(1) General rule. The annual percentage rate shall be the nominal annual percentage
rate determined by multiplying the unit-period rate by the number
of unit periods in a year.
(2) Term of the
transaction. The term of the transaction begins on the date of
its consummation, except that if the finance charge or any portion
of it is earned beginning on a later date, the term begins on the
later date. The term ends on the date the last payment is due, except
that if an advance is scheduled after that date, the term ends on
the later date. For computation purposes, the length of the term shall
be equal to the time interval between any point in time on the beginning
date to the same point in time on the ending date.
(3) Definitions
of time intervals.
(i) A period is the interval of time
between advances or between payments and includes the interval of
time between the date the finance charge begins to be earned and the
date of the first advance thereafter or the date of the first payment
thereafter, as applicable.
(ii) A common period is any period that
occurs more than once in a transaction.
(iii) A standard interval of time is
a day, week, semimonth, month, or a multiple of a week or a month
up to, but not exceeding, one year.
(iv) All months shall be considered
equal. Full months shall be measured from any point in time on a given
date of a given month to the same point in time on the same date of
another month. If a series of payments (or advances) is scheduled
for the last day of each month, months shall be measured from the
last day of the given month to the last day of another month. If payments
(or advances) are scheduled for the 29th or 30th of each month, the
last day of February shall be used when applicable.
(4) Unit period.
(i) In all transactions
other than a single-advance, single-payment transaction, the unit
period shall be that common period, not to exceed one year, that occurs
most frequently in the transaction, except that—
(A) If two or
more common periods occur with equal frequency, the smaller of such
common periods shall be the unit period; or
(B) If there is no common period in the transaction,
the unit period shall be that period which is the average of all periods
rounded to the nearest whole standard interval of time. If the average
is equally near two standard intervals of time, the lower shall be
the unit period.
(ii) In a single-advance, single-payment
transaction, the unit period shall be the term of the transaction,
but shall not exceed one year.
(5) Number of
unit periods between two given dates.
(i) The number of days
between two dates
shall be the number of 24-hour intervals between any point in time
on the first date to the same point in time on the second date.
(ii) If the unit period
is a month, the number of full unit periods between two dates shall
be the number of months measured back from the later date. The remaining
fraction of a unit period shall be the number of days measured forward
from the earlier date to the beginning of the first full unit period,
divided by 30. If the unit period is a month, there are 12 unit periods
per year.
(iii) If
the unit period is a semimonth or a multiple of a month not exceeding
11 months, the number of days between two dates shall be 30 times
the number of full months measured back from the later date, plus
the number of remaining days. The number of full unit periods and
the remaining fraction of a unit period shall be determined by dividing
such number of days by 15 in the case of a semimonthly unit period
or by the appropriate multiple of 30 in the case of a multimonthly
unit period. If the unit period is a semimonth, the number of unit
periods per year shall be 24. If the number of unit periods is a multiple
of a month, the number of unit periods per year shall be 12 divided
by the number of months per unit period.
(iv) If the unit period is a day, a
week, or a multiple of a week, the number of full unit periods and
the remaining fractions of a unit period shall be determined by dividing
the number of days between the two given dates by the number of days
per unit period. If the unit period is a day, the number of unit periods
per year shall be 365. If the unit period is a week or a multiple
of a week, the number of unit periods per year shall be 52 divided
by the number of weeks per unit period.
(v) If the unit period is a year, the
number of full unit periods between two dates shall be the number
of full years (each equal to 12 months) measured back from the later
date. The remaining fraction of a unit period shall be—
(A) The remaining
number of months divided by 12 if the remaining interval is equal
to a whole number of months, or
(B) The remaining number of days divided by
365 if the remaining interval is not equal to a whole number
of months.
(vi) In a single-advance, single-payment
transaction in which the term is less than a year and is equal to
a whole number of months, the number of unit periods in the term shall
be one, and the number of unit periods per year shall be 12 divided
by the number of months in the term or 365 divided by the number of
days in the term.
