Suppose New Zealand has three major trade partners, i.e., Australia, China, the United States, each accounting for 30% of NZ's foreign trade. The rest of the world accounts for 10% of NZ's foreign trade. NZ, Australia and the United States adopt the floating exchange rate regimes, while China pegs its currency to the US dollar.

The RBNZ constructs two trade-weighted indices (TWIs). The narrow TWI only includes the Australian dollar and the US dollar, while the broad TWI includes the Australian dollar, the US dollar and the Chinese Yuan.

• Create numerical examples and show whether the broad TWI is less volatile than the narrow TWI. Is this always the case?
• Then, explain your findings intuitively.

include numerical examples and economic arguments.