It's simple math. A 10 year mortgage for 120K will cost $1160.00 per month for P&I. That mortgage will cost you 19K in interest over the course of the 10 year term. During that same term your 120K in savings at a 5% yearly return nets you 98K in interest. That's a 78K net return on investment over 10 years or 8K per year or $666.00 per month on average. If you pay off the 120K and invest the payment of $1160.00 per month over the course of 10 years at 5% interest, your total interest earned will be 43K. By holding onto his current mortgage and letting his investment mature he will see a net increase in savings of 35K or $291.00 per month.
The problem is everyone situation is different. You assumed this person had a 10 year mortgage. Most people have a 30 year mortgage and the beginning balance is going to be much higher than $120k. Therefore the P&I is going to be higher.
You example is very simple. All you are looking at is interest. You are ignoring the $1160/mo investment every month for 10 years that the person would be making by not having a mortgage now. There is also more that can be taken into account, but that will vary upon person to person and their situation.
if you pay off mortgage and invest that $1160/mo for 120 months @5% you will end up at $173k.
If you don't pay off mortgage and put $120k into an investment that will pay you 5% for 120 months you will end up at $182k
So you made $9k more over the 120 months for an average of $75/mo
Now you have to subtract out the $19k in mortgage interest you had to pay
Suddenly I am better off by $10k. The gap would be even wider because the mortgage amount would have been even higher meaning the $1160 monthly investment would be more and the interest would have been more than $19k.