In addition, investors can borrow money at an annual interest rate of 6% or 5.5%. If investors choose to borrow money, they need to repay the principal plus interest after one year. If he borrows X dollars, he needs to pay x*( 1+6%)= 1.06x dollars. On the contrary, if investors choose to lend funds, they can recover the principal and interest after one year. If they lend Y dollars, they can recover Y * (1+5.5%) =1.055Y dollars.
If investors arbitrage, two conditions should be met:
The capital needed to buy gold is not higher than the borrowing cost.
That is, 560m ≤1.06x.
The proceeds from the sale of gold shall not be lower than the loan interest.
That is, 559m ≥1.055y.
By sorting out the above two formulas, we can get:
m/x ≤ 1.06/560
m/y ≥ 1.055/559
Since m is the quantity of gold purchased, it must be a positive number, so the first inequality requires that the left side should not be less than zero, that is, x ≥ (1.06/560) m. Take the reciprocal of the right side and multiply it by 560 to get 560x≥ 1.06m, that is, x ≥ (1.06/560).
m/( 1.055/559) ≤ y
Similarly, because Y is a loan fund, it must be a positive number, so the above formula requires that the right side should not be less than zero, so there is y≥( 1.055/559)m, and the left side is multiplied by (1.055/559) to get:
( 1.055/559)m ≤ y
Combining the above two inequalities, we can get:
( 1.055/559)m≤y≤m/( 1.06/560)
That is to say, the precondition for investors to carry out arbitrage is that the gold price fluctuates in the range of [( 1.055/559) × 560, (1.06/560) × m]. In other words, investors can arbitrage only when the price of gold rises or falls beyond this range.