YTM of a zero-coupon bond

Let the yield $y$ that makes $\$1,000$ in $5$ years to have a present value of $\$900$ . Therefore,

$\dfrac {1000}{(1+y)^5} = 900$

$\Rightarrow y = \sqrt [5] {\frac {1000}{900}} - 1 = 0.0213 = \boxed{2.13\%}$

The value will multiply by $\frac{1000}{900}$ over 5 years, and by the same factor each year, due to compound yield. The factor must thus be $\sqrt[5]{\frac{1000}{900}}=1.0213$ , so the percentage yield is $100(1.0213-1)=\boxed{2.13 \%}$