Question

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A $10 000 bond with 5% interest payable quarterly, redeemable at par on November 15, 2030,...

A $10 000 bond with 5% interest payable quarterly, redeemable at par on November 15, 2030, was bought on July 2, 2014, to yield 9% compounded quarterly. If the bond sells at 92.75 on September 10, 2020, what would the gain or loss on the sale be?

Face value = 10 000.00; b = 1.25%

Principal = 10 000.00; i = 2.25%

Interest dates are November 15, February 15, May 15, and August 15.

The interest date preceding the date of sale is August 15, 2020.

The time period August 15, 2020, to November 15, 2030, is 10.25 years: n = 41.

b < i → discount

The interest payment interval August 15, 2020, to November 15, 2020, is 92 days

The interest period August 15, 2020, to September 10, 2020, is 26 days.

Expert Solution

Period	Discounting Factor [1/(1.0225^period)]	Discounting Factor Annuity (Sum of discounting factor & all previous discounting factors)
1	0.97799511	0.97799511
2	0.956474435	1.934469545
3	0.935427321	2.869896866
4	0.914843345	3.784740211
5	0.894712318	4.679452529
6	0.875024272	5.554476801
7	0.855769459	6.41024626
8	0.836938346	7.247184607
9	0.81852161	8.065706217
10	0.800510132	8.866216349
11	0.782894995	9.649111344
12	0.765667477	10.41477882
13	0.748819048	11.16359787
14	0.732341367	11.89593924
15	0.716226276	12.61216551
16	0.700465796	13.31263131
17	0.685052123	13.99768343
18	0.669977626	14.66766106
19	0.655234842	15.3228959
20	0.640816472	15.96371237
21	0.626715376	16.59042775
22	0.612924573	17.20335232
23	0.599437235	17.80278955
24	0.586246685	18.38903624
25	0.573346391	18.96238263
26	0.560729967	19.5231126
27	0.548391165	20.07150376
28	0.536323878	20.60782764
29	0.52452213	21.13234977
30	0.512980078	21.64532985
31	0.501692008	22.14702186
32	0.490652331	22.63767419
33	0.47985558	23.11752977
34	0.469296411	23.58682618
35	0.458969595	24.04579577
36	0.44887002	24.49466579
37	0.438992684	24.93365848
38	0.429332699	25.36299118
39	0.41988528	25.78287646
40	0.41064575	26.19352221
41	0.401609536	26.59513174
42	0.392772162	26.9879039
43	0.384129254	27.37203316
44	0.375676532	27.74770969
45	0.367409811	28.1151195
46	0.359324999	28.4744445
47	0.351418092	28.82586259
48	0.343685175	29.16954777
49	0.336122421	29.50567019
50	0.328726084	29.83439627
51	0.321492503	30.15588877
52	0.314418095	30.47030687
53	0.30749936	30.77780623
54	0.30073287	31.0785391
55	0.294115277	31.37265438
56	0.287643302	31.66029768
57	0.281313743	31.94161142
58	0.275123465	32.21673489
59	0.269069403	32.48580429
60	0.263148561	32.74895285
61	0.257358006	33.00631086
62	0.251694871	33.25800573
63	0.246156353	33.50416208
64	0.24073971	33.74490179
65	0.235442259	33.98034405
66	0.230261378	34.21060543

May 15, 2014 to November 15, 2030 = 66 Quarters

Flat Price or Clean Price (i.e. Price as on May 15, 2014) = PV of All Coupons + PV of Maturity Value = [Coupon*Annuity Factor] + [Maturity Value*Discounting Factor] = [10000*1.25%*34.2106] + [10000*0.23026] = 4276.325 + 2302.6 = $6578.925

Invoice Price or Dirty Price (as on July 2, 2014) = Flat Price + Accrued Interest till Purchase Date = 6578.925 + [10000*1.25%*48/92] = 86578.925 + 65.22 = $6644.145

Therefore, Purchase Price = $6644.145

Value as on August 15, 2014 = $6644.145 + Coupon for July 2 to August 15 i.e. 44 days = 6644.145 + (10000*1.25%*44/92) = 6644.145+59.78 = $6703.925

Period	Compounding Factor [1.0225^period]	Compounding Factor Annuity (Sum of discounting factor & all previous discounting factors)
1	1.0225	1.0225
2	1.04550625	2.06800625
3	1.069030141	3.137036391
4	1.093083319	4.230119709
5	1.117677693	5.347797403
6	1.142825442	6.490622844
7	1.168539014	7.659161858
8	1.194831142	8.853993
9	1.221714843	10.07570784
10	1.249203426	11.32491127
11	1.277310504	12.60222177
12	1.30604999	13.90827176
13	1.335436115	15.24370788
14	1.365483427	16.6091913
15	1.396206804	18.00539811
16	1.427621457	19.43301957
17	1.45974294	20.89276251
18	1.492587156	22.38534966
19	1.526170367	23.91152003
20	1.560509201	25.47202923
21	1.595620658	27.06764989
22	1.631522122	28.69917201
23	1.66823137	30.36740338
24	1.705766576	32.07316996
25	1.744146324	33.81731628

August 15, 2014 to August 15, 2020 = 24 Quarters and 25 Coupons will be received.

Future Value of Purchase Price as on August 15, 2020 = Value as on August 15, 2014 as above *Compounding Factor = 6644.145*1.7058 = $11333.58

Future Value of Coupons received as on August 15, 2020 = Coupon*Annuity Factor = (10000*1.25%*33.8173) = $4227.1625

Sales Value = 10000*92.75% = $9275

Gain/(Loss) = Sales Value + FV of Coupons - FV of Purchase Price = 9275 + 4227.1625 - 11333.58 = $2168.5825

jeff jeffy answered 2 years ago

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