Question

Chapter 9 from Financial institution management question#11

- you can obtain a loan of $100,000 at a rate of 10% for two years. you have two choices of paying

i) paying the interest (10%) each year and the total principal at the end of the second year or.

ii) amortizing the loan, that is, paying interest (10%) and principal in equal payments each year. the loan is priced per year.

a) what is the duration of the loan under both methods of payments?

b) explain the difference in the two results?

Answer 1

The question can be divided into two parts. First part is to calculate the duration of a loan under two different payment methods, the second part is the analysis of the difference between the two.

Lets first calculate the duration.

Duration of a loan = Present value of a loan's cash flows, weighted by length of time to receipt and divided by the bond's current market value.

Now lets calculate the duration for $100000 at a rate of 10% paying interest at the end of each year and principal at the end of second year.

By using the present value formula, we can find PV of Cash Flows for each period. So for the first row, we'd figure in ($10000) / (1 + 0.1)^1 = 90909.09. similarly we will calculate for other periods just like mentioned in the table.

period	cash flow	present value of $1 at 10%	PV of cash flow	period * cash flow	PV of period * cash flow
1	10000	0.90909	9090.909	10000	9090.909
2	110000	0.8264463	90909.09	220000	181818.18
		total	100000		190909.08

Duration = 190909.08/100000 = 1.9090

Similarly, we can calculate the duration for $100000 at a rate of 10% paying interest and principal at the end of each year.

period	cash flow	present value of $1 at 10%	PV of cash flow	period * cash flow	PV of period * cash flow
1	60000	0.90909	54545.4	60000	54545.4
2	55000	0.8264463	45454.54	110000	181818.18
		total	100000		145454.54

Duration= 145454.54/100000= 1.4545

Now, coming to the second part, we must understand what is duration.

Duration is a measure of a bond's sensitivity to interest rate changes. The higher the bond's duration, the greater its sensitivity to changes in interest rates (also known as volatility) and vice versa.

So, from the above calculations we can conclude that a coupon payment and a principal payment each year will reduce the duration and hence the sensitivity to rate changes.