Why do we need the YTM of zero coupon bonds?

Suppose the 1-year, 2-year and 3-year zero-coupon bonds are trading at $97, $98 and $99, respectively. If the no-arbitrage condition holds, what is the market price of a 3-year coupon bond with $5 coupon payment?

Firstly, we find YTM for each year from zero coupon bonds:

YTM =(100/97)^(1/1)-1 =%3,09

YTM =(100/98)^(1/2)-1 =%1,02

YTM =(100/99)^(1/3)-1 =%0,34

Discounted Cash Flows= 5/(1+%3,09)^1=4,9

Discounted Cash Flows= 5/(1+%1,02)^2=4,9

Discounted Cash Flows= 105/(1+%0,34)^3=104