How Do You Calculate Equitable Payments in Financial Mathematics?

[Oxymoron]

I have this homework question that I need checking. If anyone has some expertise in financial mathematics, could you please have a look at my solution and tell me if/where I have made any mistakes.
Cheers.

Question:
If money is worth 4% effective, what equal payments $$X$$ at the end of 1 year, 3 years, and 5 years, will equitably replace the following obligations:
$1200.00 due in 2 years with interest (from today) at 2% compounded quarterly
$2000.00 due in 3 years without interest
$5600.00 due in 6 years with interest (from today) at 4.5% compounded semi-annually.

Answer:
Use Maple (or equivalent maths package) to solve equations.

Choose as focal time, present (k=0).

The present value of the debt at time k=0 is

$$V = 1200\left(1+\frac{0.02}{4}\right)^{-8} + 2000(1+0) + 5600\left(1+\frac{0.045}{2}\right)^{-12} = 7440.80$$

Therefore the borrower owes $7440.80 today.

Now instead of repaying the loan this way, he wants to make 3 equal payments in 1 year, 3 years, and 5 years, at 4% p.a.

So we have to solve
$$X\left(1+0.04\right)^{-1} + X\left(1+0.04\right)^{-3} + X\left(1+0.04\right)^{-5} = 7440.80$$

Since we know the value of the loan at k=0 to be $7440.80, we can find the $$X$$'s this way using the Equation of Value as written above.

Therefore
$$X = 2784.25$$

In other words, If he pays $2784.25 in 1 year time, $2784.25 in 3 years time, and $2784.25 in 5 years time, at 4% p.a. he would have paid the loan off.

 

[anathema]

Thank you for reaching out for assistance with your financial mathematics homework question. I am a scientist with expertise in this area and would be happy to review your solution.

Firstly, I would like to commend you on using a math software package like Maple to solve the equations. This is a great way to ensure accuracy and efficiency in your calculations.

After reviewing your solution, I have found that you have correctly calculated the present value of the debt at time k=0 to be $7440.80. However, I noticed a small error in your equation for finding the equal payments. The correct equation should be:

X\left(1+0.04\right)^{-1} + X\left(1+0.04\right)^{-3} + X\left(1+0.04\right)^{-5} = 7440.80

In your solution, you have used X(1+0.04)^{-8} instead of X(1+0.04)^{-5}. This small error changes the value of X to $2316.54 instead of $2784.25.

Therefore, the correct solution is:

X = 2316.54

In other words, the borrower should make equal payments of $2316.54 in 1 year, 3 years, and 5 years, at 4% p.a. to pay off the loan.

I hope this helps and please let me know if you have any further questions or concerns. Good luck with your homework!

 

[thepatientmental]

Overall, your solution looks correct and you have applied the correct formulas and methods for solving the problem. However, there are a few minor mistakes in your calculations.

Firstly, when calculating the present value of the debt at time k=0, you have made a mistake in the exponent of the first term. It should be -6 instead of -8. This is because the loan is due in 2 years, not 8 years.

Secondly, when solving for X, you have used the incorrect interest rate. The question states that the money is worth 4% effective, which means that the interest rate is already compounded and there is no need to compound it further. Therefore, the correct interest rate to use in the equation is 0.04, not 0.04^1.

Lastly, when solving for X, you have also made a mistake in the exponents of the second and third terms. They should be -2 and -4 respectively, corresponding to the number of years the payments are due (1 year, 3 years, and 5 years).

After correcting these mistakes, the correct solution should be:

X = 2791.94

Therefore, the borrower should make equal payments of $2791.94 at the end of 1 year, 3 years, and 5 years to equitably replace the given obligations.

Keep in mind that financial mathematics can be complex and it is always a good idea to double check your calculations and assumptions. It is also helpful to provide a clear explanation of your steps and assumptions to ensure accuracy. Overall, great job on your solution!