Question

a) $200,000 U.S. Treasury 7 7/8% bond maturing in 2002 purchased and then settled on October 23, 1992, at a dollar price of 105-20 (this is the clean price) with a yield to maturity of 7.083% with the bond originally being issued at 11/15/1977. Face value per unit is $1,000.

i) Calculate the clean price of the bond issue

ii) Calculate the accrued interest of the bond issue

iii) Calculate the full price of the bond issue

b) $200,000 U.S. Treasury 7 7/8% bond maturing in 2002 purchased and then settled on October 23, 1992, at a dollar price (clean price) with a yield to maturity of 7.083% with the bond originally being issued at 11/15/1977. Calculate the full price (per unit of the bond). Note: On a per unit basis, the answers to(a)and(b)should be the same.Anydifferencemustbe due to rounding error only.

Answer 1

Treasury bonds usually pay coupon payments semi-annually.

As the bonds were issued on Nov 15, 1977 , the coupon payment dates will fall on Nov 15 and May 15 of every year.

In order to calculate the full price, we need to calculate the fraction of period between the settlement date and the next coupon date.

Number of days between settlement date (Oct 23,1992) and next coupon payment date (Nov 15, 1992) = 23 days (9 days of Oct (including 23^rd) and 14 days of Nov (excluding 15^th) )

Considering 182 days in six months period, if the fractional time period between settlement date and next coupon payment date is denoted by w,

Then, w= 23/182 = 0.12637

Now, we know that the coupon rate is 7 7/8 % i.e., 7%+0.875% = 7.875%

Therefore, in dollar terms each semi-annual coupon payment c = (1/2)x(7.875%)x1,000 =$39.375

(as coupon is calculated on face value, which in this case is $1000)

Based on this information, we can calculate the full price, which will be the present value of all future (expected) cash flows, as follows:

Coupon dates	Number of periods (t)	Time period considering the fraction t-1+w	Expected cash flow ©	Present value of cash flows [= Expected cash flow/(1+i)^(t-1+w)]
15-Nov-92	1	0.1264	39.375	39.0359
15-May-93	2	1.1264	39.375	36.4539
15-Nov-93	3	2.1264	39.375	34.0427
15-May-94	4	3.1264	39.375	31.7909
15-Nov-94	5	4.1264	39.375	29.6881
15-May-95	6	5.1264	39.375	27.7244
15-Nov-95	7	6.1264	39.375	25.8906
15-May-96	8	7.1264	39.375	24.1780
15-Nov-96	9	8.1264	39.375	22.5788
15-May-97	10	9.1264	39.375	21.0853
15-Nov-97	11	10.1264	39.375	19.6906
15-May-98	12	11.1264	39.375	18.3882
15-Nov-98	13	12.1264	39.375	17.1719
15-May-99	14	13.1264	39.375	16.0361
15-Nov-99	15	14.1264	39.375	14.9754
15-May-00	16	15.1264	39.375	13.9848
15-Nov-00	17	16.1264	39.375	13.0598
15-May-01	18	17.1264	39.375	12.1960
15-Nov-01	19	18.1264	39.375	11.3893
15-May-02	20	19.1264	39.375	10.6359
15-Nov-02	21	20.1264	1039.375	262.1838
Sum of all present values				702.1802

Note: The expected cash flow for the last period also includes: maturity payment of $1000.

Therefore, the full price (or dirty price) of the issue = $702.1802 (Ans a(iii))

Fractional time period for the accrued interes = 1 - w = 1 - 0.12637 = 0.87363

In other words, the accrued interest will be 87.363% of the semi-annual coupon payment i.e.,

Accrued interest = 0.87363 x 39.375 = $34.39904 (Ans a(ii))

And, Clean Price = Full price - Accrued interest = 702.1802 - 34.399 = $667.7812 (Ans a (i))