(vii) In a single-advance, single-payment transaction in which the
term is less than a year and is not equal to a whole number
of months, the number of unit periods in the term shall be one, and
the number of unit periods per year shall be 365 divided by the number
of days in the term.
(6) Percentage
rate for a fraction of a unit period. The percentage rate of
finance charge for a fraction (less than one) of a unit period shall
be equal to such fraction multiplied by the percentage rate of finance
charge per unit period.
(7) Symbols. The symbols used to express
the terms of a transaction in the equation set forth in paragraph
(b)(8) of this section are defined as follows:
Symbols
| Ak = |
The amount of the kth advance. |
| qk = |
The number of full unit periods from the
beginning of the term of the transaction to the kth advance. |
| ek = |
The fraction of a unit period in the time
interval from the beginning of the term of the transaction to the
kth advance. |
| m = |
The number of advances. |
| Pj = |
The amount of the jth payment. |
| tj = |
The number of full unit periods from the
beginning of the term of the transaction to the jth payment. |
| fj = |
The fraction of a unit period in the time
interval from the beginning of the term of the transaction to the
jth payment. |
| n = |
The number of payments. |
| i = |
The percentage rate of finance charge per
unit period, expressed as a decimal equivalent. |
Symbols used
in the examples shown in this appendix are defined as follows:
Symbols
Figure 1. DISPLAY EQUATION
$$
\require{enclose}
\mathrm{\ddot{a}_{\enclose{actuarial}{x}}} =
$$
|
The present value of 1 per unit period for x unit periods,
first payment due immediately. |
Figure 2. DISPLAY EQUATION
$$\mathrm{=1 + \frac{1}{(1 + i)}} + \frac{1}{(1 + \mathrm{i})^2}+ \\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad....+ \frac{1}{(1 + \mathrm{i})^{\mathrm{x}-1}}
$$
- w = The number of unit periods per year.
I = wi × 100 = The
nominal annual percentage rate.
(8) General equation. The following equation sets forth the relationship among the terms
of a transaction:
Figure 3. DISPLAY EQUATION
$$
\mathrm{\frac{A_1}{(1+e_1i)(1+i)^{q_1}} + \frac{A_2}{(1+e_2i)(1+i)^{q_2}} +}
\\
\qquad\qquad...+ \mathrm{\frac{A_m}{(1+e_mi)(1+i)^{q_m}}} =
\\
\qquad \mathrm{\frac{P_1}{(1+f_1i)(1+i)^{t_1}} + \frac{P_2}{(1+f_2i)(1+i)^{t_2}} +}
\\
\qquad\qquad\qquad\qquad\qquad...+ \mathrm{\frac{P_n}{(1+f_ni)(1+i)^{t_n}}}
$$
(1) Single-advance transaction, with or without an odd first period,
and otherwise regular. The general equation in paragraph (b)(8)
of this section can be put in the following special form for this
type of transaction:
Figure 6. Display Equation
$$
\require{enclose}
\mathrm{A = \frac{1}{(1+fi)(1+i)^t}} \bigg( \mathrm{P\mathrm{\ddot{a}_{\enclose{actuarial}{n}}}} \bigg)
$$
Example (i): Monthly payments (regular first period)
- Amount advanced (A) = $5000. Payment (P) = $230.
- Number of payments (n) = 24.
- Unit period = 1 month. Unit periods per year (w)
= 12.
- Advance, 1-10-78. First payment, 2-10-78.
- From 1-10-78 through 2-10-78 = 1 unit period. (t
= 1; f = 0)
- Annual percentage rate (I) = wi = .0969
= 9.69%
Example (ii): Monthly payments (long first period)
- Amount advanced (A) = $6000. Payment (P) = $200.
- Number of payments (n) = 36.
- Unit period = 1 month. Unit periods per year (w) =
12.
- Advance, 2-10-78. First payment, 4-1-78.
