Allocating a Lump Sum Across Deferred Investment Segments

# Segmented investment to yield same monthly return in each segment


# Segmented investment to yield same monthly return in each segment












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?

## Answer by mbison (score 1)

https://quant.stackexchange.com/a/20775

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)