102 Session 1

102 Session 1 (p) (p) switch from i to d

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Jargon: interest rates Suppose you lend someone (e.g. your bank) £100 and they promise to pay you £110 in a year’s time. By convention, we could call £10 of the money returned “interest” and £100 of the money returned “capital” or “principal”. The annual interest rate would be 10 / 100 = 10% . 120 100 80 60 40 20 0

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Jargon: discount rates (1) The bank might have described this deal a different way. It might say it “sold you a bond” for £110, which you “bought” for only £100. By convention, we could say that you “paid” £10 less than the “face value” of the bond (ie you paid £100 = £110 - 10). So we could say you got a £10 “discount”, an annual rate of 10 / 110 = 9% 120 100 80 60 40 20 0

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Jargon: discount rates (2) Something pays 100, but price has been knocked down from 100 to 95

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d (discount rate) = 5 (the discount) / 100

5

100

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Jargon: interest & discount rates i (Interest rate) = interest / START d (discount rate) = interest / END

start

end

interest

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Jargon: interest convertible quarterly Would you rather get interest sooner or later? Interest may be described as “convertible quarterly”. Ie if you lend a bank £100 for 10% pa “convertible quarterly”, it will pay you (for that £100) £10 interest in four chunks: £2.50 after 3 months, 6 months, 9 months, 12 months. So, after 3 months, you have £2.50 interest plus your £100 principal. So your original £100 has “grown” to £102.50, or by a factor of 1.025. So if you reinvest the whole lot on the same terms (ie if you reinvest the interest) your £100 grows to 100 * 1.025 x 1.025 x 1.025 x 1.025 = 110.38 by then end of the year, a total growth of 10.38%. So by convention, we say 10% pa convertible quarterly ( denoted i(4) ) is equivalent to a 10.38% “annual effective” rate ( denoted i ).

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Specimen 11. Just calculate i

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Specimen 11. Just calculate i on deposit account

1

Deposit account pays ……. / 2 = ……% each six months So every six months, deposit account grows by factor of ……. So in a year, deposit account grows by factor of …………………. So annual effective rate of return on deposit account is ……….. 13

Specimen 11. Just calculate i on deposit account

1

Deposit account pays 4% / 2 = 2% each six months So every six months, deposit account grows by factor of 1.02. So in a year, deposit account grows by factor of 1.02 * 1.02 = 1.0404 So annual effective rate of return on deposit account is 4.04% 14

April 2000 3. Just calculate i

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April 2000 3. Just calculate i

1

Account pays …. / 12 = …….% each month So every month, account grows by factor of …..…. So in a year, account grows by factor of ………………..…. So annual effective rate of return on account is ……….. 16

April 2000 3. Just calculate i

1

1.05833

1.05833^2

1.05833^12

Account pays 7% / 12 = 0.58333% each month So every month, account grows by factor of 1.0058333 So in a year, account grows by factor of 1.005833^12 = 1.0723 So annual effective rate of return on account is 7.23% 17

Sep 2002 3. Just calculate i

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Sep 2002 3 (i). Just calculate i

1

Account pays ……. each month So every month, account grows by factor of ……. So in a year, account grows by factor of ……………..…. So annual effective rate of return on account is ……….. 19

Sep 2002 3 (i). Just calculate i

1

1.005

1.005^2

1.005^12

Account pays 0.5% each month So every month, account grows by factor of 1.005 So in a year, account grows by factor of 1.005^12 = 1.0617. So annual effective rate of return on account is 6.17%. 20

Sep 2002 3 (ii). Just calculate i

1

Account pays 6% pa so 2 * ….. = ……. every two years So every year account grows by factor f so that f * f = ……. So in a year, account grows by factor of ……………………... So annual effective rate of return on account is ……….. 21

Sep 2002 3 (ii). Just calculate i

1

Account pays 6% pa so 2 * 6% = 12% every two years So every year account grows by factor f so that f * f = 1.12 So in a year, account grows by factor of √1.12 = 1.0583 So annual effective rate of return on account is 5.83% 22

April 2000 3(i)

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April 2000 3(i)

1

Every month, account grows by factor of ……. (we’ve done it already) So in six months, account pays …………...…..… per £1 invested. So account pays 2 * …… per £1 invested per year = ……. So annual rate of interest convertible half yearly is …….. 24

April 2000 3(i)

1

1.05833

1.05833^6

Every month, account grows by factor of 1.05833 So in six months, account pays 1.05833^6 = 1.0355 per £1 invested. So account pays 2 * 3.55% per £1 invested per year = 7.10% So annual rate of interest convertible half yearly is 7.10% 25

April 2000 3(i)

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Sep 2002 3

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Sep 2002 3 (i)

1

Every month, account grows by factor of ……. So in a quarter, account grows by factor of ………………...…. So every quarter, accounts pays ……..…… per £1 invested. So account pays 4 * …… per £1 invested per year = ……. So annual rate of interest convertible quarterly is …….. 28

