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FINANCIAL MATHEMATICS
DEFINITIONS GENERAL
Version 10/1/
Prepared by David Forfar, MA, FFA with the assistance of David Raymont, Librarian, Institute of Actuaries, London
Interest :-the reward paid by the borrower to the lender for use of the lender s money. The borrowed money is referred to as capital.
Compound interest :- where interest is paid on the original money (capital) and on interest arising from the original capital i.e. you get interest on interest.
Simple interest :- where interest is paid only on the original money (capital) but not on interest arising from that capital i.e. you do not get interest on interest.
Time period :- the time period selected in order to solve the problem at hand- it could be one year, a half-year, a month etc. It could even be a period of two years.
Effective rate of interest i over a time period (which must be stated) :- the rate of interest i which operates over that selected time period and such that £1 invested at the beginning of that time period accumulates to £(1+i) at the end of the time period, where i is the effective rate of interest e.g. if the rate of interest is 6% effective per annum then £1 invested at the start of the year will accrue £0. interest at the end of the time period which, together with the original £1, gives £1.06. If the effective rate of interest is 3% effective per half-year then after one year an investment of £1 will have grown to £1.03*1.03 = £1.0609. This means that an effective rate of 3% per half-year is equivalent to an effective rate of 6.09% per annum (year).
Effective discount rate d over a time period (which must be stated) :- the rate such that the present/discounted value of £1 payable at the END of the stated time period is £(1-d) at the beginning of the stated time period. Thus if i is the effective rate of interest over the time period 1/(1+i) = v = (1- d). If the effective discount rate per annum is 6.5% then the present value at the start of the year of £ payable at the end of the year is £(1-.065) = £0.935. The equivalent effective rate of interest per annum is such that (1+i) =1/.935 which gives i equal to 6.95187%
Instantaneous force of interest of (t) per annum :- the instantaneous rate of growth of a fund, per unit of the fund, reflecting the operation of a force of interest of (t) per year. If the fund at time t is
F(t) then
dF t t F t dt
. This gives as a solution 0
( ) ( ) (0)
s t F t F e s^ s ds. If (t) is a constant equal
to then F (1) F (0) e so that (1+i)=e or =loge(1+i).
Nominal rates of interest :- rates of interest in name only which have to be converted into effective rates of interest over the appropriate time period.
i(2)^ :- a nominal rate of interest of i(2)^ per annum convertible half-yearly means an rate of i(2)/ effective per half-year, which is equivalent to an effective rate of i per annum where (1+i) = (1+i(2)/2)^2.
i(4)^ :- a nominal rate of interest of i(4)^ per annum convertible quarterly means an rate of i(4)/4 effective per quarter, which is equivalent to an effective rate of i per annum where (1+i) = (1+i(4)/4)^4.
i(12) :- a nominal rate of interest of i(12)^ per annum convertible monthly means an rate of i(12)/ effective per month, which is equivalent to an effective rate of i per annum where (1+i) = (1+i(12)/12)^12.
Nominal rates of discount :- rates of discount in name only which have to be converted into effective rates of discount over the appropriate time period,
d(2)^ :- a nominal rate of discount of d(2)^ per annum convertible half-yearly means an discount rate of d(2)/2 effective per half-year, which is equivalent to an effective rate discount of d per annum where v = 1/(1+i) = (1-d) = (1-d(2)/2)^2.
d(4)^ :- a nominal rate of discount of d(4)^ per annum convertible quarterly means an discount rate of d(4)/4 effective per quarter, which is equivalent to an effective rate discount of d per annum where v = 1/(1+i) = (1-d) = (1-d(4)/4)^4.
d(12)^ :- a nominal rate of discount of d(12)^ per annum convertible monthly means an discount rate of d(12)/12 effective per month, which is equivalent to an effective rate discount of d per annum where v = 1/(1+i) = (1-d) = (1-d(12)/12)^12.
v :- the discount factor 1/(1+i)
Time value of money :- takes account of the fact that £1 to-day is not the same as £1 tomorrow. If the effective rate of interest is 6% pa. effective, you are equally happy with £1 to-day or £1.06 in one year s time or £1.061/2^ in six-months time etc. This follows because £1 invested to-day at 6% pa. effective will be worth £1.06 in a year's time etc. Correspondingly you are equally happy with £1 in one year's time or £1/1.06 =£0.9434 now. You are equally happy with £1 in six-months time or £1/1.061/2^ now.
