FX Forecasting $$ Market Efficiency: Suppose you work for a U$.$S$.$ firm. You receive the following quarterly Consumer Price Index $($CPI$)$ information for the US and the UK from the first quarter of $2023$ to the second quarter of $2024.$ You also receive Spot rates of the British pound during that period. Your job is to generate estimates of the equilibrium exchange rates using PPP and make a forecast of the spot rate for the second quarter of $2024$ using the forecasted inflation rates provided by the consumer price research team in your firm:

DateCPI USCPI UKInflation US $($$)$Inflation UK $($$)$PPP Estimate $($S$)$Actual $($S$)$Forecast Error e$=$ S$-$ S$2023$:I$108\mathrm{.}6106\mathrm{.}21\mathrm{.}97542023$:II$111\mathrm{.}0108\mathrm{.}21\mathrm{.}99142023$:III$112\mathrm{.}3109\mathrm{.}31\mathrm{.}77052023$:IV$109\mathrm{.}1108\mathrm{.}41\mathrm{.}43782024$:I$108\mathrm{.}6106\mathrm{.}12024$:II$109\mathrm{.}7106\mathrm{.}91\mathrm{.}4381$

a$.$ Determine the equilibrium spot rates from $2023$:II to $2024$:II using PPP$.$

b$.$ Determine the forecast errors from $2023$:II to $2024$:I of the future spot rate based on PPP$.$ Compute the mean squared error $($MSE$)$ of these forecasts.

c$.$ You receive forward rate information for the $90$ day forwards during the period $2023$:II to $2024$:I, they are:

$1\mathrm{.}9992(2023$:II$)$; $1\mathrm{.}8123(2023$:III$)$; $1\mathrm{.}5298(2023$:IV$)$ and $1\mathrm{.}4401(2024$:I$).$ Use these rates to compute the MSE of the three$-$month forward as a forecast of the spot rate.

d$.$ Based on the MSEs you calculated in b$.$ and c$.$ what recommendation would you give your boss: to use the PPP based forecast of the future spot rate or rely on the forward to make predictions on future spot rates? Explain your reasons.