The group's margin is now 10% but, with competitors such as Barratt, Persimmon and Bellway posting 14-16%, Johnson is under pressure to improve Wimpey's returns.
Johnson said his first 10 months in the job had been "so far, so good". His first few months were taken up with the merger of the group's Wimpey Homes and McLean Homes subsidiaries, in which 435 jobs were axed and the regional set-up reorganised.
He said that raising margins was now his top priority. "We're still lagging behind our competitors but in three years our margins will be no worse than our peers.
"There is no reason why they should be any worse once we have made all the changes."
He said profit could be improved by cutting the cost of procurement and building design, buying better land and raising the average selling price by offering optional extras.
Johnson made his promise on margins as he announced Wimpey's interim results for the six months to 30 June, which were ahead of market forecasts.
Despite a warning earlier in the year that first-half performance would not match 2000, pre-tax profit rose 10.6% to £53.1m. This was affected by the £14.2 cost of the merger of Wimpey and McLean. Turnover improved £12.6m to £706.7m.
The merger is expected to save Wimpey £20m a year.
Johnson said: "When I joined, I knew this was a business that needed shaking up.
"Everyone knows Wimpey builds well, but there were savings that could be made. The staff knew how they could save on costs and had already thought about it – all I did was free them to do it."
The UK housing business, which accounts for 74% of group turnover, was affected by poor weather and fall in the number of its sites. Turnover fell 4% to £512m, but the average selling price rose 8% to £118,000.
Johnson highlighted the strong performance of Wimpey's US operation, Morrison Homes.
Morrison has businesses in six states. It completed 1142 sales, up 21% on the same period last year. Turnover rose 34% to £189m and operating profit jumped 96% to £15.3m, in part because of favourable exchange rates. The average selling price rose 5.5% to £164,000.
Johnson said: "Morrison Homes is a very good performing company that is very customer focused, and it will grow."