Equivalent payments :- means payments at different times but which, allowing for the time value of money, are identical i.e. the payments accumulate or discount to the exactly same thing at any point of time e.g. i(2)/2 payable at the end of the first six months and again at the end of the second six months is equivalent to i at the end of the year. For example the following payments are all equivalent at a rate of interest of i p.a. effective:-
i at the end of the year i(2)/2 payable at the end of each half-year i(4)/4 payable at the end of each quarter-year i(12)/12 payable at the end of each month d payable at the start of each year d(2)/2 payable at the start of each half-year d(4)/4 payable at the start of each quarter-year d(12)/12 payable at the start of each month
Hence:-
(2) a n :- represents the present/discounted value of payments of ½ after half a time period and ½ after a
whole time period repeated for n time periods. If i is the effective rate of interest per time period then
(2) a n is ½{v 0.5+v (^1) +v1.5+v (^2) +v2.5....+vn} which is equal to (1-vn)/i(2) (^) which equals (2) (^) n
i a i
where i(2)^ is
the nominal rate of interest per time period convertible per half-time period.
(2) ä n :- represents the present/discounted value of payments of ½ at the beginning of the first half time
period and ½ at the beginning of the second half time period repeated for n time periods. If i is the effective rate of interest per time period then ä n (2)is ½{1+v0.5+v^1 +v1.5+v^2 + ..+vn-1/2} which is equal
to (1-vn)/d(2)^ which equals (^) (2) n
i a d
where d (2)^ is the nominal rate of discount per time period
convertible per half-time period.
(4) a n :- represents the present/discounted value of payments of 1/4 at the end of the first, second, third
and fourth quarter time periods repeated for n time periods. If i is the effective rate of interest per
time period then a (4) n is 1/4{v0.25+v0.5+v0.75+v^1 + ..+vn} which is equal to (1-vn)/i(4)^ which equals
(4) n
i a i
where i(4)^ is the nominal rate of interest per time period convertible per quarter period.
(4) ä n :- represents the present/discounted value of payments of 1/4 at the beginning of the first,
second, third and fourth quarter time periods repeated for n time periods. If i is the effective rate of
interest per time period then ä n (4)is 1/4{1+v0.25+v0.5+v0.75+v^1 + ..+vn-1/4} which is equal to (1-vn)/d(4)
which equals (^) (4) n
i a d
where d (4)^ is the nominal rate of discount per time period convertible per
quarter period.
(12) a n :- represents the present/discounted value of payments of 1/12 at the end of the first, second,
third, fourth, fifth ..........eleventh and twelfth sub-time-periods where each sub-time-period is one- twelfth of a time period and where these payments are repeated for n time periods. If i is the effective
rate of interest per time period then a (12) n is 1/12{v1/12+v2/12+v3/12+v4/12+ ..+vn} which is equal to (1-
vn)/i(12)^ which equals (^) (12) (^) n
i a i
where i(12)^ is the nominal rate of interest per time period convertible per
one-twelfth time period.
(12) ä n :- represents the present/discounted value of payments of 1/12 at the beginning of the first,
second, third, fourth, fifth ..........eleventh, twelfth sub-time-periods where each sub-time-period is one-twelfth of a time period time and where these payments are repeated for n time periods. If i is the effective rate of interest per time period then ä n (12)is 1/12{1+v1/12+v2/12+v3/12+v4/12+ ..+vn-1/12}
which is equal to (1-vn)/d12)^ which equals (^) (12) n
i a d
where d (12)^ is the nominal rate of discount per time
period convertible per one-twelfth time period.
n :- represents the present/discounted value of a payment of £1 per annum payable continuously
throughout n years. If i is the effective rate of interest then (^) n is (1-vn)/ which equals (^) n
i a where
is the force of interest per annum.
