Question

Celeste just borrowed 47,500 dollars. She plans to repay this loan by making equal quarterly payments of 3,126.28 dollars for 19 quarters. If she makes her first quarterly payment later today, then what is the quarterly interest rate on the loan?

Answer 1

let her quarterly rate be i
then 3126.28( 1 - (1+i)^-19)/i = 47500
1 - (1+i)^-19 = 15.193777i
1 - 1/(1+i)^19 = 15.193777i
1 - 15.193777i = 1/(1+i)^19

very nasty to solve, we used to do these using interpolation
try some values:
i = .01 --> PV = 53,853.33 too low
i = .02 --> PV = 49,015.26 still too low
i = .03 --> PV = 44,780.21 too high
i = .025 -> PV = 46828.21
i = .023 -> PV = 47685.73

nibbled away at this and got
i = .02343

check:
3126.28(1 - 1.02343)^-19)/.02343
= 47,498.70 , not bad

the quarterly rate is 2.343%