Calculating accumulated values for annuities

Question

Having difficulty with this problem

Find the accumulated value at end of four years of an investment fund in which 100 is deposited at the beginning of each quarter for the first two years and 200 is deposited at the beginning of each quarter for the second two years, if the fund earned is 12% convertible monthly.

The solution is $j = (1.01)^3 - 1 = 0.030301$

$100(\ddot{s}_{\overline{16|}j} +\ddot{s}_{\overline{8|}j}) = 2999$

However, I was wondering why this doesn't work.

$100\ddot{s}_{\overline{8|}j} + 200\ddot{s}_{\overline{8|}j}$?

My reasoning is we pay 100 for the first 8 payments and then 200 for the last 8 payments

Answer 1

I recommend writing out the cash flow. If payments are made quarterly, the effective quarterly rate of interest for a nominal rate $i^{(12)} = 0.12$ convertible monthly is $$j = \left(1 + \frac{i^{(12)}}{12}\right)^3 - 1 = 0.030301.$$ Then we have the accumulated value $$AV = 100(1+j)^{16} + \cdots + 100(1+j)^9 + 200(1+j)^8 + \cdots + 200(1+j)^1.$$ This is because payments at the beginning of quarter $t+1$ have had $16-t$ quarters to accumulate (e.g., in quarter $1$, $t = 0$ and the payment of $100$ has $16-t = 16$ quarters to accumulate value). Now we can write this in actuarial notation in a number of ways: $$AV = 100\ddot s_{\overline{16}\rceil j} + 100 \ddot s_{\overline{8}\rceil j} = 100\left(\ddot s_{\overline{16}\rceil j} + \ddot s_{\overline{8}\rceil j}\right)$$ is the notation if we split up the second half of payments into two flows of $100$ each, and view the accumulated value as the sum of two annuities-due of $100$ each in which the first one begins immediately, and the second begins two years (8 quarters) after, and both are accumulated up to the end of four years. Another way to write the cash flow is: $$AV = 100(1+j)^8 \ddot s_{\overline{8}\rceil j} + 200 \ddot s_{\overline{8}\rceil j} = 100\left((1+j)^8 + 2\right) \ddot s_{\overline{8}\rceil j},$$ which takes the first $8$ payments as an accumulated annuity-due valued $8$ quarters after the last payment of $100$ (hence the additional compounding by a factor of $(1+j)^8$); plus the second annuity-due of $200$. This one I like because it only requires us to calculate a single annuity symbol. A third way to write the cash flow is $$AV = 200\ddot s_{\overline{16}\rceil j} - 100 (1+j)^8 \ddot s_{\overline{8}\rceil j}.$$ This calculates an level annuity-due of $200$ per quarter for $16$ quarters, and then subtracts out the excess of $100$ per quarter for the first $8$ quarters. Note we must accumulate the portion subtracted out for an additional $8$ quarters, else we would have calculated a cash flow in which $200$ was paid for the first half, and $100$ for the second half, which is not what the problem stated. You will find that all three expressions give the same result of $2998.86$.

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After the above, you can now understand why your expression isn't correct: it fails to account for the fact that the first two years of payments of $100$ experience additional compounding for another two years: you are missing the factor of $(1+j)^8$. Consequently, after factoring, you would find your cash flow to simply be a payment of $300$ each quarter for $8$ quarters, which is clearly not equivalent to the stated flow.

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Since there is some disagreement over what the correct answer should be, I will explicitly write out all of the accumulated values of each of the $16$ quarterly payments:

$$\begin{align*} 100(1+j)^{16} &= 161.223, \\ 100(1+j)^{15} &= 156.481, \\ 100(1+j)^{14} &= 151.879, \\ 100(1+j)^{13} &= 147.412, \\ 100(1+j)^{12} &= 143.077, \\ 100(1+j)^{11} &= 138.869, \\ 100(1+j)^{10} &= 134.785, \\ 100(1+j)^{9} &= 130.821, \\ 200(1+j)^{8} &= 253.947, \\ 200(1+j)^{7} &= 246.478, \\ 200(1+j)^{6} &= 239.229, \\ 200(1+j)^{5} &= 232.194, \\ 200(1+j)^{4} &= 225.365, \\ 200(1+j)^{3} &= 218.737, \\ 200(1+j)^{2} &= 212.304, \\ 200(1+j)^{1} &= 206.060. \end{align*}$$ The sum of the accumulated payments of $100$ is $1164.55$. The sum of the accumulated payments of $200$ is $1834.31$. Their total is $2998.86.$

Answer 2

You have a $4$-years annuity-due that pays $100$ for the first $2$ years and $200$ for the second $2$ years. This is equivalent to have an annuity due that pays 100 for 4 years and then add another annuity that pays 100 for the last 2 years: $$ 100\ddot{s}_{\overline{16|}j} + 100\ddot{s}_{\overline{8|}j} = 100(\ddot{s}_{\overline{16|}j} + \ddot{s}_{\overline{8|}j})=100\times (20.82+9.17) = 2999\tag 1 $$ This is also equivalent to have a $4$-years annuity that pays $200$ for $4$ years and then subtract a $2$-years annuity-due that pays of $100$ from the first $2$ years, that is $$ 200\ddot{s}_{\overline{16|}j} - 100\,_{8|}\ddot{s}_{\overline{8|}j} =200\times 20.82- 100\times 11.65 = 2999\tag 2 $$ where $j=\left(1+\frac{i^{(12)}}{12}\right)^{3}-1=3.03\%$ is quarterly interest rate, $i^{(12)}=12\%$ and $_{m|}\ddot{s}_{\overline{n|}j}$ is the deferred annuity $$ _{m|}\ddot{s}_{\overline{n|}j}=(1+i)^m\ddot{s}_{\overline{n|}i} $$

Another method is consider an annuity-due that pays $100$ for the first $2$ years and then $2$-years annuity-due that pays $200$, that is $$ 100\,_{8|}\ddot{s}_{\overline{8|}j} + 200\,\ddot{s}_{\overline{8|}j} = 100\times 11.65 +200\times 9.17 = 2999\tag 3 $$

As you can see, we've found the same result. Infact for $(3)=(1)$ $$ 100\,_{8|}\ddot{s}_{\overline{8|}j} + 200\,\ddot{s}_{\overline{8|}j} =100(\ddot{s}_{\overline{16|}j}-\ddot{s}_{\overline{8|}j}) + 200\,\ddot{s}_{\overline{8|}j} =100(\ddot{s}_{\overline{16|}j}+\ddot{s}_{\overline{8|}j}) $$ and for $(2)=(1)$ $$\begin{align} 200\ddot{s}_{\overline{16|}j} - 100\,_{8|}\ddot{s}_{\overline{8|}j} &=100\ddot{s}_{\overline{16|}j}+100(\ddot{s}_{\overline{16|}j}-\,_{8|}\ddot{s}_{\overline{8|}j})\\ &=100\ddot{s}_{\overline{16|}j}+100\left(\ddot{s}_{\overline{16|}j}-[\ddot{s}_{\overline{16|}j}-\ddot{s}_{\overline{8|}j}]\right)\\ &=100(\ddot{s}_{\overline{16|}j}+\ddot{s}_{\overline{8|}j}) \end{align} $$ observing that $$ _{m|}\ddot{s}_{\overline{n|}j}= \ddot{s}_{\overline{n+m|}i}-\ddot{s}_{\overline{m|}i} $$

Finally the future value is $$ \color{blue}{\operatorname{FV}=100(\ddot{s}_{\overline{16|}j}+\ddot{s}_{\overline{8|}j})} $$