In: Finance
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.
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