Question

Hands Insurance Company issued a $90 million, 1-year, zero-coupon note at 8 percent add-on annual interest (paying one coupon at the end of the year). The proceeds were used to fund a $100 million, 2-year commercial loan at 10 percent annual interest. Immediately after these transactions were simultaneously closed, all market interest rates increased 2 percent .

a. What is the true market value of the loan investment and the liability after the change in interest rates?

b. What impact did these changes in market value have on the market value of the equity?

c. What was the duration of the loan investment and the liability at the time of issuance?

d. Use these duration values to calculate the expected change in the value of the loan and the liability for the predicted increase of 2 percent in interest rates.

Answer 1

a)

Given

Bonds issued = $90,000,000

Proceeds used to fund $100,000,000

Time period n= 2 years

Annual interest rate = 10%

Market rate increased by 1.5%

So interest rate will be = 10% + 1.5%= 11.5%

Let us calculate the market value of investment

          MVA = ($100,000,000 - $90,000,000)*PVIFAn=2, i=11.5% + $100,000,000* PVIFn=2, i=11.5%

Where PVIFA = [(1+i)ⁿ – 1]/[ i * (1+i)ⁿ ]

PVIFA = [(1+0.115)² – 1]/[ 0.115 * (1+0.115)² ]

= 1.701220616

PVIF = 1/ (1+i)ⁿ = 1/ (1+0.115)² = 0.8043596292

MVA = $97,448,169.08

So, the market value is $97,448,169.08

Which is $2,551,83 0.92 worth lower than the loan.

Now, liability is calculated as

At the end of one year the coupon gives amount of $90,000,000 * (1 + 0.08)¹ = $$97,200,000

So, immediately after this transaction, the liability or the market value will be

          MVL = $97,200,000* PVIFn=1, i=9.5%

Here interest rate is changed as 10% - 1.5% = 9.5%

PVIFn=1, i=9.5% = 1/ (1 + 0.095)¹ = 0.9132420091

MVL = $97,200,000* 0.9132420091 = $88,767,123.29

The market value of the note declined by $1,232,876.71 to $88,767,123.29.

b)

change in market value of equity = Change in asset– change in liability

∆E = ∆A - ∆L = -$2,551,831 – (-$1,232,877) = -$1,318,954.

The increase in interest rates caused the asset to decrease in value more than the liability which caused the value of the net worth to decrease by $1,318,954.

c)

What was the duration of the loan investment and the liability at the time of issuance?

Two-year Loan(values in millions of $s)

Par value = $100,000,000

Coupon rate = 10%

Payments are made as annual payments

R = 10%

Maturity = 2 years

Time	Cash Flow      (value in millions)	PV of CF	CF     PV of CF x t
1	$10	$9.091	$9.091
2	$110	$90.909	$181.818
		$100.000	$190.909

Duration = $190.909/$100.00 = 1.9091

So, the duration of the loan investment is 1.9091 years. The duration of the liability is one year since it is a zero-coupon note.

d)

The approximate change in the market value of the loan for a 150 basis points change is:

Change in market value of loan investment =

-1.9091 * (0.015/1.1) * $100,000,000 = -$2,603,318.18

The expected market value of the loan using the above formula is $100,000,000 -$2,603,318.18 = $97,396,681.82

The approximate change in the market value of the note for a 150 basis points change is

Change in market value of loan investment =

-1.0 * (0.015/1.08) * $90,000,000 = -$1,250,000

The expected market value of the loan using the above formula is $90,000,000 -$1,250,000 = $88,750,000