Question

Suppose you get $500, 000 mortgage loan from a bank and you have a two options to repay.

option one: pay $620, 000 after 10 years as a one time payment

2. Repay $30,000 at the end of every year for infinite years

Which has a lower interest rate?

please describe how to get the answer, thanks

Answer 1

Present value of mortgage loan = $500, 000

Option 1: Pay $620, 000 after 10 years as a one time payment

Option 2: Repay $30,000 at the end of every year for infinite years

Step 1 : Calculation of interest rate in Option 1

Option 1: Pay $620, 000 after 10 years as a one time payment
Future value= $620,000
Time period = 10 years

PV = FV / (1+r)^t
where PV = Present Value
FV = Future Value
r = Interest rate
t = Time period

$500,000 = $620, 000 / (1+r)^10
(1+r)^10 = 1.24

Note: (1+r)^10 = 1.24, indicates Compound  Value of r for 10 years = 1.24.
Using Compound value table, we will search in time period = 10 years at which rate the compound value = 1.24 or close to 1.24

Rate	Compound  Value (r,10)
2%	1.2189944
3%	1.3439164

This means "r" lies between 2% & 3%

Using Interpolation

r = 2% + (1.24 - 1.2189944) / (1.3439164 - 1.2189944)
r = 2% + 0.021005 / 0.124922
r = 2% + 0.1681
r = 2.1681% or 2.17% (approx).

Interest rate in Option 1 = 2.17%

Step 2 : Calculation of interest rate in Option 2

Option 2: Repay $30,000 at the end of every year for infinite years

PV = Cash Flow / r
where PV = Present value
Cash flow = Monthly payment
r = Interest rate

500,000 = 30,000 / r
r = 0.06 or 6%

Interest rate in Option 2 = 6%

Therefore , We can conclude that Option 1 has a lower interest rate