I am having trouble understanding a lot of things from the following problem. It would be very helpful if I could get some explanations.
Smith purchases a Canadian bond for 1000 with an issue date of 11/1, paying an interest rate of 11.25% per year. The bond can be cashed in anytime after 1/1 the following year,and it will pay simple interest during the first year of $1 \over 12$ of the annual interest for every completed month since 11/1. The government allows purchasers to pay for their bonds as late as 11/9 with full interest paid for November. Smith pays 1000 on 11/9. What is his equivalent effective annual rate of interest for his transaction?
i have so many unclear things, but I want to understand all of them.
1), buying a bond
When solving a problem like this, I noticed that sometimes the 1000 is the future value rather than the present value. I have a feeling that this sounds like a present value because Smith pays 1000 during the purchase, but when are these transactions future values?
2), "it will pay simple interest during the first year of $1 \over 12$ of the annual interest for every completed month since ..."
Is there a grammatical error here or am I not understanding this sentence correctly? If it is simple interest, I want to say that Smith pays 1000 and keeps the money growing for 60days, so why should not the answer be
$$1000(1+\frac{60}{365}(.1125))$$
?
3), Paying a couple of days later? How on earth can someone invest 1000 on 11/1 and pay 1000 on 11/9? Does this mean that for these 8 days the Canadian bank (or whatever it is that gives Smith the money) pays interest for Smith for free? That sounds super absurd and yet, since it is mentioned in the problem it must have some significance.
4), what is the effective annual rate? This is what bothers me most. The problem clearly defines that the bond is worth 11.25% more than when it was purchased (assuming that 1000 is the present value) so why would the effective annual interest be different from 11.25%? Does this have to do with the fact that there were 8 days where Smith did not have to pay for the bond?
1 - Yup this is present value.
2 - There is no error here, it pays simple interest per calendar month, or $$1000 \times\left(1 + \frac{n}{12}\left(\frac{11.25}{100}\right) \right)$$ where $n$ is the number of complete months.
3 - A trade and settlement of the trade often take place several days apart, with settlement typically expected at $t+2$ or $t+3$, when the interest is paid from is a convention, but think about it from this perspective: Smith has contractually obliged himself to deliver $\$1,000\text{CAD}$ by a certain point in the month, unless he is able to invest that money elsewhere for that period, and then cash in the investment to settle his bond purchase, he has lost the opportunity that came with the $\$1,000\text{CAD}$ to earn an interest of some sort. So should he be compensated from the time of the commitment, or the time of settlement on the trade?
4 - The effective annual rate is not a simple interest calculation, it is a compound interest calculation, and yes you need to take into account the date of payment, and how long the bond is held for before being cashed in. Think in these terms, if you settle on the last day possible and cash out on the first day possible, how much interest have you received for the days the money is invested? What about when the settlement takes place on the first day possible? Or, if Smith cashes out on the on the last day of a long month, so he holds the bond for $31$ days without receiving any interest?