TABLE VL
Amount at the end of any number
of months, from 1 to 12, of a
Monthly Instalment of$l, paid
at the end of each month during
that period.
Months.
Amounts.
1
1.000000
2
2.009759
3
8.029371
4
4.058933
5
6.098543
6
6.148299
7
7.208299
8
8.278643
9
9.359432
10
10.450770
11
11.552757
12
12.666498
%k
38
Interest 12 per Cent, per Annum, Payable Half-yearly.
TABLE VII.
Amount at the end
of any number of
quarters, from 1 to
4, of a quarterly
luHtalmeut of %\,
paid at the end of
each quarter dar-
ing that period.
Qrs.
1
2
3
4
1
Amouiits.
1.000000
2.029562
3.089562
4.180900
— ■ — ■— -■
TABLE VIII.
FrewU Value of %\ due at the end of any
number of yearn, from 1 to 20.
Yrs.
1
2
3
4
5
6
7
8
9
10
Present Values.
0.889996
0.792094
0.704961
0.627-112
0.558395
0.496969
0.442301
0.393646
0.350344
0.311805
Yrs,
Present Values,
11
12
13
11
15
16
17
18
19
20
0.277505
0.246979
0.219810
0.195630
0.174110
0.154957
0.137912
0.122741
0.109239
0.097222
39
Intekest 13 PER Cent, per Annum, Payable Half-yearly.
TABLE I.
Instalments to repay a Loan of $1000 hi the named mimber
of years, payable
Years.
Yearly.
Half-yearly.
Quarterly.
Monthly.
Years.
1
1134.23
649 . 27
270.31
89.16
1
2
602.18
291.91
143.66
47.39
2
3
426.57
206.57
101.66
•
33 54
3
4
339.15'
164.24
80.83
26.66
4
5
287.26
139.11
68.46
22.58
5
6
253.11
122.57
60.32
19.90
6
1
229.10
110.94
54.60
18.01
7
8
211.41
102.38
50.39
16. G2
8
9
107.04
95.86
47.18
15.56
9
10
187.42
90.76
44 67
14.74
10
11
170.02
86.70
42.67
14.^8
11
12
172.22
83.40
41.05
13.54
12
Hi
166.04
80.70
39.72
13.10
13
U
162.01
78,46
38.61
12.74
14
15
158.14
76.58
37.09
12.43
15 ,
16
154.87
75.00
36.91
12.18
16
17
152.10
73.66
36.25
11.06
17
18
140.74
72.52
35.60
11.77
18
19
147. 72
71.54
35.21
11.62
19
20
145.99
70.70
34.79
11.48
20
Nominal Rates of Interest : 13 jier cent, per annum payable half-yearly ;
12.795G per cent, per annum payable quarterly ; 1^.(3012 per cent, per annum
payable uiontlilj'.
Actual Rates : 13.4225 per cent, per annum ; 6J per cent, per half-year;
3. lt»S9 ppr cent, per quarter ; 1 .05.51 per cent, per mouth,
h
i'
40
Interest 13 per Cent, per Annum, Payable Half-yearly.
]
TABLE 11.
Present value of Instalment of $1 payable
Years.
Yearly.
Half-yearly.
Quarterly.
Monthly.
Years.
1
0.88166
1.8206
3.6995
11.2160
1
2
1.65898
3.4258
6.9612
21.1047
2
3
2.34432
4.8410
9.8369
29.8231
8
. 4
2.94855
6.0888
12.3723
37.5098
4
5
3.48127
7.1888
14.6076
44.2869
5
6
3.95096
8.1587
16.5784
60.2619
6
1
4.S6506
9.0138
18.3160
55.5299
7
8
4.73015
9.7678
19.8480
60.1744
8
9
5.05204
10.4325
21.1987
64.2693
9
10
5.33584
11.0185
22.3895
67.8796
10
11
5.58605
11.5352
23.4394
71.0627
11
12
5.80665
11.9907
24.3650
73.8691
12
13
6.00115
12.39t!4
25.1812
76.3433
13
14
6.1V263
12.7465
25 . 9007
78.5248
14
15
6.32381
13.0587
26.5351
80.4481
15
16
6.45711
13.3339
27.0944
82.1438
16
n
6.57463
13.5766
27.5876
83.6388
17
18
6.67824
13 7900
28 0223
84.9569
18
19
6.76959
13.9792
28 4056
86.1191
19
20
6.85013
14.1455
28.7436
87.1437
20
41
LY.
