Difference Equation Model: Fund Operation and Management

background

The year before his death, the Nobel Prize winner ordered most of his estate (approximately US$9.2 million) to be used as a fund, and the annual interest earned would be divided into 5 parts to establish 5 institutions in physics, chemistry, physiology or medicine, literature and peace. A kind of prize (the Nobel Prize) is awarded to people from all over the world who have made significant contributions to mankind in these fields. Among them, explosives are the most famous. The artificial element Nobelium is named after Nobel. (The Economics Prize was added in 1968), awarded every year on December 10, Nobel’s death day, to commend those who have made indelible and great contributions to mankind in the above fields.

 [Model Assumptions]

1. The bonus amount is fixed every year and is paid out in one lump sum at the end of the year.
2. During the fund operation period, the annual rate of return of the fund remains unchanged.
3. The fund amount at the beginning of the current year is equal to the principal and interest of the previous year's fund amount minus the bonus amount.
4. The accumulation of the fund is calculated based on compound interest.

【Symbol Description】

 【Modeling】

According to assumptions (1)-(4), the fund amount at the end of year t is At, the fund income in year t+1 is At(1+r), and the bonus amount B is paid, which is the base amount at the end of year t+1 At+1, that is [22]

This is a first-order non-homogeneous difference equation with constant coefficients, and the general solution of the corresponding homogeneous equation is [23]

It is easy to observe that a special solution of [22] is so the general solution of [22] is

From the initial value condition A0, the special solution of [22] is [24]

Among them, as long as you know three of A0, B, r and At, you can find the remaining one. 【twenty four】

【Model Application】 

A company has established a bonus fund named after the company. It plans to pay it out once a year at the end of each year. The bonus is 10,000 yuan. If the annual rate of return of the fund is 15% and the bonus payment period is 10 years, then based on compound interest calculation, the original fund should be How many? If it is a perpetual fund, how much is the original fund?

1. The distribution period is 10 years. How about the initial fund?

The distribution period is ten years, which means that the fund balance at the end of the 10th year is 0, that is, A10=0, r=0.15, B=10000, substitute [24], we get 

 

Solution: 

2. What if it is made into a sustainable fund? 

  If it is a perpetual fund, it means that after the bonus is distributed at the end of each year, the fund balance will not be lower than the value of the fund at the beginning of the year, that is, 

We might as well take A1>=A0, that is

 Equivalent deformation income (that is, the bonus issued cannot exceed the annual interest)

Substituting B=10000 and r=0.15 into the above formula, we get

 For the convenience of management by ordinary management personnel, the year-by-year operation data of 10-year funds and perpetual funds are now listed in Table 1 and Table 2.

                                  Table 1 10-year fund operation overview: r=0.15 unit (yuan) 

years	Fund at the beginning of the year	year-end fund	Disburse bonus	Fund balance
1	50187.68	57715.84	10000	47715.84
2	47715.84	54873.22	10000	44873.22
3	44873.22	51704.20	10000	41704.20
4	41604.20	47844.83	10000	37844.83
5	37844.83	43521.55	10000	33521.55
6	33521.55	38549.78	10000	28549.78
7	28549.78	32832.25	10000	22832.25
8	22832.25	26257.09	10000	16257.09
9	16257.09	18695.65	10000	8695.65
10	8695.65	10000.00	10000	0.00

                                     Table 2 Summary of Perpetual Fund Operations: r=0.15 unit (yuan)

years	Fund at the beginning of the year	year-end fund	Disburse bonus	Fund balance
1	66666.67	76666.67	10000	66666.67
2	66666.67	76666.67	10000	66666.67
3	66666.67	76666.67	10000	66666.67
…	…	…	…	…
10	66666.67	76666.67	10000	66666.67
…	…	…	…	…

 3. Model promotion

If the founder of the fund takes inflation into account and considers that the annual bonus is not fixed, but increases by 5% every year, how should he operate this fund?

Assume the initial bonus amount is B, and the annual increase rate is 5%. Assume that the annual inflation rate is r.

1. After the first year, the bonus amount is B.
2. After the second year, the bonus amount is B + B * 0.05 - B * r.
3. After the third year, the bonus amount is B + (B + B * 0.05 - B * r) * 0.05 - (B + B * 0.05 - B * r) * r.
4. By analogy, the bonus amount after the nth year is the bonus amount of the previous year plus the bonus amount of the previous year multiplied by 0.05, minus the bonus amount of the previous year multiplied by the inflation rate r.

This difference equation can be written in the form of a recurrence:

B(n) = B(n-1) + B(n-1) * 0.05 - B(n-1) * r

Among them, B(n) represents the bonus amount after the nth year, and B(n-1) represents the bonus amount after the n-1th year.

Using this recursive formula, you can calculate the annual bonus amount and make a forecast. It should be noted that the inflation rate r may change every year, so in actual operation, you need to adjust it according to the actual situation.

4. Model comparison

Comparing the fund management model with the mortgage repayment model, what are the similarities and differences?

Similarities:

1. All are managed and planned based on mathematical models.
2. Both involve the growth or decrease of funds through planning and decision-making to achieve long-term goals.
3. All need to consider time value factors, such as interest, rate of return, etc.

The difference:

1. The objects are different: the fund management model is applied to the management of financial instruments such as investment funds, while the housing loan model is a model applied to home buyers' loan repayments.
2. The goals are different: the goal of the fund management model is usually to realize the appreciation of funds, so that investors can obtain returns; the goal of the mortgage model is to manage the borrower's repayment plan and ensure that the loan is repaid on time.
3. Risks and returns: Fund management involves more investment risks and return expectations, requiring managers to make investment decisions and risk control; the mortgage model focuses more on repayment ability and management risks to ensure on-time repayment without falling into financial difficulties .
4. Time scale: Fund management models are usually used for long-term investments, involving longer time scales, and may need to consider more market fluctuations and cyclical factors; mortgage models usually involve medium and long-term repayment plans, usually in months or years.

Although there are similarities between the fund management model and the home loan model, their models and approaches differ due to their different objectives and application areas. For each model, it needs to be customized and adapted on a case-by-case basis to meet specific management needs.