Question

1.there is an example of a three-year annuity, which the first payment is in year 1, the second payment in year 2, and the third payment is in year 3. However, if the first payment will be paid in the second year, how can I calculate the present value of such annuity.

2.Why the annuity due's result in a higher present value? If you have cash outflow, wouldn't your money be less and the present value shouldn't be decreased?

3.It is said that continuous compounding is the mathematical limit compounding interest can reach. And at that time n reaches infinity, so the formula changed from FV = P*(1+r/n)^(nt) to FV = P*e^(rt). I have no idea how these changes happened and I want to know when and how to use this formula.

4.I don't really understand that when I compound interest if I'm dealing with months or quarters, I have to divide 4 or 12.
Quarterly compounding: FV = 1000(1+(0.05/4))^(4*3)
Monthly compounding: FV = 200(1+(0.05/12))^(12*3)

could you please explain the meaning on why to divide 4 or 12?

Answer 1

1) if annuity is started in 2nd year then calculate present value of annuity thinking 2nd year as first year. Then you will get present value of annuity as on second year. Now bring the pv of annuity as on 2 nd year to first year by given formula

P.v =p.v of annuity/(1+x)

Where x is rate of interest

2)in ordinary annuity payments are made at the end of year.but in annuity due payments are made at the beginning of year so it will have highest present value according to time value of money concept time is money.we are calculating present value of annuity we don't know nature of annuity if it is an expense then it will be p.v of expenses. If it is investment then it is present value of investment (wheather it is outflow or inflow we are calculating p.v of that flow

3) compound interest formula is f.v=pv(1+r/n)^nt

In continous compounding interest is compounded by the lowest time possible so we take n as infinity

According to maths equation

Limit (1+x/n)^n when n tends to infinity is e^r

So continous compounding formula is pv×e^rt

4) monthly compounding means that interest on interest will be added every month (principle will be increased with interest every month )but we will be given annual rates. So we need to convert that rates to the period of compounding

For monthly it will be r/12 (as 12 months in year)

For quarterly it will be r/4(as four quarters per year)

Any doubts welcomed in comment section