Sep 2002 3 (i)

1

1.005

1.005^2

1.005^3

Every month, account grows by factor of 1.005 So in a quarter, account grows by factor of 1.005^3 = 1.01508 So every quarter, accounts pays 1.508% per £1 invested. So account pays 4 * 1.508% per £1 invested per year = 6.03% So annual rate of interest convertible quarterly is 6.03% 29

Sep 2002 3

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Sep 2002 3 (ii)

1

Account pays 2 * ….. = ……. every 2 years There are ….. quarters in two years. … So every quarter account grows by factor f so that f^…. = ……. So in a quarter account pays ……..…… per £1 invested. So account pays 4 * …… per £1 invested per year = ……. So annual rate of interest convertible quarterly is ……..

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Sep 2002 3 (ii)

1

Account pays 2 * 6% = 12% every 2 years There are 8 quarters in two years. … So every quarter account grows by factor f so that f^8 = 1.12 So in a quarter account pays 1.12^(1/8) – 1 = 1.427% per £1 invested. So account pays 4 * 1.427% per £1 invested per year = 5.71% So annual rate of interest convertible quarterly is 5.71%

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Sep 2002 3 (ii)

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Sep 2003 1

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Sep 2003 1

1

Discount rate per 90 days is …… % * 90/365 = …..…… So to get 1 in 90 days you must pay 1 - …….. = ………. now. Ie any investment grows by factor of 1 / ….… = ……. per 90 days. So it grows by …….. ^ (1/90) = …….... each day, and grows by factor of ….…..^365 = ………. each year. So effective annual rate of interest is …….. 35

Sep 2003 1

0.98521

1

Discount rate per 90 days is 6% * 90/365 = 1.479% So to get 1 in 90 days you must pay 1 – 1.479% = 0.98521 now. Ie investment grows by factor of 1 / 0.98521 = 1.01501 per 90 days. So it grows by 1.01501 ^ (1/90) = 1.0001656 each day, and grows by factor of 1.0001656^365 = 1.0623 each year. So effective annual rate of interest is 6.23% 36

Sep 2003 1

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Sep 2000 2

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Sep 2000 2

1

Discount rate per 90 days is …… % * 90/365 = …..…… So to get 1 in 90 days you must pay 1 - …….. = ………. now. Ie any investment grows by factor of 1 / ….… = ……. per 90 days. So it grows by …….. ^ (1/90) = …….... each day, and grows by factor of ….…..^(365/2) = ………. each half year. So in six months, account pays ……..…… per £1 invested. So account pays 2 * …… per £1 invested per year = ……. 39 So annual rate of interest convertible half yearly is ……..

Sep 2000 2

0.9877

1

Discount rate per 90 days is 5% * 90/365 = 1.233% So to get 1 in 90 days you must pay 1 – 1.233% = 0.9877 now. Ie investment grows by factor of 1 / 0.9877= 1.01248 per 90 days. So it grows by 1.01248^ (1/90) = 1.000138 each day, and grows by factor of 1.000138 (365/2) = 1.02547 each half year. So in six months, account pays 2.547% per £1 invested. So account pays 2 * 2.547% per £1 invested per year = 5.095%. So annual rate of interest convertible half yearly is 5.095% 40

Sep 2000 2

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Apr 2001 3(i)

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Apr 2001 3(i)

1

Discount rate per quarter is …… % / 4 = …..…… So to get 1 in a quarter you must pay 1 - …….. = ………. now. Ie any investment grows by factor of 1 / ….… = ……. per quarter. There are …... quarters in half-a-year. So each half-year investment grows by ……..^2 = ………. So in six months, account pays ……..…… per £1 invested. So account pays 2 * …… per £1 invested per year = ……. 43 So annual rate of interest convertible half yearly is ……..

Apr 2001 3(i)

0.98

1

Discount rate per quarter is 8% / 4 = 2%. So to get 1 in a quarter you must pay 1 – 2% = 0.98 now. Ie any investment grows by factor of 1 / 0.98 = 1.020408 per quarter. There are 2 quarters in half-a-year. So each half-year investment grows by 1.020408^2 = 1.04123 So in six months, account pays 4.123% per £1 invested. So account pays 2 * 4.123% per £1 invested per year = 8.247% So annual rate of interest convertible half yearly is 8.247%

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Apr 2001 3(i)

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Sep 2001 1

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Sep 2001 1

1

This bill grows by ……… per year. So it grows by factor of ……… ^(91/365) = ………. per 91 days. So to have 1 after 91 days you need 1 / ……… = ………. now. Ie you must pay 1 - …….% now to get 1 after 91 days. ie discount rate per 91 days is ……..% So discount rate per 365 days is ……. * 365/91 = …….%

