find the present value to the nearest dollar on January 1 of an annuity which pays $2000 every six months for fiver years. The first payment is due on the next April 1 and the rate of interest is 9% convertible semiannually
present value for annuity values on any date ?
31
Mar
Ben
March 31, 2010 at 6:58 pm
In my opinion, this would be easiest to calculate by first finding the present value as of the just-past October 1, then just accumulate that value up to the present (January 1). That way your annuity immediate formula will work just fine, using 4.5% interest per period and 10 payment periods (and of course payments of 2000).
EDIT: Actuarially, interest is always being calculated; being compounded semiannually doesn’t mean we won’t earn interest in a three-month period.
Since you asked for some workings:
PV(Oct) = 2000*a10 (4.5%)
= 2000* (1-v^10)/i
= 15825.436354220331972834023495304
And then we accumulate up to January:
*(1.045)^0.5
= 16177.590531736951018721363699918
= 16177.59
[The 0.5 exponent is since we're only accumulating for 3 months, or half of the compounding period.]
a simple man
March 31, 2010 at 7:15 pm
On the last day, it is worth 2,000
It earned 4.5% interest in the last six months to make it up to 2,000.
Meaning it was worth 2000/ (100%+4.5%)= 2000/(1.045)=1913.86
plus the additional 2000 payment six months earlier.
One year earlier it is worth 2000/(1.045)^2 + 2000/(1.045) + 2000
(Where 2000/(1.045)^2 represents the principal that will be needed to make the last payment and here 2000/1.045 represents the principal needed to make teh next to last payment.)
18 months earlier it is worth 2000/(1.045)^3+ 2000/(1.045)^2+2000/(1.045) +2000
Five years earlier it is worth:
2000/(1.045)^10 + 2000/(1.045)^9+ 2000/(1.045)^8+2000(1.045)^7+2000/(1.045)^6+2000/(1.045)^5+2000/(1.045)^4+2000/(1.045)^3+2000/(1.045)^2+2000/(1.045)^1+2000/(1.045)^0
Got it??
Answers shortened my last expression but I think you can see the trend where the last number is 2000/(1.045)^0
Since the interest is only paid every six months, you cannot assume that it earned any interest in the three months before Jan 1. So the total value on Jan 1 is the same as the value on October 1:
2000/(1.045)^11+2000/(1.045)^10……. without the 2000 on the end.
PS: The interest is paid only twice a year, hence, the 4.5% interest per period.
There are only 10 periods in five years.
If there was a payment six months before April 1, that would have been on October 1. The value of the annuity would have been the total of 11 periods instead of 10, But in this case there was no October payment. Here, the value of the annuity on April 1 (when the fist payment is made) will the sum of ten periods but three months earlier (jan 1) it is worth only what it would have been worth on October 1. The interst is paid only every six months, not by the month.