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Important Equation

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**1. **Important Equation! Discount or Premium = (Total payable to bondholders at the end) – (Cash received at the beginning, without accrued interest)
If answer is positive, then it’s a discount
If answer is negative, then it’s a premium
HINT: this equation can be used for bonds with detachable warrants: just include the value of the detachable warrants in the “Total payable to bondholders at the end” amount

**2. **Exercise 14-1 Requirement 1
$100 million x 12% x 2/12 = $2 million face annual fraction of the accrue amount interest rate annual period
Requirement 2
($ in millions)
Cash ($99 million plus accrued interest) 101 Discount on bonds ($100 million – $99 million) 1 Bonds payable (face amount) 100 Interest payable (accrued interest determined above) 2

**3. **Exercise 14-4 1. Price of the bonds at January 1, 2000
Interest $4,000,000¥ x 11.46992 * = $45,879,680 Principal $80,000,000 x 0.31180 ** = 24,944,000 Present value (price) of the bonds $70,823,680
¥ 5% x $80,000,000
* present value of an ordinary annuity of $1: n=20, i=6%
** present value of $1: n=20, i=6%
2. January 1, 2000
Cash (price determined above) 70,823,680 Discount on bonds (difference) 9,176,320 Bonds payable (face amount) 80,000,000
3. June 30, 2000
Interest expense (6% x $70,823,680) 4,249,421 Discount on bonds payable (difference) 249,421 Cash (5% x $80,000,000) 4,000,000
4. December 31, 2000
Interest expense (6% x [$70,823,680 + 249,421) 4,264,386 Discount on bonds payable (difference) 264,386 Cash (5% x $80,000,000) 4,000,000

**4. **Exercise 14-5
1. January 1, 2000
Interest $4,000,000¥ x 11.46992 * = $45,879,680 Principal $80,000,000 x 0.31180 ** = 24,944,000 Present value (price) of the bonds $70,823,680
¥ 5% x $80,000,000
* present value of an ordinary annuity of $1: n=20, i=6%
** present value of $1: n=20, i=6%
Bond investment (face amount) 80,000,000 Discount on bond investment (difference) 9,176,320 Cash (price determined above) 70,823,680
2. June 30, 2000
Cash (5% x $80,000,000) 4,000,000 Discount on bond investment (difference) 249,421 Interest revenue (6% x $70,823,680) 4,249,421
3. December 31, 2000
Cash (5% x $80,000,000) 4,000,000 Discount on bond investment (difference) 264,386 Interest revenue (6% x [$70,823,680 + 249,421) 4,264,386

**5. **Exercise 14-6
1. Price of the bonds at June 30, 2000
Interest $58,500¥ x 15.04630 * = $880,209 Principal $900,000 x 0.09722 ** = 87,498 Present value (price) of the bonds $967,707
¥ 6.5% x $900,000
* present value of an ordinary annuity of $1: n=40, i=6%
** present value of $1: n=40, i=6%
2. June 30, 2000
Cash (price determined above) 967,707 Bonds payable (face amount) 900,000 Premium on bonds (difference) 67,707
3. December 31, 2000
Interest expense (6% x $967,707) 58,062 Premium on bonds payable (difference) 438 Cash (6.5% x $900,000) 58,500
4. June 30, 2001
Interest expense (6% x [$967,707 – 438) 58,036 Premium on bonds payable (difference) 464 Cash (6.5% x $900,000) 58,500

**6. **Exercise 14-14
Requirement 1
($ in millions)
Limbaugh (Issuer)
Cash (104% x $30 million) 31.2
Discount on bonds payable (difference) 3.6 Bonds payable (face amount) 30.0
Paid-in capital – stock warrants outstanding
($8 x 20 warrants x [$30,000,000 ÷ $1,000] bonds) 4.8
Interstate (Investor)
Investment in stock warrants ($4.8 million x 20%) 0.96 Investment in bonds (20% x $30 million) 6.00 Discount on bonds (difference) 0.72 Cash (104% x $30 million x 20%) 6.24

**7. **Problem 14-7
Requirement 1
Cash (price given) 5,795,518 Discount on bonds (difference) 12,204,482 Bonds payable (face amount) 18,000,000
Requirement 2
The discount rate that “equates” the present value of the debt ($5,795,518) and its future value ($18,000,000) is the effective rate of interest:
$5,795,518 ÷ $18,000,000 = .32197 – the Table 6A-2 value for n = 10, i = ?
In row 10 of Table 6A-2, the value .32197 is in the 12% column. So, this is the effective interest rate. A financial calculator will produce the same rate.

**8. **Critical Thought Case 14-7 The company's accountant is incorrect in valuing the note at $200,000. The note should be valued at the present value of the receivable using the prevailing market rate and the difference between the present value and the cash given is regarded as an addition to the cost of products purchased during the contract term.
In this case, the note would be valued at $136,602, computed as follows:
PV = $200,000 x .68301
PV of $1:
n=4, i=10% (from Table 6A-2)
PV = $136,602