Deferred annuity :- an annuity where there is a deferred period i.e. a period of time when nothing is payable i.e. the period before the annuity comes into effect.
Immediate annuity :- means an annuity where there is no deferred period.
m | an^ :-^ represents the present/discounted value of the payments represented by the symbol to the
right of the bar but in the case where these payments are deferred m years. The symbols (^) m | än ,
|^ (2) m an^ ,^
m än^ ,^
m an^ ,^
m än^ ,^
m an^ ,^
m än^ ,^ m | n are^ defined^ similarly.^ For^ example, m |^ an^ v am n^ am^ n am
s n :- represents the accumulated value of payments of £1 at the end of each time period repeated for n
time periods. Where the effective rate of interest is i per time period, the formula for sn is
1 + (1+i)+(1+i)^2 +(1+i)^3 + +(1+i)n-1^ which equals ((1+i)n-1)/i.
.. s n (i.e. with double dot over the s) :- represents the accumulated value of payments of £1 at the
beginning of each time period repeated for n time periods. Where the effective rate of interest is i per
time period, the formula for
.. s (^) n is^ (1+i)+(1+i) (^2) +(1+i) (^3) +... +(1+i)n (^) which equals ((1+i)n-1)/d which
equals (^) n
i s d
(2) s n :- represents the accumulated value of payments of ½ after half a time period and ½ after a whole
time period repeated for n time periods. If i is the effective rate of interest per time period then s (2) n is
½{1+(1+i)0.5+(1+i)^1 +(1+i)1.5+(1+i)^2 + ..+(1+i)n-1/2} which is equal to ((1+i)n-1)/i(2)^ which equals
(2) n
i s i
where i(2)^ is the nominal rate of interest per time period convertible per half-time period.
..(2) s n (i.e. with double dot over the s) :-represents the accumulated value of payments of ½ immediately and ½ after half a time period repeated for n time periods. If i is the effective rate of interest per time
period then
..(2) s n (i.e. with double dot over the s) is ½{(1+i)0.5+(1+i)^1 +(1+i)1.5+(1+i)^2 + ..+(1+i)n}
which is equal to ((1+i)n-1)/d(2)^ which equals (^) (2) n
i s d
where d (2)^ is the nominal rate of discount per
time period convertible per half-time period.
(4) s n :- represents the accumulated value of payments of 1/4 at the end of the first, second, third, fourth
quarter time periods repeated for n time periods. If i is the effective rate of interest per time period
then s (4) n is 1/4{1+(1+i)0.25+(1+i)0.5+(1+i)0.75+(1+i)^1 + ..+(1+i)n-1/4} which is equal to ((1+i)n-1)/i(4)
Iän :-represents the present/discounted value of payments of £1 at the beginning of the first time
period, £2 at the beginning of the second time period .... and £n at the beginning of the nth time period. Where the effective rate of interest is i per time period, the formula for Iän is
1+2v+3v^2 +4v^3 +.......+nvn-1^ which equals
( ä n nvn ) d
or (^) n
i Ia d
. Where the payments are twice in
every time period, four times, twelve times then the corresponding symbols are Iä n (2) , Iä n (4) , Iä (12) n and
(2) Iä n = (^) (2) (^) n
i Ia d
etc.
I (^) n :- represents the present/discounted value of payments of £1pa. payable continuously during the
first time period, £2pa. payable continuously during the second time period .... and £n pa. payable continuously during the nth time period i.e. the rate of payments are step functions.
_ I (^) n :- represents the present/discounted value of payments at a rate of t pa. at time t i.e. the rate of
payments increases continuously for n years.
Is n :- represents the accumulated value of payments of £1 at the end of the first time period, £2 at the
end of the second time period .... and £n at the end of the nth time period. Where the effective rate of interest is i per time period, the formula for Isn is (1+i)n-1+2(1+i)n-2+3(1+i)n-3+.....+n. Where the
payments are twice in every time period, four times or twelve times then the corresponding symbols are Is (2) n , Is (4) n , Is (12) n.
