Question

You have a mortgage with a remaining balance of $160,000 and 200 more monthly payments.
The bank charges you 5% APR with monthly compounding on the loan. The current payments are $1,200 per month.

Due to covid-19 you have lost your job and need to skip the payments in months 1 to 5 .The bank offers you two alternatives.
In alternative 1 you will start making payments again, in month 6 but with new payments that take into account the interest the bank charges you over the months you skipped. You would have these new payments in months 6 to 205

In alternative 2, you keep the same payments of $1,200 and make these in months, 6 to 205 but pay a "delay fee" of $2,000 in month 5
Which alternative is cheaper in present value terms as of month 5 and by how much if you earn 7% on your investments?

Go with Alt 2. It is cheaper by $2,618.83

Go with Alt 1. It is cheaper by $1,354.31

Go with Alt 1. It is cheaper by $2,618.83

Go with Alt 2. It is cheaper by $1,354.31

Answer 1

Alternative I

Remaining Balance today = 160000

Interest rate per month = 5%/12 =0.004166666667

Number of interest accrued = 5

So balance at end of 5 month = Present value*(1+i)^n

160000*(1+0.004166666667)^5

$163,361.2271

Now this is balance at month 5 (P) =$163,361.23

Number of periods of repayment (n) = 200

Interest rate per month = 5%/12 =0.004166666667

so equal monthly payment payable for loan =Monthly payment formula = P*i/(1-((1+i)^-n))

163361.23*0.004166666667/(1-((1+0.004166666667)^-200))

1205.478326

Now we will calculate present value of $1205.48 for us.

Number of months (n) = 200

Opportunity cost of capital permonth (i)= 7%/12 =0.005833333333

Present value of annuity = =monthly payment *(1-((1+i)^-n))/i

1205.48*(1-((1+0.00583333333)^-200))/0.00583333333

142082.5815

Alternative II

delay fee paid at end of month 5 = $2000

Present value will be same at that time = $2000

Monthly payment made for 200 months = $1200

Number of months (n) = 200

Opportunity cost of capital permonth (i)= 7%/12 =0.005833333333

Present value of annuity = =monthly payment *(1-((1+i)^-n))/i

1200*(1-((1+0.00583333333)^-200))/0.00583333333

=141436.6873

Alternative II present value total = 2000+141436.6873

143436.6873

Alternative I present value is 142082.5815

Alternative II present value is 143436.6873

So alternative I is cheaper by 1354.11

Answer is II Go with Alt 1. It is cheaper by $1,354.31