This goes on for 10 years

If you have to find a present value of payments:

$30 being paid at the end of each month for the **first** year

$40 being paid at the end of each month for the **Second** year

$50 being paid at the end of each month for the **third** year

Now, consider you have been given a nominal rate of 12% convertible monthly, meaning .12/12 = 0.01 is being applied each month and to find the annual effective for this case you can simply say — i = (1.01)^12 and you will have (1.1268 – 1).

I need to find the present value for the condition above, I need help to understand if in such cases can I use the 0.1268 and compute this increasing annuity problem annually by converting all the payments above to annual payments?

Making them increase by 120 each year…

UPDATE:

I figured out that the increasing annuity can be calculated yearly after converting the interest rate, and at the same time annuity for money increasing monthly can be separately calculated as monthly.

(Ia)10 = 120*[(((1.126825^-10) -1)/0.11255075) – 10(1.1268)^-10]/0.126825 + 20 as pmt, 1% as interest monthly and n = 120 you get a total of 4556.71

## Best Answer

I assume that the total term is $10$ years, with payments increasing by $10$ per year. Then the series of monthly payments in the final year is $120$.

Note that the periodic payments are monthly, and the interest rate is specified as a nominal (annual) rate convertible monthly. I put "annual" in parentheses because this is not explicitly stated, but implied. Then this means the effective monthly rate is simply $j = i^{(12)}/12 = 0.01$, and the monthly present value discount factor is $$v_j = \frac{1}{1+j} = \frac{1}{1 + 0.01} \approx 0.990099.$$ The

annualpresent value discount factor is $$v_i = \frac{1}{1+i} = \frac{1}{(1+j)^{12}} = v_j^{12}.$$Then the equation of value is

$$\require{enclose} \begin{align} PV &= 30(v_j + v_j^2 + \cdots + v_j^{12}) + 40(v_j^{13} + \cdots + v_j^{24}) + \cdots + 120(v_j^{109} + \cdots + v_j^{120}). \\ &= (30 + 40v_j^{12} + \cdots + 120v_j^{108})a_{\enclose{actuarial}{12} j} \\ &= \left((30 + 30v_i + \cdots + 30v_i^9) + 10(v_i + 2v_i^2 + \cdots 9v_i^9)\right) a_{\enclose{actuarial}{12} j} \\ &= \left(30 \ddot a_{\enclose{actuarial}{10} i} + 10(Ia)_{\enclose{actuarial}{9} i} \right) a_{\enclose{actuarial}{12} j} \\ &= \left(30 (1+i)\frac{1 - v_i^{10}}{i} + 10\frac{\ddot a_{\enclose{actuarial}{9}i} - 9v_i^9}{i} \right) \frac{1 - v_j^{12}}{j} \\ &\approx 4556.8695965. \end{align} $$

You could also have written the equation of value as

$$PV = (1+i) \left(20 a_{\enclose{actuarial}{10}i} + 10 (Ia)_{\enclose{actuarial}{10}i} \right) a_{\enclose{actuarial}{12}j}$$

which yields the same result. Your answer is slightly off probably because of rounding.