Interest 13 pek Cent, per Annum, Payable Half-yearly.
i*
TABLE III.
Present value af $1 due after
ani; number of months, from
1 to 12.
Months.
Present Value.
1
0.9895592
2
0.9792270
3
0.9690032
4
0.9588859
5
0.9488743
6
0.9389672
Y
0.9291635
8
0.9194623
9
0.9098622
10
0.9003621
11
0.8909618
12
0.8816594
TABLE IV.
Tlie present valve of an Indal-
vicnt of $1, pnynhle at the end
or each month dnrhig the named
number of moiithft.
Months.
Present Values.
1
989559
2
1.968786
8
2.937789
4
3.896675
6
4.845550
6
5.784517
1
6.713680
8
7.633143
9
8.543005
10
9.443367
11
10.334329
12
11.215988
!i
42
Interest 13 per Cent, per Annum, Payable IIalf-ye
ARLY.
TABLE VL
AN
Amount c
moi
TABLE V.
/ |1 in any nnmhc
dhs from, \ to 1 2.
Amount at the end of any number
of montha, from 1 to VI, of a
^ of Month} y Inkahnoit of s ] , paid
at the end of each month during
- that period.
^fontlis
1
Amounts.
«
1.010551
li
Moiitlis
Amounts.
1
/
2
1 021213
1
1.000000
8
l.Uai9S9
2
2.010551
•
4
1.042876
3
3.031704
5
1.05.S8SI
4
4.063753
^
. 6
1.065000
5
5.106620
1
1.076237
6
6.160510
8
1.087593
7
7.225510
9
3.0990G8
8
8.301747
10
1.110665
9
9.389340
»
11
1.122383
10
10.488408
12
1.134225
1
1
11
12
11.599073
12.721450
43
LY.
Interest 13 prr Cemt. per Axnum, Payable Half-yearly.
■
ber
* a
lid
TABLE VI r.
Amount at the end
of mn/ number of
quartcm, from i to
4, of a qnarterhi
Insfahneni of ^1,
paid at the end. of
each quarter dur-
ing that period.
Qrs.
Amounts.
1
1.000000
2
2.031989
3
3.096989
4
4.19G057
TABLE Vin.
Prcseent value o/$l due at the end of any
number of years from 1 to 20.
Yrs.
Present Values.
1
0.881639
2
0.'77Y.'}23
3
0.G85334
4
O.G04231
5
0.532726
6
0.469683
1
0.414100
8
0.365095
9
0.321890
10
0.283797
Yrs.
Pre.sent Values.
11
12
13
14
15
16
17
18
19
20
0. '-'50212
0.220602
0.194496
0.171479
0.151186
0.133295
0.117520
0.103611
0.091351
0.080541
' *^
i i»
APPENDIX.
' *^
<.
Note I. — If i is the yearly rate of interest per unit, and B =
1 -|- i, the amount (J/) of a sum P in n years is given by
M= PR""
and the present value (P) of a sura (i/) due n years hence is given
by
P=J/P-".
In these formulas n may be integral or fractional ; thus, the amount
of 1 in the j^th part of a year is Rp, and the rate of interest per
the pi\\ part of the year is Rp — 1.
Hence the rate per half-year is P — 1.
((
" quarter is P — 1.
" month isP"— 1.
