Cessna 172 Instructor Pilot reported engine power loss on short final and landed off-field short of the runway. Aircraft touched down in a nose down attitude; instructor and student were not injured and exited the aircraft.

Narrative

On Day 0; at approximately XA15; I was riding right-seat in a C-172 providing instrument instruction to Student when a loss of power occurred at approximately 500 ft. AGL on final approach for Runway XX into ZZZ. This resulted in field landing short of Runway XX. The landing did not result in any injuries and both Student and I were able to exit the aircraft through the pilot-side door and walk to the airport. Because the aircraft landed in a snow covered field; there was damage to the aircraft as the nose gear entered the snow and pulled the plane down to the ground. The plane remained upright. The flight began at approximately XA40 leaving from ZZZ. We filed an IFR flight from ZZZ-ZZZ1-ZZZ2-ZZZ3-ZZZ. This flight was the long-distance flight to meet the IFR training requirement for Student. Prior to the flight; a preflight inspection was completed verifying we had full-tanks of fuel to ensure we had the proper fuel onboard to complete the flight with 90 minutes of reserve fuel landing in ZZZ. During the preflight briefing; we discussed the route; weather conditions and frequencies to be expected along the route. All information suggested a routine flight with no concerns for weather. We taxied to Runway XX; successfully completed the runup; and agreed to depart. We departed ZZZ and climbed to an elevation of 5;000 MSL. We contacted ATC in the air and were cleared for the IFR flight as filed. The first leg of the flight from ZZZ to ZZZ1 ended in a low approach and we continued from ZZZ1 to ZZZ2. After the low approach at ZZZ1; we contacted ATC and continued to ZZZ2 for another low approach and continued to ZZZ3. As we approached ZZZ3; we were cleared for the RNAV XX approach and handed off to ZZZ3 Tower. We completed the low approach at ZZZ3 and cancelled the IFR flight plan. We were cleared to depart northwest directly to ZZZ. Conditions were VFR with unlimited ceilings and visibility. Temperature was approximately -5 F so the flight from ZZZ3 to ZZZ was conducted visually. We climbed to approximately 3;000 MSL and continued until we visually confirmed the airport. We reported on the CTAF; XXX.X approximately 10 miles to the southwest inbound for a left base for Runway XX to ZZZ. The winds at ZZZ were reporting 300 at 06 kts. Prior to entering the traffic pattern at ZZZ; Student conducted the C-GUMPS checklist (as we had previously done for each low approach prior to the FAF). As we entered the left base for Runway XX; Student made a radio call announcing inbound for XX and we performed the C-GUMPS checklist for a second time. We turned on a 2.5 mile final for Runway XX and we conducted and verified C-GUMPS for a final time. Student reduced power to 1800-1900 RPMS and trimmed for a 80 MPH; 500 FPM descent. At approximately; 500 ft. AGL; Student called a loss of engine power. He pulled the throttle back slowly and pushed forward slowly to get the engine to respond. The throttle was slightly retarded and then slowly pushed forward since we feared putting too much fuel and choking the engine. The engine didn't respond and didn't increase in power. With no increase in power and already losing 200 ft. since loss of power was notice; I determined the plane would not be able to reach the runway. I assisted on the yoke and noted that the plane needed to complete a landing in the snow field ahead and the most important consideration was clearing a fence in our path. The plane cleared the fence and I felt the plane touch down in the snow. Immediately upon landing in the snow the plane abruptly obtained a nose-down attitude and stopped. I confirmed there was no injuries and neither of us impacted the yoke; dashboard or windshield. We shut the master switch off and turned the key to off. We both unbuckled our seatbelts and had to exit the plane from the pilot door. The seats were in a forward position and gravity made it difficult to push the seats back. Outside of the aircraft; I confirmed that there were no obvious injuries andno pair was reported by either Student and myself. After calling the owner/operator of the aircraft we walked the remaining distance to the airport. Throughout the entire flight; I never observed a loss of oil pressure or any engine instrument reading outside of normal operations. We used carb heat on each low approach and on the final landing; but never observed any loss of power in the flight due to observed carburetor icing. Except for the unexpected loss of power on final; the flight was uneventful and there was never any obvious indication of the expected loss of power.