Company DEF is considering the following three investments:

$$ Investment X is a $50$year zero$-$coupon bond

$$ Investment Y is a perpetuity with annual payments

$$ Investment Z is a geometrically increasing perpetuity with annual payments that increase at a

rate of $3\%$ per year.

For both perpetuities, the first payment occurs one year from now.

The convexities of Investments X$,$ Y $,$ and X are x$,$ y$,$ and z$,$ respectively, based on an annual effective

interest rate of $5\%.$ Determine an expression that represents the relationship between the convexity

of the three investments. Hint: for example you might determine that x $<$ y $<$ z$.$