SOA Exam P Question: Exponential Distribution?

Question

> Suppose you can borrow money at an annual interest rate of $8 \\%$ but can save money at an annual interest rate of only $5\\%$. If you start with zero capital and if the yearly cash flows of an investment are $$(-1000, 900, 800, -1200, 700)$$ should you invest?To solve this problem, I attempted to compute present value of the cash flow sequence. Here, $-1000$ must represent borrowing money because we're starting with $0$ money. So I got$$-1000 + \\frac{900}{1 + 0.05} + \\frac{800}{(1 + 0.05)^{2}} - \\frac{1200}{(1 + 0.08)^{3}} + \\frac{700}{(1 + 0.05)^{4}} \\approx 206.059483 > 0,$$so my answer is yes. Did I solve this problem correctly? Or, perhaps, are my interest rates in the denominators swithced?Thanks

Answer 1

all of the dollar signs and backslashes confuse the heck out of me.

Your cash Flows look something like this.

Year Cash Flow

0 -1000

1 900

2 800

3 -1200

4 700

At the end of year one you will have a loan balance of 1080 (the loan accrues at 8%). in year 1 you have a cash flow of 900. if you put the money on deposit you would only earn 5%. If you paid down 900 of your loan balnce you would be reducing your 8% interest charges. You will be effectively earning 8% by doing so.. That will leave you with a loan balance of 180 which will accrue at 8% for another year.

In year 2 you receive 800 dollars if you pay off your existing debt you can put the balance of $605.6 on deposit which will grow at 5%.

In year 3 you have an outflow of 1,200. Part of this can be funded by your deposit (effectively borrowing at 5%) and the balance funded at 8%.

In year 4 you pay off the second loan with the 700 cash flow and are left with $90.75. If you discount this back to present value at 5% it is how much you would have to put on deposit to have 90.75 in 4 year. If you discount it at 8% it is how much additional money you could borrow using the Future value of the cash flows as collateral..

Year Cash flow Balance

0 -1000 -1000

1 900 -180

2 800 605.6

3 -1200 -564.12

4 700 90.75

I hope this helps. Please feel free to reach out if you need help with this material.