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Not an investment specialist, so please excuse the very basic math.

Given a lump sum, I need to distribute this lump sum over (x) segments, each lasting (y) years (years can be different for each segment). Each segment could potentially be deferred from paying out, during which time it will earn (z) interest rate during the deferral period, but once payments begin, will earn (w) interest rate during the payment period (p) years.

What I assumed was that the amount of money in the first segment, which will have no deferral and start paying immediately, will have a larger portion of the lump sum. Whereas the last segment, which has the greatest deferral time, will get a smaller portion on the lump sum.

In my very naive manner, I tried to solve this by doing an average of the number of deferral years (y), an average of (z), (w) and (p) - then I took the lump sum and divided that by (x) number of segments and used that value to calculate a monthly return for that value.

To calculate a payment:

monthly_payment = ((lump_sum / x) * (1 + z)^y) * (1 + w)^p) / (p * 12)

I get really close, but the utilization of the lump sum is off by a good 5%.

I have been looking up Optimization and Solvers - but being new to this, I could use a little help in being pointed in the right direction as well as understanding some of the concepts.

Can anyone give me names of formulae that could solve this or point me in the right direction?

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  • $\begingroup$ incognos, i don't fully understand your problem. Are the $w_i$ the payments the same for all 3 segments or can they differ? Are the payment periods $p$ different for each segment? Suppose that the $p$ are different, what happens if one segment finishes paying and the other segments are still ongoing? Is your objective simply to pick the distribution such that your deterministic payout is maximized? $\endgroup$
    – mbison
    Sep 17, 2015 at 13:23
  • $\begingroup$ My objective is to get the same payout on each segment. The periods may differ but will not overlap. It could be as few as 2 segments and as many as 9. $\endgroup$
    – Sina Khelil
    Sep 17, 2015 at 15:11

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Not 100% sure if i understood your question. But below is solution to what I understood from your explain.

let $N_i(t)$ be the notional allocated to segment i at time t, i = 1,..., x. Suppose segment i is deferred for period $T^{def}_i$ during which it earns interest z. Suppose that segment i pays $w_i$ for period $T^{payment}_i$.

At time t = 0 (now) you have allocated $N_i(0)$ to segment i. After the deferral period ends, the notional will have grown to: $N_i(T^{def}_i) = N_i(0) * exp(z*T^{def}_i)$. This amount will earn the rate $w_i$ going forward.

So at the end of contract segment i will now have accumulated to $N_i(0) * exp(z*T^{def}_i) * exp(w_i*T^{payment}_i)$. To simplify notation, let s define $c_i := exp(z*T^{def}_i) * exp(w_i*T^{payment}_i)$.

What you want is to pick $N_i(0)$ for all i=1,...x in such a way that at the end we have:

$N_i(0) c_i = N_j(0) c_j$ for all i and j. Furthermore we have the condition that $\sum_{i=1}^x N_i(0) = lumpsum$. You want to solve this for $N_i(0)$. What we described above is linear system of the form Ax = b. I have provided example of A,x, b in case the number of segments is 3 just as an example.

With

\begin{equation} A = \left( \begin{array}{ccc} 1 & 1 & 1 \\ c_1 & -c_2 & 0 \\ c_1 & 0 & -c_3 \\ 0 & c2 & -c_3 \end{array} \right) \end{equation} \begin{equation} x = \left( \begin{array}{c} N_1(0) \\ N_2(0) \\ N_3(0) \end{array}\right) \end{equation} \begin{equation} b = \left( \begin{array}{c} lumpsum\\ 0\\ 0\\ 0 \end{array}\right) \end{equation}

The solution to this system (if it exists) is given by $x = A^{-1}b$. details on matrix algebra and such can be found here:

https://en.wikipedia.org/wiki/Matrix_(mathematics)

(edit: added wiki link)

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  • $\begingroup$ Just as extra background info: The first row in matrix A is condition that you invest total amount equal to lumpsum. Second row means total return on segment1 equals total on segment 2. etc. $\endgroup$
    – mbison
    Sep 17, 2015 at 18:31
  • $\begingroup$ Not sure if I 100% understand your solution... and I think the link you may have been referring to in your solution is missing (at the end) I am trying to solve this using the numpy kit in python (docs.scipy.org/doc/numpy/reference/routines.linalg.html) not being a very strong mathematician, I get lost in your solution easily. $\endgroup$
    – Sina Khelil
    Sep 21, 2015 at 2:28
  • $\begingroup$ @incognos: copy paste mistake. forgot the wiki link. here you go: en.wikipedia.org/wiki/Matrix_(mathematics) i am sure that numpy can do matrix algebra. If you tell me what part of the proposed sol you don't understand let me know, i ll try to clarify my thoughts $\endgroup$
    – mbison
    Sep 23, 2015 at 8:55

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