But if i is the nominal yearly rate of interest per unit, payable/)
times a year, meaning thereby that - is the interest payable at
the end of each jt>th part of a year, then the amount of 1 in a year
is [ 1 + ) , and the true yearly rate of interest is [1 + - j — 1.
Note II. — If A is an instalment payable at the end of each year
for n years from the present time, and P its present value, then
1 - 7^-"
in which formula n is necessaril}'' a whole number.
If A is payable at the end of each of p equal intervals in a year,
and the payments continue for n years, then
1— P-'*
P=A. —
Pi^— 1
where n may be fractional, but such thnt the whole time contains
an exact number of the intervals.
46
APPENDIX.
Thus if the instalment A is pn3-nblo
half-yearly, then F = A.
\—R
-n
quarterly,
monthly,
F = A.
P=:A.
E''- 1
Those formulas give the relation between the loan and the in-
stalment, and also give the present value of an instalment having
any exact nuiiibor of periods t ^ run.
If the time to run is not an exact number of ]ioriods, the present
value will be found by taking the present value for the whole
nuMiber of periods less than the given period, discounting this for
the broken interval, and adding the present value of the instalment
pa3'able at the end of the broken period. Thus, if a yearly instal-
ment A has n 3'ears and m mortths to run, its present value ia
given by
m
+ AE r^
P 47? 12- ^
NoTK III. — If V is the present value of I due at the end of any
interval, and A is an instalment paj-able at tlie end of each of u
such intervals, then
P = A (v -{- V- -\- v'^ -^ ... to n terms.)
If the interval is a year, than v = R ; if a half year, v = R ',
if a quarter, v — R ; if a month, v = R ^ -
Tables III. and VIII. give the values of ?^ for niontlis and years.
Tables II. and IV. have been formed by the addition of the v'a.
Table I. has been formed by taking the reciprocals of the num-
bers in Table II., and are given exact to the next cent above the
true value when the difference exceeds -^^t\\ of a cent. In all the
other tables, .the values are made true to the nearest decimal.
IfoTE IV. — When the loan {P), instalment (A), and time are
given, and it is required to find the rate of interest, R nuiat be
obtained from the equation
f\
<
in-
nc;
int
)le
or
nt
il-
is
n
APPENDIX.
, , 1—7?-**
2 =: J. —
47
where n is the number of payments at the end of each interval, and
7? = 1 -f- interest per unit for one interval =!-{-?'.
The solution of this equation, by Horner's method, though theo-
retically possible in all cases, becomes impracticable when n is not
a small number, on account of the length of the operations, and in
general it3is better to proceed by Newton's approximation, ■which,
however, converges but slowly and is often very troublesome to
apply. Several approximate formulas have been devised to save
this labor, of which the following may be noticed.
(i.) Ilalley's formula.
Calculating
A^Yi+i—i,
w— 1 \
and calling this value /3, then
\
•r/3)
(ii.) Using the same notation,
(iii.) Baily's formula with the same notation : •
«•=: (12-n-li3)^
12 — 2 n — 13
3
Of these formulas, (i.) fails altogether when /3 exceeds :
and becomes less correct the nearer ^ approaches to this limit ;
(ii.) is of more easy application than (i.), as the successive terms
are readily found by aid of a table of squares and cubes ; it be-
comes more and more erroneous " ider the same circumstances as
(i.), the degree of error being indicated by the less convergence
of the second and third terms relatively to the first term, (iii.) is
G
also a modification of (i.); it fails when /3 gets be3-ond •.
and becomes less correct as j3 approaches this lii *. It is more
troublesome to work tlian either (i.) or (ii.), but its great disad-
vantage is that it givea no indication iu itself of its degree of
error. • . ,
48
APPENDIX.
None of the throe give good approximations when the period or
the rate is Inrgo. For example, A = l,2' = 9-9992'7484, 71 =x 100,
/3=:*01'762. Here (1.) fails; in (ii.) the terms diverge; (iii.)
gives i = '13477, which is quite astray, the true value being i =
•10000.
