A five-year annuity of ten $10000 semi-annual payments will begin 9 years from now, with the first payment coming 9,5 years from now. If the discount rate is 16% per annum compounded monthly, what is the value of this annuity five years from now? The answer should be $35 106,79 but I have no idea how it is gotten. Please help…thanks!
Step 1) Calculate a meaningful discount factor.Originally Posted by Beto
We have an annual rate. It is compounded monthly. All the payments are semi-annual. It is a mass of confusion.
i = 0.16
$\displaystyle i^{(12)} = 0.16/12 = 0.0133333...$
$\displaystyle v^{(12)} = \frac{1}{1+i^{(12)}} = 0.98684211$
$\displaystyle v^{(2)} = \left(v^{(12)}\right)^{6} = 0.92360447$
There, now we have a semi-annual discount factor.
Step 2) BEFORE all else fails, consider basic principles. Practice this. Get good at it.
From here on out, I will just use 'v'. Understand this to mean the semi-annual discount factor.
Just write it down.
$\displaystyle 10000(v^{19}+v^{20}+...+v^{28})*v^{-10}$
There. It is done. That is the present value you seek.
Of course if you would like to simplfy it, feel free.
$\displaystyle 10000*v^{9}(1+v+...+v^{9})$
You should recognize the right-most piece.
$\displaystyle 10000*v^{9}\frac{1-v^{10}}{1-v}$
How are we doing?
$10,000*0.48907362*7.17698305 = $35,100.73
Note: I used machine rounding. I would guess this is 15 decimal places. If I use the numbers I actually typed here, rounding each intermediate result to 8 decimal places, I get:
$\displaystyle v = 0.92360450$
$\displaystyle v^{9} = 0.48907376$
$\displaystyle v^{10} = 0.45171073$
$\displaystyle \frac{1-v^{10}}{1-v} = 7.17698392$
$10,000*0.48907376*7.17698392 = $35,100.75
This demonstrates rather clearly that your "answer" is using a different rounding convention. In any case, you should see the idea. When you implement your rounding conventions, you should be in the neighborhood.
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