Accrued Interest and Bond Prices

EXERCISE 01

A bond pays semiannual coupons of €4 and redeems €100 at month 24. Tizio buys at month 8 at a clean price of €97.60, then sells at month 15. Use proportional monthly accrued interest and effective annual yields. Coupons occur at months 6, 12, 18 and 24.

A.Find the dirty purchase price.
B.Find the sale price giving the buyer a 6% yield to maturity.
C.Find the sale price giving Tizio a 5% annualised holding-period IRR.

A.

P_{dirty}=P_{clean}+Accrued\ interest

P_{dirty,8}=97.60+4\frac26

€98.93

B.

P_{15}=\frac4{1.06^{3/12}}+\frac{104}{1.06^{9/12}}

Dirty sale price: €103.50. Accrued interest at month 15 is €2, so the corresponding clean price is €101.50.

C.

P_{dirty,8}=\frac4{1.05^{4/12}}+\frac{P_{15}}{1.05^{7/12}}

P_{15}=\left(P_{dirty,8}-\frac4{1.05^{4/12}}\right)1.05^{7/12}

€97.74 dirty; €95.74 clean.

These are alternative target prices. A single sale price need not simultaneously give buyer and seller their different target returns.

Accrued Interest and Bond Prices

EXERCISE 02

A bond has a clean quote of 97.30 per €100 face value immediately after the coupon on 1 April 2014. Coupons are €6 every four months. Three months later, the clean quote remains 97.30. Use proportional month-based accrual.

A.Find clean and dirty prices at both dates.

A.

P_{dirty}=P_{clean}+Coupon\frac{Elapsed\ months}{Coupon\ interval}

P_{dirty,0}=97.30+0=97.30

P_{dirty,3m}=97.30+6\frac34=101.80

The clean price is 97.30 at both dates. The dirty price rises by the €4.50 coupon accrual.

Accrued Interest and Bond Prices

EXERCISE 03

A €1,000 bond has a 12% annual coupon rate, paid semiannually (€60 each half-year), and matures in 2 years and 3 months at par. The gross effective annual yield is 13%. The next coupon is in 3 months. Use linear accrual and a hypothetical 12.5% coupon withholding tax. Calculate gross market prices; for the net current yield, apply withholding only to coupon income and keep the quoted clean price unchanged.

A.Find dirty and clean prices.
B.Find gross and net annual current yields.

A.

P_{dirty}=\sum_k\frac{CF_k}{(1+y)^{t_k}}

Years from today	Cash flow
0.25	60
0.75	60
1.25	60
1.75	60
2.25	1,060

P_{dirty}=60\sum_{k=0}^{4}1.13^{-(0.25+0.5k)}+1{,}000(1.13)^{-2.25}

Dirty price: €1,018.04.

AI=60\frac36=30

P_{clean}=P_{dirty}-30

€988.04

B.

Current\ yield=\frac{Annual\ coupon\ income}{Clean\ price}

y_{current,gross}=\frac{120}{P_{clean}}

12.145255% per year.

y_{current,net}=\frac{120(1-0.125)}{P_{clean}}

10.627098% per year.

Using €60 in the numerator produces a semiannual coupon-to-price ratio, not an annual current yield. Current yield is also different from yield to maturity.