老师，课上不是说interest rate的变化不影响 amortizatized cost 嘛，那为什么这题要这样算？-有问必答-品职教育专注CFA ESG FRM CPA 考研等财经培训课程

老师，课上不是说interest rate的变化不影响 amortizatized cost 嘛，那为什么这题要这样算？

NO.PZ2019120301000278

问题如下：

Question On 1 January, a corporation issues ten-year notes with a face value of €10,000,000 and with annual interest payments made each 31 December. The coupon rate is 2.0 percent, and the effective interest rate is 3.0 percent. Using the effective interest rate method, the amortized discount at the end of year 1 is closest to:

选项：

A.€ 74,409.00 B.€ 274,409.00 C.€ 82,035.00

解释：

Solution

A is correct. To determine the amortized discount at the end of year 1, first calculate the bond’s price at issuance.

The bond at issuance is priced at: €9,146,979.72, calculated as:

=C[1−(1+i)−Ni]+M(1+i)−N=€200,000[1−(1+0.03)−100.03]+€10,000,000(1+0.03)−10=€1,706,040.57+€7,440,939.15=€9,146,979.72≈€9,146,980 $\begin{array}{l} = C [\frac{1 - {(1 + i)}^{- N}}{i}] + M {(1 + i)}^{- N} \\ = € 200, 000 [\frac{1 - {(1 + 0.03)}^{- 10}}{0.03}] + € 10, 000, 000 {(1 + 0.03)}^{- 10} \\ = € 1, 706, 040.57 + € 7, 440, 939.15 \\ = € 9, 146, 979.72 \\ \approx € 9, 146, 980 \end{array}$

Using a financial calculator:

FV = €10,000,000
PMT = €200,000
N =10
I/Y = 3.00%
CPT PV = €9,146,979.72

Next, determine the interest expense (based on the effective interest rate and amortized cost).

The effective interest rate is 3.0%, so the interest expense is calculated as:

Interest expense = €9,146,979.72 * 0.03 = €274,409.

Next, determine the interest payment (based on the coupon rate and face value).

The interest payment is calculated as:

Interest payment = 2.0% × €10,000,000 = €200,000.00.

Finally, the difference between the interest expense and the interest payment is the amortized discount: Amortized discount = €274,409 – €200,000.00 = €74,409

C is incorrect because €82,035 represents the amortized premium if the effective rate is 2% and the coupon rate is 3% and the premium is incorrectly calculated by subtracting interest expense from interest payment, instead of the opposite difference, to determine the amortization.

The bond at issuance is priced at: €10,898,258.50, calculated as:

=C[1−(1+i)−Ni]+M(1+i)−N=€300,000[1−(1+0.02)−100.02]+€10,000,000(1+0.02)−10=€2,694,775.50+€8,203,483.00=€10,898,258.50≈€10,898,259 $\begin{array}{l} = C [\frac{1 - {(1 + i)}^{- N}}{i}] + M {(1 + i)}^{- N} \\ = € 300, 000 [\frac{1 - {(1 + 0.02)}^{- 10}}{0.02}] + € 10, 000, 000 {(1 + 0.02)}^{- 10} \\ = € 2, 694, 775.50 + € 8, 203, 483.00 \\ = € 10, 898, 258.50 \\ \approx € 10, 898, 259 \end{array}$

Using a financial calculator:

FV = €10,000,000
PMT = €300,000
N =10
I/Y = 2.00%
CPT PV = €10,898,258.50

The effective interest rate is 2.0%. The interest expense is calculated as:

Interest expense = €10,898,258.50 × 0.02 = €217,965.

The interest payment is calculated as:

Interest payment = 3.0% × €10,000,000 = €300,000.00.

The calculated difference between the interest payment and the interest expense is the misinterpreted amortized discount:

€300,000.00 – €217,965 = €82,035.

B is incorrect because €274,409 represents the interest expense (not the amortized discount).

老师，课上不是说interest rate的变化不影响 amortizatized cost 嘛，那为什么这题要这样算？

嗨，爱思考的PZer你好：

课上说的interest rate是市场利率，这道题中2%是coupon rate，3%是市场利率， effective interest rate，我们看一下这一道题为什么是解答中的计算方式。

首先，由于本题中，债券的实际利率（3%）大于coupon rate（2%），所以该债券是折价发行的（发行价小于面值）。此时债券的发行价和面值之间会存在一个差值——这就是未来几年需要被慢慢摊销的折价的总的值。

我们知道，无论折价发行或是溢价发行的债券，随着到期日的临近，债券的价值一定会回归到面值的部分。但是期初的时候，由于债券是折价发行的，所以期初我们账面上记录的债券的carrying value，就是期初的发行价（而非面值）

而这个差值就会在未来的几年里慢慢以一定的方式计增加到债券的carrying value里面，目的是使最终债券到期的时候，carrying value = 债券面值。

具体来说，以本题为例，期初刚发行该债券的时候，该债券的发行价=carrying value = 9146979.42

而第一年年末的时候，如果我们再计算一次债券的市场价值（PMT=10,000,000*2%, I/Y =3, N =9[因为债券还有9年就到期了], FV=10,000,000),可以得到此时的债券价值=9221389.11

9221389.11大于9146979.42，因为债券距离到期日近了一年，所以价值会向面值更靠近一步，而此时9221389.11-9146979.42=74409.11

这个黄色的两年债券价值的差值，就是债券折价的摊销部分

会计上，第一年年末债券的账面价值就会由原来的9146979.42向上摊销至9221389.11，摊销部分74409.11计增加第一年的利息费用支出

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老师，课上不是说interest rate的变化不影响 amortizatized cost 嘛，那为什么这题要这样算？

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