United Kingdom
Debt Management
Office
Eastcheap Court
11 Philpot Lane
London EC3M 8UD
Public Works Loan Board
ABCDEFGHIJ
Tel. 0845 357 6610
Fax. 0845 357 6509
e-mail: pwlb @dmo.gsi.gov.uk
www.pwlb.gov.uk
Formula For Calculating Premium/Discount On Half-Yearly Fixed Rate Loans
KEY
id = discount rate / 200
il = loan rate / 200
D = number of days since previous interest payment date
N = number of half years (or parts of half years) in unexpired period
ANNUITY LOANS
(1 + id (D)) (
182.5
Annuity factor for unexpired period at loan rate
) - (1 + il(D) )
Annuity factor for unexpired period at discount rate
182.5
EIP LOANS
(1 + id (D) ) (
182.5
1
( 1 - il ) + il ) - (1 + il(D) )
N (Annuity factor for unexpired period at discount rate)
id
id
182.5
MATURITY LOANS
(1 + id(D) ) (
il
+ Present value of one at discount ) - ( 1 + il(D) )
182.5
Annuity factor for unexpired period at discount rate
POSITIVE RESULT = PREMIUM
rate for unexpired period
NEGATIVE RESULT = DISCOUNT
182.5
Calculation of DISCOUNT on premature repayment of £12,000,000.00 MATURITY loan on 2 October 1989.
Date of final payment scheduled for 18 September 1992, therefore, the unexpired period is 3 years.
Loan rate = 8 3/8%.
Discount rate = 11 1/2%.
Number of days since previous interest payment date (18 September 1989) = 14 days.
+ 0.7150192657) - (1 + 0.041875 x 14)
(1 + 0.0575 x 14) (0.041875
182.5
0.2017680253
182.5
=
(1 + 0.0044109589) (0.2075403173 + 0.7150192657) - (1 + 0.0032123288)
=
(1.0044109589) (0.9225595830) - 1.0032123288
=
0.9266289554 - 1.0032123288
=
- 0.0765833734
DISCOUNT = 0.0765833734 x £12,000,000.00 = £919,000.48
Calculation of PREMIUM on a premature repayment of £2,000,000.00 MATURITY loan on 5 August 1988.
Date of final payment scheduled for 15 January 2011, therefore, the unexpired period is 22 1/2 years.
Loan rate = 10 1/4 %.
Discount rate = 9 5/8 %.
Number of days since previous interest payment date (15 July 1988) = 21 days.
+ 0.1206177998) - (1 + 0.05125 x 21)
(1 + 0.048125 x 21) (0.05125
182.5
0.0547259201
182.5
=
(1 + 0.0055376712) (0.9364849400 + 0.1206177998) - (1 + 0.0058972603)
=
(1.0055376712) (1.0571027398) - 1.0058972603
=
1.0629566272 - 1.0058972603
=
0.0570593669
PREMIUM = 0.0570593669 x £2,000,000.00 = £114,118.73.
Calculation of DISCOUNT on premature repayment of £1,592,183.10 ANNUITY loan on 2 October 1989.
Date of final payment scheduled for 18 March 1991, therefore, the unexpired period is 1 1/2 years.
Loan rate = 8 3/8 %.
Discount rate = 12 3/4 %.
Number of days since previous interest payment date (18 September 1989) = 14 days.
(1 + 0.06375 x 14) (0.3616316252) - (1 + 0.041875 x 14)
182.5
0.3767082822
182.5
=
(1 + 0.0048904110) (0.9599778988) - (1 + 0.0032123288)
=
(1.0048904110) (0.9599778988)
=
0.9646725853 - 1.0032123288
=
- 0.0385397435
- 1.0032123288
DISCOUNT = 0.0385397435 x £1,592,183.10 = £61,362.33.
Calculation of PREMIUM on premature repayment of £705,502.02 ANNUITY loan on 5 August 1988.
Date of final payment scheduled for 31 March 2006, therefore, the unexpired period is 18 years.
Loan rate = 13 1/2 %.
Discount rate = 9 3/4 %.
Number of days since previous interest payment date (31 March 1988) = 127 days.
(1 + 0.04875 x 127) (0.0746042775) - (1 + 0.0675 x 127)
182.5
0.0594673534
182.5
=
(1 + 0.0339246575) (1.2545417483) - (1 +0.0469726027)
=
(1.0339246575) (1.2545417483)
=
1.2971016474 - 1.0469726027
=
0.2501290447
- 1.0469726027
PREMIUM = 0.2501290447 x £705,502.02 = £176,466.55.
Calculation of DISCOUNT on premature repayment of £1,776.00 EIP loan on 20 September 1989.
Date of final payment scheduled for 17 April 2013, therefore, the unexpired period is 24 years.
Loan rate = 4 1/4 %.
Discount rate = 10 1/8 %.
Number of days since previous interest payment date (17 April 1989) = 156 days.
(1 + 0.050625 x 156) (
182.5
1
(1 - 0.02125) +(0.02125
48 (0.0558426500) 0.050625 0.050625
) - (1 + 0.02125 x 156)
182.5
=
(1 + 0.0432739726) (
) (
=
(1.0432739726) (0.3730720754 (0.5802469136) + 0.4197530864) - 1.0181643836
=
(1.0432739726) (0.2164739203 + 0.4197530864) - 1.0181643836
=
(1.0432739726) (0.6362270067) - 1.0181643836
=
0.6637590768 - 1.0181643836
=
- 0.3544053068
1
(1 - 0.4197530864) + 0.4197530864 - 1 + 0.0181643836
2.6804472
DISCOUNT = 0.3544053068 x £1,776.00 = £629.42.
)
Calculation of PREMIUM on premature repayment of £200,000.00 EIP loan on 5 August 1988.
Date of final payment scheduled for 31 March 1992, therefore, the unexpired period is 4 years.
Loan rate = 13 1/2 %.
Discount rate = 10 1/8 %.
Number of days since previous interest payment date (31 March 1988) = 127 days.
(1 + 0.050625 x 127) (
182.5
) (
1
(1 - 0.0675) + 0.0675
- 1 + 0.0675 x 127
8 (0.1551129519)
0.050625 0.050625
182.5
)
=
(1 + 0.0352294521) (
) (
=
(1.0352294521) (0.8058643619 (- 0.3333333333) + 1.3333333333) - 1.0469726027
=
(1.0352294521) (- 0.2686214539 + 1.3333333333) - 1.0469726027
=
(1.0352294521) (1.0647118794) - 1.0469726027
=
1.1022210956 - 1.0469726027
=
0.0552484929
1
(1 - 1.3333333333) + 1.3333333333 - 1+.0469726027
1.2409036152
PREMIUM = 0.0552484929 x £200,000.00 = £11,049.70.
)