$($Running PV example$)$ A $($yearly$)$ cash flow stream is $x=(-40,10,10,10,10,10,10).$

The spot rates are those of Exercise $2.$

$($a$)$ Find the current discount factors $d_{0k}$ and use them to determine the $($net$)$ present value

of the stream

$($b$)$ Find the series of expectations dynamics short$-$rate discount factors and use the running present value method to evaluate the stream

spot rates: s $=(50,53,56,58,60,61)$