The current spot gold price is $\$1788$ per ounce. The storage cost is $\$24$ per ounce per year, to be paid quarterly at the beginning of each quarter. Suppose the current $3$-month and $6$ month spot rates are $0.15\%$ and $0.20\%$ (both are annualized with quarterly compounding) respectively. A trader decides to enter into a forward contract to purchase $5000$ ounces of gold in $6$ months.
For the forward price for 6 month contract, I got $\$1801.44$ for $F0$.
The next part of the question I'm unsure what happen when the spot rates changes.
Suppose three months later the spot gold price has risen to $\$1888$ per ounce, and the $3$-month and $6$-month spot rates at that time are $0.12\%$ and $0.15\%$ (both are annualized with quarterly compounding) respectively. If the storage cost remains the same, determine the value of the forward contract to the trader at that time.
What I have gotten is $\$123.10$. However, I'm not sure if my understanding is correct. I only use the new $3$-month spot rate in this part.
Will appreciate if someone can explain in detail for the second part.