please explain this question and answer, "Kai Lung has $$9,700.$ He wants to

buy a T$-$bill with a face value of $$10,000$ maturing on Nov. $15,2023.$ The

simple interest rate is $3\mathrm{.}875\%$ and the daycount convention is ACT$/365.$

What is the last date on which he can purchase the T$-$bill? How much does

he actually pay?

The unknown variable in our equation F $=$ P $(1+$iT$)$ is the time T$.$ To solve

this question, first we solve for T $($in years$)$ exactly: T $=($A$($T$)/$A$(0)1)/$i $=$

$($F$/$P $1)/$i $=0\mathrm{.}79814.$ The problem is this doesn$$t correspond to an exact

number of days before Nov. $15.$ So the next step is to convert this into

a number of days using the daycount convention $($which must always be

specified in the question$).$ T$($days$)=$ T$($years$)($number of days in a year$).$

In this case $($ACT $/365)$ that means mutiplying by $365$: T $=0\mathrm{.}79814365=$

$291\mathrm{.}32.$ As advertised, this does not $($usually$)$ correspond to an integer

number of days, so we have to decide whether to round up$,$ or round down.

$2$

We saw last time $($see the spreadsheet L$3$Examples$)$ that the T$-$bill price

increases as we get closer to maturity. So the longer Kai waits the more

expensive the T$-$bill gets. In theory, he has enough money to buy the T$-$bill

with $291\mathrm{.}32$ days to maturity, but in practice he has to buy it on either day

$291$ or day $292.$

We could just compute the price on both days and see which one is less than

$$9,700.$ Or we could reason it out: Remember, this is days until maturity. If

Kai waits until there are only $291$ days to maturity, he will not have enough

$($because the price goes up each day$),$ so he should buy it when there are

$292$ days to maturity. On that day he will pay

P $=$ F$/(1+$ iT$)=10,000/(1+\mathrm{.}03875292/365)=9,699\mathrm{.}32$

If he waits one more day, so T $=291/365,$ the price would be P $=9,700\mathrm{.}32$

which is more than he has!

The final step is to figure out what day this is$.$ Remember we want $292$ days

before Nov. $15,2023.$ It turns out this is Jan $27,2023.$ With a spreadsheet,

this is easy. On tests$/$the final exam you will have to figure out a counting

method that works for you. $($Check Jan $27,2023->$ Nov $15,2023=4+28$

$($not a leap year$)+31+30+31+30+31+31+30+31+15=292.)$

Another variation on this type of question is to ask on what date you should

sell a T$-$bill. $($Remember$,$ T$-$bills are freely tradeable at any time: a T$-$bill

is just a piece of $($electronic$)$ paper that says the government will pay the

holder the face value F on the maturity date.$)$