# Why YTM is higher than current yield in discount bond

I learned that in discount bond,

``````coupon rate < current yield < YTM
``````

and the relationship will always hold.

I can figure out why `coupon rate` < `current yield`. But just cannot don't understand why does `current yield` < `YTM` hold anyway?

• the YTM includes the fact that the bond's principal is paid back on maturity, the other two don't. – serakfalcon Sep 26 '14 at 9:18
• principal is discounted by YTM and coupon is not discounted in current yield. How could we know the yield after discount is large than current yield? – Marcus Thornton Sep 26 '14 at 12:59

## 1 Answer

Say you buy a bond that currently costs \$950, and matures in one year, at \$1000 face value. It has one coupon (\$50 interest payment) left.

The coupon, \$50, is 50/950 or 5.26%, but you get the face value, \$1000, for an additional \$50 return. This is why the yield to maturity is higher than current yield.

If the maturity were in two years, the coupons still provide 5.26%, and the extra 1000/950 is another 5.26% over 2 years, or (approx) 2.6%/yr compounded, for a total YTM of 7.86%.

This is a back-of envelope calculation, the real way to calculate is with a finance calculator. Entering PV (present value) FV (future value) PMT (coupon payment(s)) and N (number of periods). With no calculator or spreadsheet, my estimate will be pretty close.

• In this case YTM is `(50/950 + 1000/950) -1` so it's easy to understand YTM is larger than current yield. But what if it matures in two years? I cannot generalized to two years because the computation is more complicated and I cannot make sure YTM is larger than current yield. – Marcus Thornton Sep 26 '14 at 13:10
• I updated/edited to address multiple years. – JTP - Apologise to Monica Sep 26 '14 at 13:38
• It seems that the key point is that no matter how long the maturity is, the discounted principal still will contribute to a positive yield. This pluses the contribution of current yield will hold `current yield < YTM` always true. – Marcus Thornton Sep 26 '14 at 14:18