That untrue. If they held the bonds to maturity they would be worth more than their face value. However due to time value of money other bonds with higher yield

That untrue. If they held the bonds to maturity they would be worth more than their face value. However due to time value of money other bonds with higher yields are in more demand making the current price lower. A private entity wouldn’t take the risk but a public one would. In fact the fed (quasi public) has been buying precisely these assets for over a decade. They actually made substantial returns by simply holding these MBS to maturity.