The method of " reversion of series" may also be used for the
solution, but it is liable to the same objections as above unless the
series is carried to so large a number of terms as to render the
operation too laborious to be practically useful.
Note V. — The foregoing Tables will be found useful in enabling
Building Societies to keep a proper account with each borrower.
Those accounts should be kept on the principle that whenever an
instalment is paid, the borrower is charged with interest at the
Society's rate on the sum in his possession since the date of the
last payment, and credited with the amount paid ; or, which comes
to the same thing, instead of balancing an account whenever
an instalment is paid, the borrower may at the end of the year
be debited with the amount due at the beginning of the year, with
interest on the same, and credited with the instalments paid in
tills interval with interest from tlie times of payment. This
method, which is the one to bo adopted in practice, is illustrated
in the following oxamplej. The multiplications that occur in the
calculations are most conveniently performed by means of the
Arithmometer of M. Thomas, of Colmar, France, an instrument
which should be in the possession of every Building Society.*
In these examples the books of the Society are closed on the
first day of December in each year, payments made on that day
being included in the accounts, and all loans are supposed to be
contracted on the first day of a month.
* London agents : C. & J. Layton, 150 Fleet Street. A 12-flgure machine
will be found sufBcient. In multiplying a sum by a factor, it will be better to
begiD from the left of the multiplier, a;* follows ; Having set the multiplicand
with the buttons, raise the plate and carry it to the right, so as to leave at
least one hole to the left of the left-hand button. Multiply by the left hand
llgure of the multiplier ; raise the plate and cany it one stage to the left ; then
multiply by the next figure, and so on. In this way the operation can bo
stopped without taking in unnecessary figures, when aufllcieut accuracy has
bicu attained.
APPENDIX.
49
i\^
Example 1.— August 1, 1872. A. B. borrows $1,000.00, to bo
re-paid in C years by yearly instalmonts of $231 .80 each.
The amounts oil the Dr. side are calculated by means of Table
V. by multiplying the sum at the head of a column by the factor
corresponding to the number of months between the dates at the
head and foot of the column.
Thus in the first column 1,000 is multiplied by the factor
1.088062 corresponding to the interval 4 months, giving 1033.00.
The amounts on the Cr. side are calculated in like manner.
Thus in the second column the instnl-nacnt, 231 .30, is multiplied
by the same factor, giving 238.95.
No instalment having been paid before Dec. 1, 1S'J2, the bor-
rower is charged with $1,033.06 in opening the account for the
next year. The differenco between $1,138.95 and $238.95 is
$900.00, the present value of the mortgage on Dec. 1, 1873, and
this is the amount with which the borrower is properly charged
in opening the account for the next year. The account is closed
Aug. 1, 1878, when there is a difTerence of nine cents between the
Cr. and Dr. sides. This difference arises from thf> fact that the
instalment, $231.30, is somewhat greater than the true value. If
the more accurate value, $231,292, be taken, the account will bo
found to balance exactly. Omitting dates, Ac, the account would
then stand as follows : , -
Dr.
1,000.000
1,033.062
1,033.062
1,138.951
900.012
992.263
753.324
830.540
591.601
652.240
413.301
455.664
216.725
231,292
Cr,
4-
0.000
o.ooo
1,033.062
1,033.062
231.292
238.939
900.012
1,138.951
231.292
288.939
753.324
992.263
281.292
238.939
691.601
880.540
231.292
238.939
231.292
238.939
413.301 216.725
652.240 455.664
231 292
231.292
0.000
281.292
50
APPENDIX.
Example 2.— March 1, 1872. C. D. borrows $1,000.00 to bo
repaid in 5 years by half-yearly instalments of $129.51 each.
The amounts on the Dr. side are calculated as in Example 1.
The amounts on the Cr. side are found by niultiplyinj? each
instalment by the factor corresponding to tiio proper iutervul, and
adding together the products so obtained.
Thus for 1873 the first instahncnt is multiplied by the factor fur
months, and the second instalment by the factor for 3 months.
