A bond is quoted at 0.97 of par. What is 1/0.97 to four decimal places in your head, and how many terms of your expansion do you need?

A bond is quoted at 0.97 of par. What is 1/0.97 to four decimal places in your head, and how many terms of your expansion do you need?

Approach: Write the divisor as one minus a small number and use the geometric expansion of the reciprocal, then size the first omitted term to justify stopping.

1.0309. Write 0.97 as 1 - 0.03 and use the geometric expansion 1/(1 - x) = 1 + x + x^2 + x^3 and so on, giving 1 + 0.03 + 0.0009 + 0.000027 = 1.030927. The true value is 1.0309278, so three terms are enough, because the next term is 0.03 to the fourth power, which is 0.00000081 and cannot move the fourth decimal. The reciprocal of a number just below 1 therefore reads off as roughly 1 + d + d^2 for a discount d, so the first order answer 1.03 is 0.09% low and the second order correction 1.0309 is right to four places. That rule prices the yield pickup on a discount instantly, since a 3% discount inverts to a 3.09% uplift.

Follow-up: What is 1/1.03 to four decimal places, and why does the expansion alternate in sign there?

Key concepts: geometric expansion, reciprocal, second order correction.