.. Is n (with double dot over the s) :-represents the accumulated value of payments of £1 at the
beginning of the first time period, £2 at the beginning of the second time period .... and £n at the beginning of the nth time period. Where the effective rate of interest is i per time period, the formula
for
.. Is n (double dot) is (1+i) n+2(1+i)n-1+3(1+i)n-2+.......+n(1+i). Where the payments are twice in
every time period, four times or twelve times then the corresponding symbols are
.. (2) Is n ,
.. (4) Is n ,
.. (12) Is n
(with a double dot over the a in each case).
_ I s (^) n :- represents the accumulated value of payments of £1pa. payable continuously during the first time period, £2pa. payable continuously during the second time period .... and £n pa. payable continuously during the nth time period i.e. the rate of payments are step functions.
_ _ I s (^) n :- represents the accumulated value of payments at a rate of t pa. at time t i.e. the rate of payments increases continuously for n years.
Da n :- represents the present/discounted value of payments of £n at the end of the first time period,
£(n-1) at the end of the second time period .... and £1 at the end of the nth time period. Where the effective rate of interest is i per time period, the formula for Dan is nv+(n-1)v^2 +(n-2)v^3 +.....+vn.
Perpetuity :- an annuity of infinite term, it just goes on being paid for ever! The value is obtained by
letting n go to so that, for example,
a i
Equation of value:- An equation formed by taking the present value ( discounted value ) of the cash inputs and equating them algebraically to the present value of the cash outputs and solving for the effective rate of interest which gives equality between the two sides of the algebraic equation. Alternatively take the present value of all payments after allowing inputs to have a positive sign and outputs a negative sign (or vice versa) and the equating to zero. A corresponding equation can also be formed by equating the accumulated value of the inputs and outputs and this equation is equivalent to the first one and gives the same solution. Example:- you invest £1000 in a project and get back £ at the end of the first year and £627.20 at the end of the second year. The equation of value is 1000=560v + 627.20v^2 and the positive solution to this quadratic equation is i=.12 or 12%.
Discounted payback period :- you wish to invest in a project and borrow money (a loan) from the bank in order to do so. From the monies you are paid as a result of the project being successful you gradually pay back to your banker both interest on the loan and the loan capital itself thus gradually paying off the loan. The time it takes you to pay off the loan is the discounted payback period and any further monies you are paid from the project thereafter are pure profit since you have no outstanding loan interest or loan capital to pay off after the discounted payback period. [note:- the payback period is the discounted payback period calculated assuming you are able to borrow from the bank at 0% i.e. zero percent interest, but in reality is an almost meaningless number because you will almost always be charged interest on loans].
Repayment schedule for a loan :- a table showing in full detail how a loan is repaid. The schedule will show (1) the loan outstanding just before a repayment is made (2) the amount of the repayment that is used to pay interest on the outstanding loan (3) the balance of the repayment which pays off part of the outstanding loan itself and (4) the amount of the outstanding loan after the repayment has been made. For example if a loan of £ an is made, repayable by instalments of £1p.a. yearly in arrear,
then the interest content of the first repayment (of £1) is ia (^) n or (1-vn) and the amount going to repay
the loan (the capital content) is vn^ leaving the outstanding loan as a (^) n 1. T he next year the interest
content of the second repayment of £1 is ia (^) n 1 or (1-vn-1) and the amount going to repay the loan (the
capital content) is vn-1^ leaving the outstanding loan as an (^) 2 etc.
Negative rates of interest :- normally when you invest say £1 you expect it to grow say to £1.06 at the end of a year thus giving you an effective rate of interest of 6% pa.effective. However if your investment has not been successful and is now worth only £0.95 pence then your effective rate of interest is negative and is -5%pa. effective.
Linked-internal rate of return :- this return is produced by calculating the internal rates of return for successive periods and compounding them to give an average annual rate of return over the total period. If the total time period is n years and the rates of investment return effective over the first, second, third .. time periods (not necessarily exact years) are i 1 , i 2 , i 3 , im then (1+i)n^ = (1+i 1 )(1+i 2 )(1+i 3 ) (1+im) where i is the internal linked rate of return per annum.