Example 3.— October 1, 1872. E. F. borrows $1,000 00 to be
repaid in 3 years by quarterly instalments of $97.31 each.
The amounts on the Dr. side are calculated as in previous
Examples.
The amounts on the Cr. side may bo obtained by multiplying
each instalment $97.31 by the factor corresponding to the proper
interval, and adding together the products bo obtained.
Thus for 1873 the first instalment is multiplied by the factor for
1 1 months, the second by the factor for 8 months, the third by the
factor for 5 months, and tiie fourth by the factor for 2 months, thus
giving 410.52.
But tliis result is more easily obtained with the aid of Table
VII by multiplying the instalment by the factor corresponding
to the number of payments in Table VII, and then multiplying
this result by the factor in Table V corresponding to the interval
between the last payment and the end of the year.
Thus in the above example, the instalment 97.31 is multiplied
by the factor 4.150621 in Table VII corresponding to four pay-
ments, and then by the factor 1 .016397 in Table V, corresponding
to the remaining period '^ Tionths, giving 410.52.
And in the last co^- 1875 the amount of the instalments
at date of last pa" j given at once from Table VII by multi-
plying the instah. J7.31 by the factor corresponding to 4 pay-
ments, 4 . 150624, giving 403 . 90.
Example 4.— September 1, 1872. G. II. borrows $1,000.00 to
be repaid in 8 years by monthly instalments of $32. 18 each.
The amounts on the Dr. side are obtained as before.
The amounts on the Cr. side are obtained by means of Table
VI, by multiplying the instalment 82.18 by the factor corres-
dondiog to as many months as there have beon instalments paid.
4^
APPENDIX.
01
lor
bo
Thu8 m the first column 32.18 is niultiplica by tlio factor
3 0245G2 corresponding to 3 months, giving 97.33. In the second
column 32.18 is multiplied by the factor 12.553819 correspondin-
to 12 months, giving 403.98 ; and in the fourth column 3'^ 18 U
multiplied by ti.e factor 9.299G04 corresponding to 9 months
giving 299.20. '
As in the previous examples, the small difforcnce between tl»o
final amounts on the two sides 299.25 and 298.91 arises from the
fact that the instalment ^^as only given to the nearest cent above
the true value; if the more correct value 32. 17235 had been used
the balance would liave come out exact, the final sums on each'
side of the account being 299. ir2. In the great majority of cases
a small balance will bo found to the advantage of the society
which may be carried to Profit and Loss. '
f»
«p
.0> \ ' r vJ:..R,f:y I :■,'!' • " w»^;'^( i" <i » WJ.Wj""-^
1 , ^ . , y,, i im.|m ii »« ^|i ^i i Mi,» i ii ii ( i Mni>, ii n il '
1 1 ,11111 ■ .J 1 4) i i» ii i« i.. I*
"T
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v
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. 187V.
IT.
A. B.
rroscut vnluo of I 1SV2. I 187'..
inortjjngo f A\ig. 1 . . . . C;l ,000 . 00 I Dec. 1 .... $1 ,033 . 06
Amount to date . . Dec. 1 $1,033.00
Cr.
187t.
Dec. 1 $900.00
1«74.
Dec. 1 M20.25
1874.
Dec. 1 $768,30
A. B
1875.
Dec, 1 $;830,51
1876.
Dec, 1 ,
187C.
Dec. 1 .
. $691.68
, $C52.19
1870.
Dec. 1 . ,..$413.24
1877.
Dec. 1 $465. CO
1877.
Dec. I $210.05
1878.
Aug. 1 .
,$231.21
1872.
Instnlment
$0.00
Amount to tlate . , Dec, 1 $0,00
Ualnnce 1,033.00
S!,030 00
1873.
Aug. 1 $231.30
Dec. 1 $238.95
illlO.OO
§1,138.95
1871.
Aug, 1 . .