- From 3-1-78 through 4-1-78 = 1 unit period. (t = 1)
- From 2-10-78 through 3-1-78 = 19 days. (f = 19/30)
- Annual percentage rate (I) = wi = .1182
= 11.82%
Example (iii): Semimonthly payments (short first period)
- Amount advanced (A) = $5000. Payment (P) = $219.17.
- Number of payments (n) = 24.
- Unit period =½ month. Unit periods per year (w) =
24.
- Advance, 2-23-78. First payment, 3-1-78. Payments
made on 1st and 16th of each month.
- From 2-23-78 through 3-1-78 = 6 days. (t = 0; f =
6/15)
- Annual percentage rate (I) = wi = .1034
= 10.34%
Example (iv): Quarterly payments (long first period)
- Amount advanced (A) = $10,000. Payment (P) = $385.
- Number of payments (n) = 40.
- Unit period = 3 months. Unit periods per year (w)
= 4.
- Advance, 5-23-78. First payment, 10-1-78.
- From 7-1-78 through 10-1-78 = 1 unit period. (t =
1)
- From 6-1-78 through 7-1-78 = 1 month = 30 days. From
5-23-78 through 6-1-78 = 9 days. (f = 39/90)
- Annual percentage rate (I) = wi = .0897
= 8.97%
Example (v): Weekly payments (long first period)
- Amount advanced (A) = $500. Payment (P) = $17.60.
- Number of payments (n) = 30.
- Unit peiod = 1 week. Unit periods per year (w) =
52.
- Advance, 3-20-78. First payment, 4-21-78.
- From 3-24-78 through 4-21-78 = 4 unit periods. (t
= 4)
- From 3-20-78 through 3-24-78 = 4 days. (f = 4/7)
- Annual percentage rate (I) = wi = .1496
= 14.96%
(2) Single-advance transaction, with an odd first
payment, with or without an odd first period, and otherwise regular. The general equation in paragraph (b)(8) of this section can be
put in the following special form for this type of transaction:
Figure 7. DISPLAY EQUATION
$$
\require{enclose}
\mathrm{A} = \frac{1}{(1+\mathrm{fi})(1+\mathrm{i})^\mathrm{t}}
\Bigg[ \mathrm{P}_1 + \frac{\mathrm{P}\ddot{\mathrm{a}}_\enclose{actuarial}{\mathrm{n}-1}}{(1+\mathrm{i})} \Bigg]
$$
Example (i): Monthly payments (regular first period
and irregular first payment)
- Amount advanced (A) $5000. First payment (P1) = $250.
- Regular payment (P) = $230. Number of payments (n)
= 24.
- Unit period = 1 month. Unit periods per year (w)
= 12.
- Advance, 1-10-78. First payment, 2-10-78.
- From 1-10-78 through 2-10-78 = 1 unit period. (t
= 1; f = 0)
- Annual percentage rate (I) = wi = .1008
= 10.08%
Example (ii): Payments every four weeks (long first
period and irregular first payment)
- Amount advanced (A) = $400. First payment (P1) = $39.50.
- Regular payment (P) = $38.31. Number of payments
(n) = 12.
- Unit period = 4 weeks. Unit periods per year (w)
= 52/4 = 13.
- Advance, 3-18-78. First payment, 4-20-78.
- From 3-23-78 through 4-20-78 = 1 unit period. (t =
1)
- From 3-18-78 through 3-23-78 = 5 days. (f = 5/28)
- Annual percentage rate (I) = wi = .2850
= 28.50%.
(3) Single-advance transaction, with an odd final
payment, with or without an odd first period, and otherwise regular. The general equation in paragraph (b)(8) of this section can be
put in the following special form for this type of transaction:
Figure 8. DISPLAY EQUATION
$$
\require{enclose}
\begin{align*}
\mathrm{A} = \frac{1}{(1+\mathrm{fi})(1+\mathrm{i})^\mathrm{t}}
\Bigg[ \mathrm{P}\ddot{\mathrm{a}}_\enclose{actuarial}{\mathrm{n}-1} +&\\
&\frac{\mathrm{P}_\mathrm{n}}{(1+\mathrm{i})^{\mathrm{n}-1}} \Bigg]
\end{align*}
$$
Example (i): Monthly payments (regular first period
and irregular final payment)
- Amount advanced (A) = $5000. Regular payment (P) =
$230.
