Consider the following exercise:
Exercise. A financial institution issues bonds with maturities of $13$ weeks, $26$ weeks and $52$ weeks, at zero coupon, and with a discount value $B_1(0) = 98€$, $B_2(0) = 96.5€$ and $B_3(0) = 95€$ such that all these bonds have a facial value of $100€$. Determine the continuous interest rate for the three bonds and what conclusions can you take about the value of money depending on the maturity of the bond?
My attempt. I was able to determine the continuous interest rate for each of the three bonds by using a continuous interest rate modulation of a zero coupon bond (instead of the usual simple interest modulation of a zero coupon bond) and obtained the following results:
$$ r_1^{\text{weekly}} = 0.155\%, \quad r_2^{\text{weekly}} = 0.137\% \quad \text{ and } \quad r_3^{\text{weekly}} = 0.0986\%.$$
Please note that $r_i^{\text{weekly}}$ should be read as the weekly interest rate associated to the $i$-th bond.
I also computed the interest rates without considering them weekly (i.e., seeing each bond as a simple payment, disregarding the maturity time of each of them) for which I obtained $r_1 = 2\%$, $r_2 = 3.6\%$ and $r_3 = 5.1\%$. I believe that my first computation is more useful than the latter because in the second part of the exercise I am asked to compare the maturity times of the bonds.
As for the second part itself, I really don't know what I am expected to conclude. Any hints for this part are highly apreciatted.
Thanks for any help in advance.
When the question uses the phrase "continuous interest rate," that to me suggests it is asking for the force of interest $\delta$, not a weekly periodic rate. As such, the force of interest of the first zero-coupon bond is given by $$\delta = \log(1+i)$$ where $i$ is the effective annual rate of interest. Assuming $52$ weeks in a year, this would imply $$i = \left(\frac{100}{98}\right)^{4} - 1 \approx 0.0841658$$ and $\delta = \log(1+i) \approx 0.0808108$.
The forces of interest on the other bonds are computed in the same way.
In this manner, you will be able to compare the relative yields of each bond--i.e., which bond is better to purchase because the force of interest is higher.
Your second calculation doesn't help you do this comparison because the maturities are different: if I offered you a choice between two bonds, both with face value $100$, and the first bond matures in $1$ year with a discount value of $95$, and the second matures in $10$ years with a discount value of $94$, does it make any sense to choose the $10$-year bond? The effective interest rate over the same time period is actually much worse. Another way to think about it is that assuming the $1$-year bond could be repurchased every year, you could earn at least $5$ per year for $10$ years, whereas you only get $6$ after $10$ years with the second bond. That's why your second interest calculation isn't useful: those are rates of return over the maturities of each bond.