Dec, 1
$231.80
$238,95
753.30
$992.25
1875.
Aug. 1 $231.30
Doc. 1 . . ,
. $238.95
. 691.50
$830.61
1870.
Aug. 1 $231.30
Dec. 1 $238.95
413.24
$062.19
1877. I 1878.
Aug. 1 ^231.80 1 Aug. 1 $231.30
Dec. 1 $238.95 Aug. 1 $231.30
216.05 0.00
$155. CO i
$231.30
Dr.
C. D.
1872.
Preaent value uf !
mortgage ) Mar. 1 $1,000.00
Amount to date . . Dec. 1 $l,u75.93
1872.
Doc, 1 $943.22
1S73.
Dec. 1 $1,039.90
1873.
3C. 1 . .
..$707.85
1874.
Deo. 1 . .
..?:574.60
187.-.
Dec. 1 .
..$3C1..34
1871.
;c. 1 . .
..$840.55
1876.
Dec. 1 . .
..$033.39
18711.
Dec. 1 . .
..$398.38
1870.
Dec. 1 .... $128.33
1877.
Mnr. 1 $129.45
Cr.
CD.
1872. 1873.
Instalment Mar. 1 $0.00 Mar. 1 . ,
Sept. 1.... 129.15 Sept. 1.,
Amount to date . . Dec. 1 . . . . $132.71 Dec. 1
Balance 943.22
$1,076.93
$129.51
129.51
$272,05
707.85
$1,039.90
1874.
Mar. 1.... $129.51
Sept. 1 129.51
Dec. 1 $272.05
574.60
$840.55
1875.
Mar. 1...
Sept. 1 . . .
$129.61
129.51
Dee. 1 $292.05
361.34
$03S.39
1870.
Mar. 1...
Sept. 1 , . ,
Deo. 1.
,$129.51
, 129.61
, $272.05
126.33
$398.38
1877.
Mar. 1.
.. $129.61
Mai-. 1 $129.61
0.00
$129.51
Dt.
E. F
Present value of \
00
' ) 1872.
J Oct. l....*l,000
\mountto date . . Doc. 1 .. *l,Ul6.4ii
1872.
Dec. l...,.«:i,010.40
187."..
Dec. 1 ...SI, 120, 68
1873,
Dec. 1 .$710.00
1871.
Dee. 1 $782.84
1874.
Dec. 1.... $372.32
1875.
Oct. 1 $403,80
Cr,
1872,
[natnlmcnt $0 , 00
Amount to date $0,00
Balance 1,016.40
$1,010.40
E. F.
187.>.
Jan. 1 . . . .
Apl. 1....
July 1....
Oct. 1 . . .
I 1871,
$07.31 .Ian. 1.
97.31 Apl. 1.
97.31 July 1
97.81 Oct. 1.
897.31
97.31
97.31
97.31
Dec. 1... $410.52 Dec. 1
710.00
$1,120.68
$110.52
, 372.32
$782.84
1875.
Jan. I .
Apl. 1.
July 1.
Oct. 1.
Oct. 1
$97.81
97.81
97.31
97.81
. $1C3.90
O.OO
$103 . 90
Pc.
G. II.
I'resont value of j^ 1872.
mortgage . . . ) Sept. 1 .SI ,000 . 00
Amount to dale . . Dec. 1 . . . .i?l,024.C.9
1872.
Dec. 1.... $927.36
1873.
Dec. 1.... $1,022. 42
1873.
Dec. 1 $018.43
1874.
Dec 1 $881.82
1874.
Dee. 1 $277.84
Oct. 1 ... $298.94
Cr.
G. II.
Instalment
1872.
.Ian. 1 ..
Feb. 1 .,
Mar. 1 . .
Apr. 1 ..
May 1 . .
Juno 1 . .
July 1 . .
Aug. 1 . .
Sep. 1 ..
Oct. 1 ..
Nov. 1 . .
Dec. 1 ..