- Final payment (Pn) = $280. Number of payments
(n) = 24.
- Unit period = 1 month. Unit periods per year (w)
= 12.
- Advance, 1-10-78. First payment, 2-10-78.
- From 1-10-78 through 2-10-78 = 1 unit period. (t
= 1; f = 0)
- Annual percentage rate (I) = wi = .1050
= 10.50%
Example (ii): Payments every two weeks (short first
period and irregular final payment)
- Amount advanced (A) = $200. Regular payment (P) =
$9.50.
- Final payment (Pn) = $30. Number of payments
(n) = 20.
- Unit period = 2 weeks. Unit periods per year (w)
= 52/2 = 26.
- Advance, 4-3-78. First payment, 4-11-78.
- From 4-3-78 through 4-11-78 = 8 days. (t = 0; f =
8/14)
- Annual percentage rate (I) = wi = .1222
= 12.22%
(4) Single-advance transaction, with an odd first
payment, odd final payment, with or without an odd first period, and
otherwise regular. The general equation in paragraph (b)(8) of
this section can be put in the following special form for this type
of transaction:
Figure 9. DISPLAY EQUATION
$$
\require{enclose}
\begin{align*}
\mathrm{A = \frac{1}{(1+fi)(1+i)^t}}
\Bigg[ \mathrm{P_1} + &\mathrm{\frac{P\ddot{a}_\enclose{actuarial}{n-2}}{(1+i)}} +\\
&\mathrm{\frac{P_n}{(1+i)^{n-1}}} \Bigg]
\end{align*}
$$
Example (i): Monthly payments (regular first period,
irregular first payment, and irregular final payment)
- Amount advanced (A) = $5000. First payment (P1) = $250.
- Regular payment (P) = $230. Final payment (Pn) = $280.
- Number of payments (n) = 24. Unit period = 1 month.
- Unit periods per year (w) = 12.
- Advance, 1-10-78. First payment, 2-10-78.
- From 1-10-78 through 2-10-78 = 1 unit period. (t
= 1; f = 0)
- Annual percentage rate (I) = wi = .1090
= 10.90%
Example (ii): Payments every two months (short first
period, irregular first payment, and irregular final payment)
- Amount advanced (A) = $8000. First payment (P1) = $449.36.
- Regular payment (P) = $465. Final payment (Pn) = $200.
- Number of payments (n) = 20. Unit period = 2 months.
- Unit periods per year (w) = 12/2 = 6.
- Advance, 1-10-78. First payment, 3-1-78.
- From 2-1-78 through 3-1-78 = 1 month. From 1-10-78
through 2-1-78 = 22 days. (t = 0; f = 52/60)
- Annual percentage rate (I) = wi = .0730
= 7.30%
(5) Single-advance, single-payment transaction. The general equation in paragraph (b)(8) of this section can be
put in the special forms below for single advance, single payment
transactions. Forms 1 through 3 are for the direct determination of
the annual percentage rate under special conditions. Form 4 requires
the use of the iteration procedure of paragraph (b)(9) of this section
and can be used for all single-advance, single-payment transactions
regardless of term.
