Lesson 19 of 25 · 7 min

Reconstructing a year of twelve statements, in outline

Twelve statements are twelve products in a chain; the year is the sum, not a twelfth of a quoted rate.

A year of this facility is twelve closing balances, each the next opening, and twelve interest products that do not share a rate or a day count. Start from the sample February and carry the chain through an invented rate path to January. No draws, no repayments, interest capitalised each month, Act/360 throughout.

The twelve products, each already rounded:

February 28 days at 6.97 percent: $9,959.99, close $1,847,220.00.
March, two legs: $11,343.47, close $1,858,563.47.
April 30 days at 7.22 percent: $11,182.36, close $1,869,745.83.
May 31 days at 7.22 percent: $11,624.63, close $1,881,370.46.
June 30 days at 7.05 percent: $11,053.05, close $1,892,423.51.
July 31 days at 7.05 percent: $11,488.59, close $1,903,912.10.
August 31 days at 6.80 percent: $11,148.46, close $1,915,060.56.
September 30 days at 6.80 percent: $10,852.01, close $1,925,912.57.
October 31 days at 6.55 percent: $10,862.68, close $1,936,775.25.
November 30 days at 6.55 percent: $10,571.56, close $1,947,346.81.
December 31 days at 6.40 percent: $10,732.04, close $1,958,078.85.
January 31 days at 6.40 percent: $10,791.19, close $1,968,870.04.

The twelve charges add to $131,610.03. February's opening plus that sum is $1,837,260.01 + $131,610.03 = $1,968,870.04, which is January's close. The chain holds for the year because it held for each month. Twelve times March's $11,343.47 would have been $136,121.64, which is $4,511.61 too high. A single 7.22 percent taken on the March opening for a 360-day year would have been $1,847,220 × 0.0722 = $133,369.28, a different wrong number.

The year is the sum of twelve four-factor products, each with its own days and its own rate, each written onto a balance the previous product enlarged. Outline reconstruction is naming those twelve and adding them. It is not dividing a quoted rate by twelve.

Key terms

Annual chain: Twelve monthly identities in sequence. The year's interest is their sum; the year's close is the last of their closes.
Quoted-year product: Balance × stated rate, as if a year were 360 days at one rate on an unchanging base. Here $133,369.28, which matches none of the twelve months.

Worked example

The year against two shortcuts

Opening $1,837,260.01 on 1 February. The twelve products listed above.

Sum of twelve charges: $131,610.03
Opening + sum: $1,968,870.04
January close: $1,968,870.04
Twelve times March: $136,121.64
March opening × 7.22%: $133,369.28
Gap, quoted-year versus chain: $1,759.25

The year that happened is $131,610.03. Both shortcuts invent a year that did not, one by repeating March and one by freezing the March rate and the March base.

Check yourself

The twelve charges add to $131,610.03 and February opened at $1,837,260.01. What is January's close, and why is $1,847,220 × 0.0722 = $133,369.28 the wrong year?

Saved in this browser only. No account, nothing sent anywhere.

Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.