# Accrued interest

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YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE

September 1999

Quoted Rate Treasury Bills [Called Banker's Discount Rate]

P 1 - P 0 ] * [ 360 ] d=[ N P1 d = Bankers discount yield

P 1 = face value P
0

= Price

N = number of days until maturity

P1 * N * d = P1 - 1 P 0 360 P0

d * N  P1  1 + P1 = holding period return  = P 0 360  P 0 
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The invoiceon a bond (what you pay) is quoted price plus accrued interest.

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Government Bonds and Notes Calculating Accrued Interest Accrued interest is actual days/actual days

Last coupon date
x

settlement date

next coupon date

y
Accrued interest = x/y times interest payment Example: (1) 10% coupon (2) \$100 interest annually or \$50 semi-annual per \$1000 face y = 183 x = 100

100 [50] =\$27.32 183

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Yield To Maturity 1. One coupon left [like T bill] At coupon paying date 0 1 2 3 4

2.

C Po

C

C

C

C + Prin

P0 =

C r [1 + ] 2

+

C + Prin C C + + r r r [1 + ] 2 [1 + ] 3 [1 + ] 4 2 2 2

3.

Between Coupon Dates
C + prin C C C + + + r r r r [1 + ] w [1 + ] 1+ w [1 + ] 2 + w [1 + ] 3+ w 2 2 2 2

P0 + A =

5

w = fraction of year to firstcoupon actual days over actual days 0 w 1 2 3 4

Note: Ignore that intervals may be of uneven size because of Saturdays or end of month.

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Eurobonds - will examine only bonds issued in dollars - interest paid annually usually

Calculating accrued interest uses 30 day months 360 day years Example 1: 1. 2. days Issue date Settlement date January February March January 28 March 5 [29, 30] 30day [1,2,3,4,5] 2 30 5 37

37 x interest = accrued interest 360

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Example 2: 1. May 14 2. Sept 17 May 16 J, Ju Aug 90 Sept 17 123 issue date settlement date

The 31st is the same as the 30th.
123 x interest = accrued interest 360

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Computing Yield To Maturity

1.

With one payment remaining like T-bill but usually with 30-day 360 calendar. Multiple payments

2.

Price +accrued =

C + Prin C C + ... + [1 + r ] n - 1+ v [1 + r ] v [1 + r ] v + 1

r is an annual rate v is fraction of year until payment and is done on 360 days in year 30 day month calendar

Most have options. We will discuss this later.

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GNMA Definition: bonds issued with mortgages as their backing.

Guaranties 1. Issuer. If borrower fails to make a scheduled amortization payment inany month, issuer makes good. If borrower defaults, issuer must promptly remit remaining mortgage. 2. Government. Makes good payments if issuer fails. Rate is mortgage rate - 50 basis points e.g. 13% mortgages 12.5% pool rate 44 basis points to issuer 6 basis points to gov.
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Timing of Payments

January [Home owners]

February [Home owners] Pass through

March

Pass through

If.No Prepayments constant amount paid each month

determining constant amount

100 =

M M M + + ... + r r 2 r 360 [1 + ] [1 + [1 + ] ] 12 12 12

M is scheduled amortized payment r 8 12 m .7338 1.0286

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Quoted Prices on GNMA Quoted as % of remaining principal balance Assume quoted price is 95 1 million original Principal value x .8 % still outstanding x .95 Quoted Price \$760,000Price

Accrued interest on GNMA Settlement day. Two business days after trade date but first settlement is usually third Wednesday for GNMA less than 9.5% and following Monday for GNMA 9.5%. Reason: Pool factors not available until 10th of month. Accrued interest

coupon number of days from 1st until settlement x 12 30

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Example: 13% GNMA 5 million face Feb. l5 settlement date .8 Poolfactor

Accrued Interest Note:

=

14 13 x [.8x5x ] = \$20,222 30 12

Always use 30 days irrespective of days in the month.

Yield to Maturity

Price + Accrued =

M M M + .. + + r 2 r r 3 ) ) (1 + (1 + ) (1 + 12 12 12

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LIBOR • Interest rates are annual using “simple interest” Interest payment = principal x LIBOR x actual days to payment 360 Example: One million dollar deposit...