formulating an optimization problem

[Jimbrisky]

I am trying to write the linear programming model for the optimization problem below. Though, I understand the problem, I am a bit stuck coming up with the mathematical equations and will really appreciate any help.

An investor can invest in exchange-traded equity with an expected return of 8% per year. Alternatively the investor can invest in private equity in the secondary market and expect to earn a return of 12% per year, but this investment requires a 3 year holding period. Finally, the investor also has the opportunity to invest in a primary private equity offering with an expected return of 18% per year, but this investment has a 7 year holding period. Each of these investment opportunities is available in the marketplace every year.

The investor has $100,000 to invest. Assume that investment decisions are decided at the beginning of each year (therefore exchange-traded equity should be treated as though it has a 1 year holding period) and that all investments must be liquidated at the end of year 10. In addition to the budget relationships that relate purchases to available funds, the investor requires an average holding period no greater than two years. What strategy (i.e., choice of investments each year) should the investor follow over the next 10 years to maximize the expected portfolio value at the end of year 10?

Formulate this problem as a linear program. Indicate and define all decision variables and model constraints.

 

[Jimbrisky]

I am trying to write the linear programming model for the optimization problem below. Though, I understand the problem, I am a bit stuck coming up with the mathematical equations and will really appreciate any help.

An investor can invest in exchange-traded equity with an expected return of 8% per year. Alternatively the investor can invest in private equity in the secondary market and expect to earn a return of 12% per year, but this investment requires a 3 year holding period. Finally, the investor also has the opportunity to invest in a primary private equity offering with an expected return of 18% per year, but this investment has a 7 year holding period. Each of these investment opportunities is available in the marketplace every year.

The investor has $100,000 to invest. Assume that investment decisions are decided at the beginning of each year (therefore exchange-traded equity should be treated as though it has a 1 year holding period) and that all investments must be liquidated at the end of year 10. In addition to the budget relationships that relate purchases to available funds, the investor requires an average holding period no greater than two years. What strategy (i.e., choice of investments each year) should the investor follow over the next 10 years to maximize the expected portfolio value at the end of year 10?

Formulate this problem as a linear program. Indicate and define all decision variables and model constraints.

So after doing a lot of think I have been able to come up with the budget constraints. I am now working on getting the objective function and holding period constraint and solving the problem in excel. Any help is still appreciated. Thanks.

 

[Ishuda]

Just talking about the problem: Obviously, in this case, we would like to do the 18% vessel for the maximum time, then the 12% vessel for as much time as we can since we can't do two holdings of 7 years. That takes up our ten years. However we can not do just that because we need the 2 year average. Well 7 and 3 and a set of of the 8% vessel each year would be 12 investments lasting a total of 20 years or an average 1.67 years each. So a minimum investment in the 1 year holding time reinvested annually with the remainder in the 7 year holding time to be reinvested in the 3 year holding time when the seven year comes due.

How about we start with, we can only get one seven year term and we would like that, i.e. max investment return for maximum time subject to 2 year average, then maximum invest return for remaining time, .... So the average is
A = (7 + 3 m + 1 n) / (n+m+1) \(\displaystyle \le\) 2
where m is the number of terms of 3 year money (other than the reinvest of the 7 year) and n is the number of terms of 1 year money so
7 + m \(\displaystyle \le\) n
Now we also would want that if we invest in the 3 year, we don't invest in the 1 year for three years, that is
n = 10 - 3m \(\displaystyle \ge\) 7 + m
or
3 \(\displaystyle \ge\) 4 m

Since we are dealing in integers, m = 0 and n = 10 and we are back at the original minimum investment in the 1 year holding time reinvested annually with the remainder in the 7 year holding time to be reinvested in the 3 year holding time when the seven year comes due.

Got to go now but I think that might work.

Edit (again): I'm also assuming here that the minimum investment in the 3 year vessel is the same (or higher) than that for the 1 year vessel or we could possibly have both a 3 year and a 1 one year investment.

 

[Ishuda]

I thought holding period was basically the period from when you bought the item until you sold the item and had nothing to do with the amount of money the item cost. For example the IRS defines long term (longer than 1 year usually) and short term (a year or less in most cases) holding periods in terms of time the item was held. The value of the item has nothing to do with holding period.

 

[Ishuda]

Agree. Did I say something to the contrary?

No, but your h is 'normalized' by 100000 and doesn't takes into account the number of investments:

...
c = 100000 - a - b
h = (a + 3b + 7c) / 100000
p = (a*.08 + b*.12 + c*.18) / 100000
...

If h is your holding period then you only have 3 investments, 1 of type a, one of type b, and one of type c. What about the money returned at the end of year 1? It would seem that you should reinvest it.

 

[Ishuda]

You're missing my point...or I wasn't clear...
All I was doing was showing that IN YEAR1 ONLY, the MAXIMUM yield
was 10%, and that this could be achieved ONLY by investing 50,000
at 8% and 50,000 at 12%.
I wanted to see the OP's reaction to that; more to see if I understood
what's going on...as this is completely new to me...

Anyway, I think we're both losing our time here, as the OP has not
responded

You are right, I did misunderstand. Sorry