Financial Calculations with Force of Interest

[henricksteele]

Homework Statement

You wish to invest $10,000 over 10 years. Available to you are 5 different funds. At any time you may transfer money from one fund to another without penalty or transaction fee. To optimize the money you receive at the end of 10 years you realize that you want all of your money in the fund with highest force of interest at any give time. Remember the force of interest δ(t) = a'(t)/a(t)

Conversely, given δ(x), you can find that investing $1 from t₁ to t₂ equals

exp(_t1∫^t₂δ(x)dx)

Fortunately, for each of the five funds you know either its accumulation or
force of interest function. These are given as

1) a₁(t) = (1+ln(1+t)/35)^t
2) a₂(t) = (1-.05t)^-1
3) a₃(t) = (1.08)^t
4) δ₄(t) = 1/(√(2π*15)*exp(-(x-5)²/30)
5) a₅(t) = 1+.25ln(1+t)

Homework Equations

There are three parts to this problem

(a) List the time intervals where you should invest your money in each fund
to maximize earnings.

(b) Calculate the amount of money you have at the end of 10 years using the
optimal strategy.

(c) Calculate the amount of money you have at the end of 10 years if you
invested your money exclusively in one fund. Do this for all five funds.

P.S.: You probably need to draw a graph of the 5 force of interests with time t and always pick up the highest force of interest to invest your money.

The Attempt at a Solution

This isn't exactly a homework problem, I'm mainly trying to learn how to use financial calculators to solve investing problems similar to this one so if someone knows how to use Excel or something similar, please describe the process or recommend a good site to learn.

Thanks.

 

[Ray Vickson]

Homework Statement

You wish to invest $10,000 over 10 years. Available to you are 5 different funds. At any time you may transfer money from one fund to another without penalty or transaction fee. To optimize the money you receive at the end of 10 years you realize that you want all of your money in the fund with highest force of interest at any give time. Remember the force of interest δ(t) = a'(t)/a(t)

Conversely, given δ(x), you can find that investing $1 from t₁ to t₂ equals

exp(_t1∫^t₂δ(x)dx)

Fortunately, for each of the five funds you know either its accumulation or
force of interest function. These are given as

1) a₁(t) = (1+ln(1+t)/35)^t
2) a₂(t) = (1-.05t)^-1
3) a₃(t) = (1.08)^t
4) δ₄(t) = 1/(√(2π*15)*exp(-(x-5)²/30)
5) a₅(t) = 1+.25ln(1+t)

Homework Equations

There are three parts to this problem

(a) List the time intervals where you should invest your money in each fund
to maximize earnings.

(b) Calculate the amount of money you have at the end of 10 years using the
optimal strategy.

(c) Calculate the amount of money you have at the end of 10 years if you
invested your money exclusively in one fund. Do this for all five funds.

P.S.: You probably need to draw a graph of the 5 force of interests with time t and always pick up the highest force of interest to invest your money.

The Attempt at a Solution

This isn't exactly a homework problem, I'm mainly trying to learn how to use financial calculators to solve investing problems similar to this one so if someone knows how to use Excel or something similar, please describe the process or recommend a good site to learn.

Thanks.

Software---EXCEL, or whatever---cannot help you until you have decided WHAT needs to be computed!

Ask yourself this: if at time $t_0$ you are invested in fund X, how would you decide whether or not to keep your money in fund X over the small time interval $t_0 < t < t_0 + \Delta t$ or switch to another fund, Y for that next little bit of time? Would you compare $a_i$ values or $\delta_i$ values, or would you do something else? Some of the data is given to you in terms of $a_i(t)$ while others give you $\delta_i(t)$, so first of all you need to decide which type of data to use, and make sure you have that data for all the funds. Then you need to know what to do with the data. Then, and only then should you concern yourself with what type of software to apply.

 

[haruspex]

first of all you need to decide which type of data to use

I think the OP makes it clear that you have to pick the highest δ at each instant.
henricksteele, assuming you have developed all the equations for the δ functions, you're simply asking how to use a spreadsheet to solve it.
I would set a heading row (1) to contain the texts "time", "fund 1" etc.
A2 = 0 (initial time)
A3=A2+.1 (or whatever increment of time you want), A4=A3+.1, etc. up to whatever end time you think is appropriate.
Populate the rest of the array with the equations you have for the δ functions. So in B2 you have a function using the value of A2 as time. Having filled in row 2 for each fund, you can just copy these down through the array and Excel (or whatever) will update the row numbers for you.
Now you can generate a chart, and you will see immediately which fund gives the highest force of interest at each instant.
You'll need to check you've taken time far enough. For that, see if you can determine algebraically which fund should give the highest result asymptotically. if that's not what you see in the chart then maybe you need to extend the time.
If you want to pin down the cross-over points more precisely, you can either make the time steps smaller or compare pairs of funds algebraically.

 

[Ray Vickson]

I think the OP makes it clear that you have to pick the highest δ at each instant.
henricksteele, assuming you have developed all the equations for the δ functions, you're simply asking how to use a spreadsheet to solve it.
I would set a heading row (1) to contain the texts "time", "fund 1" etc.
A2 = 0 (initial time)
A3=A2+.1 (or whatever increment of time you want), A4=A3+.1, etc. up to whatever end time you think is appropriate.
Populate the rest of the array with the equations you have for the δ functions. So in B2 you have a function using the value of A2 as time. Having filled in row 2 for each fund, you can just copy these down through the array and Excel (or whatever) will update the row numbers for you.
Now you can generate a chart, and you will see immediately which fund gives the highest force of interest at each instant.
You'll need to check you've taken time far enough. For that, see if you can determine algebraically which fund should give the highest result asymptotically. if that's not what you see in the chart then maybe you need to extend the time.
If you want to pin down the cross-over points more precisely, you can either make the time steps smaller or compare pairs of funds algebraically.

I agree that the OP wrote "you have to pick the highest δ at each instant", but I really wanted the OP to understand WHY that is the case, essentially by answering the question I asked. Also: there is the issue of discrete-time vs. continuous time; using continuous time is suggested by some of the integration notation used in the question. So, part (a) could involve solving several nonlinear equations numerically, while parts (b) and (c) can involve integrations, some of them being "non-elementary". However, even for the non-elementary cases, EXCEL does have some built-in functions that can handle the calculations. Again, though, the OP really needs to clarify (to himself, if not to us) exactly what type of analysis is wanted. Some finance and accounting courses want students to be thoroughly familiar with continuous-time analysis, using the full spectrum of calculus tools (at least at the level of Calculus 101).