The question is from Marcel P Finan's "A Basic Course in the Theory of Interest and Derivatives Markets: Preparation for the Actuarial Exam FM/2".
Suppose that $100$ is deposited into a savings account, earning at a discount rate of $0.15\%$ biweekly, at the beginning of year $2006$. (a) Find the nominal annual discount rate.
My main attempt at (a):
$(1 - 0.0015)^{-1} = (1 - d^{26}/26)^{-26}$
$1.001502253 = (1 - d^{26}/26)^{-26}$
$1.001502253^{-(1/26)} = (1 - d^{26}/26)^{-26}$
$0.999942266 = 1 - d^{26}/26$
$d^{26} = 0.001501083$
However, this is not the correct answer in the book. :(
The correct answer in the book: $3.9$%
Any suggestions? Thank you!
The yearly nominal discount rate d is
$$d=m\cdot ((1+e)^{\frac{1}{m}}-1),$$
where $m=26$ is the number of compounding periods in a year and $e$ the effective yearly discount rate. The formula is derived from the well known formula $e=(1+\frac{d}m)^m-1$.
Now the yearly effective discount rate is $e=(1+b)^{m}-1$, where $b$ is the biweekly nominal discount rate. Next we insert the term for $e$ into the formula for $d$.
$$d=m\cdot ((1+(1+b)^{m}-1)^{\frac{1}{m}}-1)=m\cdot ((1+b)^{m})^{\frac{1}{m}}-1)=m\cdot (1+b-1)=m\cdot b$$
Comprehensible? If yes, can you proceed?