$32.18
32.18
82.18
Amount to date . . Dee. 1 .. $97.33
Balance 927.36
$1,024 09
1873.
Jan. 1
Feb. 1
Mar. 1 ,
Apr. 1 ,
May 1
Juno 1 ,
July 1
Aug. 1 ,
Sep. 1 ,
Oct. 1 .
Nov. 1 ,
Dec. 1
,. $32.18
32.18
.. 32.18
32.18
, . 32.18
32,18
32,18
32.18
32,18
32,18
, . 32.18
82.18
$403.98
018 43
$1,022.41
1874.
.Tan. 1
Feb. 1
Mar. 1 ,
Apr. 1
May 1 ,
Juno 1
July 1
Aug. 1
Sep. 1 ,
Oct. 1 ,
Nov. I
Dec. 1
$32.18
32.18
32.18
32.18
32.18
82.18
32.18
32.18
32.18
82.18
32.18
82.18
$403.98
277.84
$681.82
1875.
Jan. 1
Feb. 1
filar. 1
Apr 1
May 1
Juno 1
July 1 ,
Aug. 1
Sep. 1
Oct. 1
$32.18
32.18
32,18
32.18
32.18
32.18
32.18
32.18
32.18
32.18
$299.20
0.00
$299.20
' "'g ! '.ff>»-''*w viy^ ' f.ji^j* * '
I
I
i.
V . i' . f
w. • .« - i.
? ,
i..)
T N D E X .
To find the present value of a sum due after a nnmber
of years. p. 10; s. (7); Table VIII
To find the present value of a sum duo after a number
of months less than a year. p. ^0; s. (7); Table III.
To find the present value of a sum due after a broken
period of years and some mr ilhs.
p. 10; 8. (7); Tables III., VIII.
To find the amount, with interest, of a sura, after any
number of months, from I to 12. p. 8; s. (5); Table V.
To find the amount, with interest, of a sum, after a period
of years and some months.
p. 12; s. (9); Tables III , VITI., or V., VIII.
[Divide the sum by the factor corresponding to the number of years
in Table VJIl. ; then divide this quotient by the fa,rtor corresponding
to the number of months in Table III., or multiply this quotient by the
Jactor corresponding to the number of months in Table V.)
To find the yearly, half-yearly, quarterly, or monthly
instalment required to repay a loan in a given
number of years. p. 5 ; 8. (1); Table I.
To find the present value of an instalment payable at the
end of each year, half-year, quarter or month, during
a given number of years. p. 5; s. (2); Table II.
To find the present value of a yearly, half-yearly, or
quarterly instalment, in payment of a loan having a
broken period of years and some months to run.
First method (best) p. 6; s. (3); Tables II., III.
Second method, p. 8; s. (6); Tables II., V
INDEX.
To find the present value of a monthly instalment, In
payment of a loan having a period of years and
some months to run.
First method, p. 7 ; s. (4); Tables II., III., IV.
Second method, p 9; s. («); Tables II., V., VI.
To find the present value of an ordinary mortgage.
p. 10: 8. (8); Tables II., III., VIII.
To find the reduction in the instalment or in the period
to run, produced by the payment of an additional^
rn sum at any time during the period. ^^ pp. 12, IS ; s. (9).
To find the rate of interest when the instalment, loan,
and period are given. -' " '
j;. , .., p^ 13; y. (10); and Appendix. Note IV., p 46.
How the account of a borrower should be kept.
Appendix, Note V., p. 48.
V-vVj^,
Xvlr
I
Vi
-■u. pt', i> ^ ^''-'i ':i
= >■ -
i-iV
OOPP, CLARK & CO. PRINTERS, COLBORNE STREET, TORONTO.
Qt, in
} and
es II., III.
los II., V.
,iv.
, vr.
II., III.,
VIII.
seriod
tional
. 12, IS; i
5. (9).
, loan,
!^ote IV., p 46.
Note v., p. 48.
;i:! . ■■■ V
i-nr
TORONTO.