Form 1—Term less than one year:
Figure 10. DISPLAY EQUATION
$$
\mathrm{I} = 100\mathrm{w} \Bigg( \mathrm{\frac{P}{A}} -1 \Bigg)
$$
Form 2—Term more
than one year but less than two years:
Figure 11. DISPLAY EQUATION
$$
\begin{align*}
\mathrm{I = \frac{50}{f}}
\Bigg\{\Bigg[ & \mathrm{(1+f)^2} + \\[8pt]
& \mathrm{4f} \Bigg( \mathrm{\frac{P}{A} -1} \Bigg) \Bigg]^{1/2} - (1 + \mathrm{f}) \Bigg\}
\end{align*}
$$
Form 3—Term equal to exactly a year or exact multiple of a year:
Figure 12. DISPLAY EQUATION
$$
\mathrm{I = 100}
\Bigg[ \Bigg(\mathrm{\frac{P}{A} \Bigg)^{1/\mathrm{t}} -1} \Bigg]
$$
Form 4—Special
form for iteration procedure (no restriction on term):
Figure 13. DISPLAY EQUATION
$$
\mathrm{A = \frac{P}{(1+fi)(1+i)^t}}
$$
Example (i): Single-advance, single-payment (term
of less than one year, measured in days)
- Amount advanced (A) = $1000. Payment (P) = $1080.
- Unit period = 255 days. Unit periods per year (w)
= 365/255.
- Advance, 1-3-78. Payment, 9-15-78.
- From 1-3-78 through 9-15-78 = 255 days. (t = 1; f
= 0)
- Annual percentage rate (I) = wi = .1145
= 11.45%. (Use form 1 or 4.)
Example (ii): Single-advance, single-payment (term
of less than one year, measured in exact calendar months)
- Amount advanced (A) = $1000. Payment (P) = $1044.
- Unit period = 6 months. Unit periods per year (w)
= 2.
- Advance, 7-15-78. Payment, 1-15-79.
- From 7-15-78 through 1-15-79 = 6 mos. (t = 1; f =
0)
- Annual percentage rate (I) = wi = .0880
= 8.80%. (Use form 1 or 4.)
Example (iii): Single-advance, single-payment (term
of more than one year but less than two years, fraction measured in
exact months)
- Amount advanced (A) = $1000. Payment (P) = $1135.19.
- Unit period = 1 year. Unit periods per year (w) =
1.
- Advance, 7-17-78. Payment, 1-17-80.
- From 1-17-79 through 1-17-80 = 1 unit period. (t =
1)
- From 7-17-78 through 1-17-79 = 6 mos. (f = 6/12)
- Annual percentage rate (I) = wi = .0876
= 8.76%. (Use form 2 or 4.)
Example (iv): Single-advance, single-payment (term
of exactly two years)
- Amount advanced (A) = $1000. Payment (P) = $1240.
- Unit period = 1 year. Unit periods per year (w) =
1.
- Advance, 1-3-78. Payment, 1-3-80.
- From 1-3-78 through 1-3-79 = 1 unit period. (t =
2; f = 0)
- Annual percentage rate (I) = wi = .1136
= 11.36%. (Use form 3 or 4.)
(6) Complex single-advance transaction.
Example (i): Skipped-payment loan (payments every
four weeks)
- A loan of $2135 is advanced on 1-25-78 . It is to
be repaid by 24 payments of $100 each. Payments are due every four
weeks beginning 2-20-78. However, in those months in which two payments
would be due, only the first of the two payments is made and the following
payment is delayed by two weeks to place it in the next month.
- Unit period = 4 weeks. Unit periods per year (w) =
52/4 = 13.
- First series of payments begins 26 days after 1-25-78.
(t1 = 0; f1 = 26/28)
- Second series of payments begins nine unit periods
plus two weeks after start of first series. (t2 = 10;
f2 = 12/28)
- Third series of payments begins six unit periods
plus two weeks after start of second series. (t3 = 16;
f3 = 26/28)
- Last series of payments begins six unit periods plus
two weeks after start of third series. (t4 = 23; f4 = 12/28)
- The general equation in paragraph (b)(8) of this section
can be written in the special form:
Figure 14. DISPLAY EQUATION
$$
\require{enclose}
\begin{align*}
2135 = & \mathrm{\frac{100\ddot{a}_\enclose{actuarial}{9}}{(1+(26/28)i)} +} \\[8pt]
& \mathrm{\frac{100\ddot{a}_\enclose{actuarial}{6}}{(1+(12/28)i)(1+i)^{10}} + } \\[8pt]
& \mathrm{\frac{100\ddot{a}_\enclose{actuarial}{6}}{(1+(26/28)i)(1+i)^{16}} +} \\[8pt]
& \qquad\qquad \mathrm{\frac{100\ddot{a}_\enclose{actuarial}{3}}{(1+(12/28)i)(1+i)^{23}} } \\
\end{align*}
$$
- Annual percentage rate (I) = wi = .1200
= 12.00%
Example (ii): Skipped-payment loan plus single payments
- A loan of $7350 on 3-3-78 is to be repaid by three
monthly payments of $1000 each beginning 9-15-78, plus a single payment
of $2000 on 3-15-79, plus three more monthly payments of $750 each
beginning 9-15-79, plus a final payment of $1000 on 2-1-80.
