Effective Borrowing Cost

EXERCISE 01

Borrow €30,000 to purchase a machine. Alternative A repays €26,000 at years 5 and 10. Alternative B pays €2,400 at the end of each half-year for ten years. There are no fees or taxes. Compare effective annual borrowing rates.

A.Calculate each financing IRR and identify the cheaper loan under this criterion.

A.

Loan_0=\sum_k\frac{Repayment_k}{(1+y)^{t_k}}

30{,}000=26{,}000(1+y_A)^{-5}+26{,}000(1+y_A)^{-10}

x=(1+y_A)^{-5},\qquad x^2+x-\frac{30}{26}=0

x=\frac{-1+\sqrt{1+4(30/26)}}2

y_A=x^{-1/5}-1

7.865400% annually.

a_{n|i}=\frac{1-(1+i)^{-n}}{i}

30{,}000=2{,}400a_{20|i_h}

i_h\approx0.049643189

y_B=(1+i_h)^2-1

10.175082% annually.

Choose A under the borrowing-rate criterion: the lower IRR is the lower cost to the borrower. The repayment timing and liquidity requirements also differ; this is not an investment-return ranking.