# Financial maths, present value annuity

The proposed salary package (per employee) commencing 1 July 1995 and lasting for exactly 3 years is as follows.

$10 000$ per month in arrear for the first year

$12 500$ per month in arrear for the second year

$16 000$ per month in arrear for the third year

Find the present value as at 1 June 1995 of a single package, with the following interest rates:

$24%$ per annum payable monthly for the period $1$ June $1995$ to $30$ June $1996$

$21%$ per annum convertible $3$ times p.a. for the period $1$ July $1996$ to $30$ June $1997$

$15%$ per annum for the period $1$ July $1997$ to $30$ June $1998$

Answer $= 320 153.07$

I tried taking each amount back by 12 using an annuity then "V" but I'm confused don't payments continue after the last interest period. My answer gets close but not exactly I don't know what I'm doing wrong. Please help

• What is the correct answer? I get something like $316,676$. – callculus Jun 7 '18 at 14:43
• The correct answer is given above. 320153.07 – KombatWombat Jun 7 '18 at 14:46

The answer to this question is the sum of the present value of three separate annuity-immediates, discounted back to the date in question.

Year 1 - Since the rate of interest is 24% compounded monthly, the periodic rate (per month) is 2%. We find the present value of a 12-period annuity-immediate at a rate of 2% with a payment of $10,000 as follows: $$10,000 * \frac{1 - 1.02^{-12}}{0.02} = 105,753.4122$$ Year 2 - Since the rate of interest is 21% compounded three times in one year, the periodic rate (per month) is $$1.07^{1/4} - 1$$. We find the present value of a 12-period annuity immediate at this rate, with a payment of$12,500.00, and discount it back one year (to the same start date as the annuity for year one):

$$(1.02^{-12}) * 12,500 * \frac{1 - 1.07^{-12/4}}{1.07^{1/4} - 1} = 106,140.3832$$

Note that the denominator of 4 in 12/4 is determined by the fact that there are four months per compounding period.

Year 3 - Since the rate of interest is 15% compounded annually, the periodic rate (per month) is $$1.15^{1/12} - 1$$. We find the present value of a 12-period annuity immediate at this rate, with a payment of 16,000.00, and discount it back two years:

$$(1.15^{-1}) *(1.02^{-12}) * 16,000 * \frac{1 - 1.15^{-1}}{1.15^{1/12} - 1} = 114,662.3375$$

The sum of the present value of these annuities adds up to \$326,556.1329. Since this is the present value of the three annuities as of July 1, 1995, we must discount it back one period further to June 1, 1995; the amount we are left with is:

$$1.02^{-1} * 326,556.1329 = \320,153.07$$