Hey guys,

I was going through this reading and came across this example:

" Consider a similar situation in which cash flows of $6 per year begin at the end of the 4th year and continue at the end of each year thereafter, with the last cash flow at the end of the 10th year. From the perspective of the end of the third year, we are facing a typical seven-year ordinary annuity. We can find the present value of the annuity from the perspective of the end of the third year and then discount that present value back to the present. At an interest rate of 5 percent, the cash flows of $6 per year starting at the end of the fourth year will be worth $34.72 at the end of the third year (t = 3) and $29.99 today (t = 0). "

I understood how to get $34.72 since I entered N = 7 because by end of 10th year it literally means beginning of 11, which is how i drew it on the time line, however I cant understand how it got $29.99 for t = 0 because to get that it would have had to put N = 3 to get that answer however the remaining periods from end of 3rd year are N = 4.

Could somebody please help me understand this?

Thanks a lot

At time t=0, I deposit $29.99 into a bank account earning 5% compounded annually. I let it roll up with compound interest and at the end of the 3rd year, there is $34.72 in the account ( a little math: 29.99 * (1.05)^3 = 34.72). Oh joy, oh rapture, $34.72 at the end of year 3 is just enough to fund the immediate annuity!!!

You could also do this on the CF and NPV worksheets on the BAII:

Cf0=0 C01=0 F01 =3 C02=6 F02=7

NPV I 5 ENTER CPT NPV 29.9909214

1 Like

I too, was wondering how they got to that result, of $34.72. The solution is to pay more attention.

It says: From the perspective of the end of the third year, **we are facing a typical seven-year ordinary annuity.**

The formula of: PV = A/r is only used when you have a perpetuity with identical payments, and here we know that there are a total of 10 periods, so it doesn’t go to infinity( to be considered a perpetuity).

After this, you can use this formula: PV= A*[(1-(1/(1+r)^N))/r]

And you get 6*[(1-(1/(1+0.05)^7))/0.05] = 34.72

And to get the final result, meaning PV at t = 0 you use either of these formulas:

PV = FV/(1+r)^3 = 34.72/1.05^3 = 29.99

or

PV = FV * (1+r)^-3 = 34.72*1.05^-3 = 29.99