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Sep 2001 1

0.98791

1

This bill grows by 1.05 per year. So it grows by factor of 1.05^(91/365) = 1.012238 per 91 days. So to have 1 after 91 days you need 1 / 1.012238 = 0.98791 now. Ie you must pay 1 – 1.209% now to get 1 after 91 days. ie discount rate per 91 days is 1.209% So discount rate per 365 days is 1.209% * 365/91 = 4.849%

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Sep 2001 1

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Apr 2001 3(ii)

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Apr 2001 3(ii)

1

To get 1 in a quarter you must pay 1 - ……./4 = ………. now. Ie any investment grows by factor of 1 / ….… = ……. per quarter. There are …... months in a quarter. So each month investment grows by …….^(1/3) = ……. So to have 1 after a month you need 1/ ………. now. Ie you need 1 - …… % now to have 1 after a month. Ie discount rate per month is ……..% So discount rate pa (convertible monthly) is 12 * ……= …….% 51

Apr 2001 3(ii)

0.98

1

To get 1 in a quarter you must pay 1 – 8%/4 = 0.98 now. Ie any investment grows by factor of 1 / 0.98 = 1.02041 per quarter. There are 3 months in a quarter. So each month investment grows by 1.02041^(1/3) = 1.00676 So to have 1 after a month you need 1/ 1.00676 = 0.9933 now. Ie you need 1 – 0.67% % now to have 1 after a month. Ie discount rate per month is 0.67% So discount rate pa (convertible monthly) is 12 * 0.67%= 8.05% 52

Apr 2001 3(ii)

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Specimen 11

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Specimen 11

1

Deposit account pays ……. / 2 = ……% each six months So every six months, deposit account grows by factor of ……. Equivalently, in 3 months, deposit account grows by factor of …….^(3/6) = ………… So to get 1 in 3 months on deposit account (or on Bill) you need to pay 1 / …….. = ………. = 1 - …… now. So discount rate per 3 months is …. ie ….. * 4 = ……… % pa 55

Specimen 11

1

Deposit account pays 4% / 2 = 2% each six months So every six months, deposit account grows by factor of 1.02 Equivalently, in 3 months, deposit account grows by factor of 1.02^(3/6) = 1.00995. So to get 1 in 3 months on deposit account (or on Bill) you need to pay 1 / 1.00995 = 0.990148 = 1 – 0.9852% now. Discount rate per 3 months is 0.9852% ie 0.9852% * 4 = 3.941% pa

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Specimen 11

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April 2000 3(ii)

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April 2000 3(ii)

1

Account pays …. / 12 = …….% each month So every month, account grows by factor of …..…. So to get 1 in a month, you need 1 / ….. = …….. = 1 - ..…% now. So discount rate per month is ……..% and discount rate pa convertible monthly is 12 * …….% = ………..% 59

April 2000 3(ii)

0.9942

1

Account pays 7% / 12 = 0.58333% each month So every month, account grows by factor of 1.005833 So to get 1 in a month, you need 1 / 1.005833 = 0.99420 = 1 -0.58% now. So discount rate per month is 0.58% and discount rate pa convertible monthly is 12 * 0.58% = 6.96% 60

April 2000 3(ii)

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Jargon for next session : “total force of interest” (1) If i is constant annual effective rate of interest, the accumulated amount at time t is ………… So we need 1 / ……… now to have 1 at time t so the present value of 1 payable at time t is …….…… Define δ as ln(1+i). So e δt = (1+i)^….. δt is the “total force of interest” from time 0 to time t. So e^(total force of interest) = accumulated amount at time t And present value of 1 payable at time t is ….. ^-(total force of interest) 62

“total force of interest” (2) If i is constant annual effective rate of interest, the accumulated amount at time t is (1+i)^t So we need 1 / (1+i)^t now to have 1 at time t so the present value of 1 payable at time t is (1+i)^-t Define δ as ln(1+i). So e δt = (1+i)^t δt is the “total force of interest” from time 0 to time t. So e^(total force of interest) = accumulated amount at time t And present value of 1 payable at time t is e ^-(total force of interest) 63

“total force of interest” (3)

E.g. (1) if i = 5%, then δ = …………. After two years, 1 accumulates to (1 + … )^2 = e^(2 * ….) = e^ (total force of interest) E.g. (2) if δ = 20%, then i = ………. and to pay 1 in three years time you need e^-(total force of interest) = e^-(3 * …….) = (1 + ……)^-3

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“total force of interest” (4)

E.g. (1) if i = 5%, then δ = ln(1.05) = 4.88% After two years, 1 accumulates to (1.05)^2 = e^(2 * 4.88%) = e^ (total force of interest) E.g. (2) if δ = 20%, then i = e^20% -1 = 22.14% and to pay 1 in three years time you need e^-(total force of interest) = e^-(3 * 20%) = (1.2214)^-3

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END of session 1

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