- Unit period = 1 month. Unit periods per year (w)
= 12.
- First series of payments begins six unit periods plus
12 days after 3-3-78. (t1 = 6; f1 = 12/30)
- Second series of payments (single payment) occurs
12 unit periods plus 12 days after 3-3-78. (t2 = 12; f2 = 12/30)
- Third series of payments begins 18 unit periods plus
12 days after 3-3-78. (t3 = 18; f3 = 12/30)
- Final payment occurs 22 unit periods plus 29 days
after 3-3-78. (t4 = 22; f4 = 29/30)
- The general equation in paragraph (b)(8) of this section
can be written in the special form:
Figure 15. DISPLAY EQUATION
$$
\require{enclose}
\begin{align*}
7350 = & \mathrm{\frac{1000\ddot{a}_\enclose{actuarial}{3}}{(1+(12/30)i)(1+i)^6} +} \\[8pt]
& \mathrm{\frac{2000}{(1+(12/30)i)(1+i)^{12}} + } \\[8pt]
& \mathrm{\frac{750\ddot{a}_\enclose{actuarial}{3}}{(1+(12/30)i)(1+i)^{18}} +} \\[8pt]
& \qquad\qquad \mathrm{\frac{1000}{(1+(29/30)i)(1+i)^{22}} } \\
\end{align*}
$$
- Annual percentage rate
- (I) = wi = .1022 = 10.22%
Example (iii): Mortgage with varying payments
A loan of $39,688.56 (net)
on 4-10-78 is to be repaid by 360 monthly payments beginning 6-1-78.
Payments are the same for 12 months at a time as follows:
Example: Mortgage
with varying payments
| |
| Year |
Monthly payment |
Year |
Monthly payment |
| 1 |
$291.81 |
16 |
$383.67 |
| 2 |
300.18 |
17 |
383.13 |
| 3 |
308.78 |
18 |
382.54 |
| 4 |
317.61 |
19 |
381.90 |
| 5 |
326.65 |
20 |
381.20 |
| 6 |
335.92 |
21 |
380.43 |
| 7 |
345.42 |
22 |
379.60 |
| 8 |
355.15 |
23 |
378.68 |
| 9 |
365.12 |
24 |
377.69 |
| 10 |
375.33 |
25 |
376.60 |
| 11 |
385.76 |
26 |
375.42 |
| 12 |
385.42 |
27 |
374.13 |
| 13 |
385.03 |
28 |
372.72 |
| 14 |
384.62 |
29 |
371.18 |
| 15 |
384.17 |
30 |
369.50 |
| |
- Unit period = 1 month. Unit periods per year (w)
= 12.
- From 5-1-78 through 6-1-78 = 1 unit period. (t = 1)
- From 4-10-78 through 5-1-78 = 21 days. (f = 21/30)
- The general equation in paragraph (b)(8) of this section
can be written in the special form:
Figure 16. DISPLAY EQUATION
$$
\require{enclose}
\begin{align*}
39&,688.56 = \mathrm{\frac{\ddot{a}_\enclose{actuarial}{12}}{(1+(21/30)i)(1+i)} } \\[8pt]
& \Bigg[291.81 + \frac{300.18}{(1+\mathrm{i})^{12}} + \frac{308.78}{(1+\mathrm{i})^{24}} + \\[8pt]
& \qquad\qquad\qquad\qquad... + \frac{369.50}{(1+\mathrm{i})^{348}} \Bigg]
\end{align*}
$$
- Annual percentage rate
- (I) = wi = .0980 = 9.80%
(7) Multiple-advance transactions.
