You invested 968 710 in a treasury bill with the face value of 1 000 000 with 91 days left till maturity. After 60 days you have the option to sell it for 989 250. Which option is more profitable?
My solution:
$$r_1 =\frac{1000000-968710}{\frac{91}{360}\cdot968710}=12,78%$$
$$r_2=\frac{989250-968710}{\frac{60}{360}\cdot968710}=12,72%$$
I used the formula for rate of profit where you take profit and divide it by money invested and time it took to make the profit. But in a textbook I found a different solution which gives a different answer.
$$d_1=\frac{1000000-968710}{\frac{1000000\cdot91}{360}}=12,38%$$
$$d_2=\frac{989250-968710}{\frac{989250\cdot60}{360}}=12,46%$$
Where I believe $d_1$ and $d_2$ are discount rates. Which one of these solutions is right?
It is not really a mathematical question -- the answer depends on something the information you've given doesn't say, namely how well you think you can invest the \$989,250 between day 60 and day 91 if you take the option.
If your only possibility is to hold the \$989,250 in cash during the last month, your choice is between having \$1,000,000 on day 91, and having \$989,250 on day 91, and this choice should be easy to make.
But if you have some way to make the \$989,250 draw an interest in the interim such that they will have grown to more than \$1,000,000 by day 91, you should sell the bill on day 60.
In either case, it is completely irrelevant when you bought the treasury bill and what you paid for it. That is a sunk cost and doesn't influence the outcome of your decisions.
Moral: It does not make sense to compare interest rates on investments over different time periods -- at least not unless you have a reason to assume that new, equivalent, investment opportunities will keep being available indefinitely, and even so you'd need to correct for differences in compounding period in the scenario where you keep reinvesting the proceeds from the short-term deal in identical deals later on.