Question

A one-year, $100,000 loan carries a coupon rate and a market interest rate of 12 percent. The loan requires payment of accrued interest and one-half of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the year.

A. what will be the cash flows at the end of six months and at the end of the year?

B. What is the present value of each cash flow discounted at the market rate? what is the total present value?

C. What proportion of the total present value of cash flows occurs at the end of six months? what proportion occurs at the end of the year?

D. What is the duration of this loan?

Answer 1

A. what will be the cash flows at the end of six months and at the end of the year?

Amount of loan =$100,000

Coupon rate = 12% per year; 12%/2 =6% per six months

Therefore,

The cash flows at the end of six months = Amount of loan * coupon rate/2 = $100,000 * 12%/2 = $6000

And the cash flows at the end of the year = six-monthly coupon payment + principal

= Amount of loan * coupon rate/2 + Amount of loan

=$100,000 * 12%/2 + $100,000 = $106,000

B. What is the present value of each cash flow discounted at the market rate? what is the total present value?

We have market interest rate = 12% per year; six-monthly interest rate = 12%/2 = 6%

Time period of cash flows = 1 (for six –months) & 2 (for one year)

Therefore,

Present value of the cash flows at the end of six months

= cash flow/ (1+six-monthly market interest rate) ^ time period = $6000/ (1+6%) ^1 = $5,660.38

Present value of the cash flows at the end of the year

= cash flow/ (1+six-monthly market interest rate) ^ time period = $106,000/ (1+6%) ^2 = $94,339.62

Total present value = $5,660.38 + $94,339.62

= $100,000

C. What proportion of the total present value of cash flows occurs at the end of six months? what proportion occurs at the end of the year?

The proportion of the total present value of cash flows occurs at the end of six months = $5,660.38/$100,000 = 0.0566 or 5.66%

Proportion occurs at the end of the year = $94,339.62/$100,000 = 0.9434 or 94.34%

D. What is the duration of this loan?

Duration of this loan = (Present value of the cash flows at the end of six months * time period in years + Present value of the cash flows at the end of the year * time period in years)/ Total present value

= ($5,660.38 * 0.5 + $94,339.62 *1)/ $100,000

= $ 97,169.81 /$100,000

= 0.97 years