Example (i):: Construction loan
- Three advances of $20,000 each are made on 4-10-79,
6-12-79, and 9-18-79. Repayment is by 240 monthly payments of $612.36
each beginning 12-10-79.
- Unit period = 1 month. Unit periods per year (w) =
12.
- From 4-10-79 through 6-12-79 = (2+2/30) unit periods.
- From 4-10-79 through 9-18-79 = (5+8/30) unit periods.
- From 4-10-79 through 12-10-79 = (8) unit periods.
The general equation in paragraph
(b)(8) of this section is changed to the single-advance mode by treating
the second and third advances as negative payments:
Figure 17. DISPLAY EQUATION
$$
\require{enclose}
\begin{align*}
20,0000 &= \mathrm{\frac{612.36\ddot{a}_{\enclose{actuarial}{240}}}{(1+i)^8} } -\\[8pt]
& \frac{20,000}{(1+(2/30)\mathrm{i})(1+\mathrm{i})^2} - \\[8pt]
& \qquad\qquad \frac{20,000}{(1+(8/30)\mathrm{i}(1+\mathrm{i})^5}
\end{align*}
$$
- Annual percentage rate
- (I) = wi = .1025 = 10.25%
Example (ii):: Student loan
- A student loan consists of eight advances: $1800
on 9-5-78, 9-5-79, 9-5-80, and 9-5-81; plus $1000 on 1-5-79, 1-5-80,
1-5-81, and 1-5-82. The borrower is to make 50 monthly payments of
$240 each beginning 7-1-78 (prior to first advance).
- Unit period = 1 month. Unit periods per year (w)
= 12.
- Zero point is date of first payment since it precedes
first advance.
- From 7-1-78 to 9-5-78 = (2+4/30) unit periods.
- From 7-1-78 to 9-5-79 = (14+4/30) unit periods.
- From 7-1-78 to 9-5-80 = (26+4/30) unit periods.
- From 7-1-78 to 9-5-81 = (38+4/30) unit periods.
- From 7-1-78 to 1-5-79 = (6+4/30) unit periods.
- From 7-1-78 to 1-5-80 = (18+4/30) unit periods.
- From 7-1-78 to 1-5-81 = (30+4/30) unit periods.
- From 7-1-78 to 1-5-82 = (42+4/30) unit periods.
Since the zero point
is date of first payment, the general equation in paragraph (b)(8)
of this section is written in the single-advance form below by treating
the first payment as a negative advance and the eight advances as
negative payments:
Figure 18. DISPLAY EQUATION
$$
\require{enclose}
\begin{align*}
-&240 = \mathrm{\frac{240\ddot{a}_\enclose{actuarial}{49}}{(1+i)}}
- \mathrm{\frac{1800}{(1+ (4/30)i}} \\[8pt]
& \Bigg[ \mathrm{\frac{1}{(1+i)^2}} + \mathrm{\frac{1}{(1+i)^{14}}} + \mathrm{\frac{1}{(1+i)^{26}}} \\[8pt]
& \qquad + \mathrm{\frac{1}{(1+i)^{38}}} \Bigg] - \mathrm{\frac{1000}{(1+(4/30)i)}} \\[8pt]
& \qquad\qquad \Bigg[ \mathrm{\frac{1}{(1+i)^6}} + \mathrm{\frac{1}{(1+i)^{18}}} + \\[8pt]
& \qquad\qquad\qquad \mathrm{\frac{1}{(1+i)^{30}}} + \mathrm{\frac{1}{(1+i)^{42}}} \Bigg]
\end{align*}
$$
Annual percentage
rate
(I) = wi = .3204
= 32.04%
