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If I pay 4 half-yearly installments in an investment scheme or I pay monthly installments of the same time (2 years). Interest rate in both cases is 9% compounded monthly. What will be my future value of annuity? I divided 9% by 12 in order to get the interest rate. For monthly installments, my compounding period is 2*12=24 months but what will be the compounding period (n) if the interest is still monthly compounded? Will there be any difference?

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You're missing some key information but I'll take a stab at it since no one else is.

Let's break it into two sections:

===Monthly payments first. ===

Cash flow:   $1,000 per year, with first monthly payment due in 30 days.  
Term:        2 years, with final value computed at end of 24th month,  
                      including the final payment due at that time.  
Annual rate: 9.00%  
Compounding: 12 periods per year (monthly)  

Cash flow  = $1,000 / 12 = $83.33   per month; first payment due in 30 days.  
Rate       =  0.09  / 12 =   0.0075 per month (in decimal)  
Num period = 12/yr x 2yrs = 24  

FVA = $2,182.37  [= $83.33 x ((((1 + 0.0075)^24)-1)/0.0075)]  

Annuity = $1,000 x 2yrs = $2,000  
Value   = $2,182.37 - $2,000 = $182.37

If you get $2,182.28 or so then don't worry. That's just rounding due to the cash flow (it's actually $83.3333 etc.)

===Semi-Annual payments, but monthly compounding===

Cash flow:   $1,000 per year, with first payment due in 183 days.  
Term:        2 years, with final value computed at end of 24th month,  
                      including the final payment due at that time.   
Annual rate: 9.00%  
Compounding: 12

We need to calculate the effect of compounding during each 6-month period.

Rate       =  0.09  / 12 =   0.0075 per month (in decimal)  
Num period = 12/yr / (2/yr) = 6  

Rate       = (1 + 0.09/12)^6 - 1
           = 0.04585 semi-annual
Num period = 1

Now we can calculate the annuity value:

Cash flow  = $1,000 / 2 = $500 semi-annual, with first payment due in 183 days.
Rate       = 0.04585  (in decimal)
Num period = 2/yr x 2yrs = 4

FVA = $2,141.81  ($500 x ((((1 + 0.04585)^4)-1)/0.04585)]

Annuity = $1,000 x 2yrs = $2,000  
Value = $2,141.81 - $2,000 = $141.81

The difference between the two annuities is $2,182.37 - $2,141.81 = $40.56 To be candid, that's chump change. What's your real cash flow?

Anyone want to confirm the math is good?

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  • The difference between the two annuities is 2 ½ months interest, plus compound interest on top of that interest. It is due to the fact the semi-annual savings plan does not require its first payment until the end of the first six months, whereas the monthly savings plan receives that payment over the course of the first six months.
    – Jasper
    Dec 20, 2015 at 3:33

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