) Solution. Let us indicate the unknown instant profit d t( ), then .
0 0 d t K t K t K v K vt v vt ( ) ( )/ ( ) / (1 ) /(1 ) Thus, the yield
changes with time. It is also clear the capital gain per
value D3 and with the termination of this time interval, at the time
point , the sum R3 is paid. For a balanced transaction, the amount of
payment R3 must be so that the debt be discharged. Figure 4.1
Consequently, the sums accumulated with compound interest are: 1210
=1100=1210, 1210+0.11000=1000+100=1100, 1100+0.11000,
1000+0.1 1331.1 . Example 1.6. The annual rate of compound
interest equals 8%.
condition must be observed: .A A 1 2 Ex a m ple s of c onve r si on 1.
Replacing the immediate annuity with the deferred one. Let there be an
annual immediate annuity with the parameters of 1 1 R n i
price index for 1 [ , ] k k t t equals 1 1 ( ) ( ) 1, ( ) (1 ) k k k k t t t t k k k k
j t t j h 1,2,., . , k m By the price index definition we
have 1 2 1 1 ( ) ( ) 1 2 ( ) m m t t t t t J n j j jm
determined from the equation P d S T (0)(1 ) ( ) or ( ( ) (0)/ (0) ( )/
(0) 1. The value d S T P P S T P S T P ( )/ (0) is called accumulation
factor or multiplier. It is obvious that . S T P d ( )/ (
calculation, it is sufficient to calculate and compare the effective rates
for each of them. Example 1.7. The bank calculates the interest out of
the nominal value of j =12% per annum. Calculate the e
the accounting equation will be e ( )(1 ) (1 ) n n D G i D i , (4.7)
from which an expression for the effective rate follows: 1/ 1 1 (1 ) e n i i
g . (4.8) The effective rate e i does not appear in th
of discounting at a compound discount rate. 20.Determine the formula
of discounting at a nominal discount rate. 21.How is discounting made
at a continuous discount rate? 22.How is discounting made at
means, the actually issued loan equals . The deal( ) D G stipulates
charging simple interest at the rate i . The total yield rate will be defined
by e i . When determining the yield of this transactio
ln(1 ) n t i i n i . 2. Changing the duration and the
term structure of annuity. Let us consider an ordinary annual annuity
with the parameters 1 1 R n i , , . This annuity is replaced with another
on
0 t t t K q R q R q R 2 3 3 0 1 2 3 ( ) 0 T t t t K q R q R q R , (4.2)
Where i i . T t From the latter equation it is obvious that the
financial and credit transaction may be conventionally divided
t t t s n n , and in every interval interest calculation is made at the
rates 1 2 , ,., s i i i correspondingly. Then the formula of
accumulation with compound interest will be: 1 2 1 2 (1 ) (1 ) (1 )
continued, by the end of the term we will obtain an amount of (1 / )m
R j m for the payment received at the end of the year. The last
payment will come in at the end of the year( 1) n n , it will not
calculated for this term, it is sufficient to use the formula for compound
interest accumulation. This problem is solved most simply for the
annual annuity with interest calculation once a year. Such
Thus, S P i n ( ) (1 ) (1 ) 0 n . S n P i g g Example 3.1. When
granting a credit for 2 years with an annual compound interest rate of
0.08, the creditor retains the commission fee at the rate of 0.5%
payments and inflows arranged for different time moments a payment
stream. The payment stream whose elements are positive values and
time intervals between two consequent payments are constant are
cal
n i n i . (2.8) 2.3. Accumulated Sum of Annual Annuity with
Interest Calculation m Times a Year Let payments come in once at the
end of the year (meaning, the annuity interval equals one year), and
in
interest rate at interest calculation m times a year. The higher the
number m , the quicker the process of the original sum accumulation. E
ffe c ti ve Ra te To compare various conditions of interest
formula for the present value of payment stream. 9. Determine the
formula for the present value of payment stream with interest
calculations m times a year. 10.Determine the formula for the present
va
R i S R s p n i p i , (2.13) where ( ) , p ni s the
accumulation factor of the p -due annuity with interest calculation once
a year, which equals 1 (1/ ) 1 / 0 1 (1 ) 1 ( , , ) (1 ) (1 ) 1) np n t p p
transaction yield would be 0.08271. Inflation depreciation of money
that is revealed in the growth of the prices for goods and services that is
followed by the reductions of purchasing capacity of mon
term end we will obtain the sum R i (1 ) for the payment coming at
the end of the year.( 1) n The last payment will come in at the end of
the n year, it will not be charged with interest. To determine
moment Rj to the term end. On the basis of accounting equations, it
is possible to measure the financial and credit transaction yield. An
accounting equation must be formed for this where accumulation
Find the amount of repayment, if the interest rate is 60%. Solution.
2000 (1 0.6) 3200. S The rates T i , T d determined above refer to
the whole period of the deal. In practice, another type of inter
beginning of the payment stream or that predicts it. The accumulated
sum is determined, for instance, in order to be aware of the total
amount of indebtedness at a certain time point, the total volume
, with commission, (1.052)360/160 1 = 0.1208, i.e. 12,08%
without commission. 3.4. Instant Profit Let at the moment t the capital
is K t( ), and after some time t the capital equals , then the averag
interest calculation period, and the interest at the discount rate d at
the beginning of the interest calculation period. Simple and compound
interest types are distinguished. When calculating the sim
with the deferred one? 18.How should one annual annuity be replaced
with another? Chapter 3 Financial Transaction Yield A transaction is
called financial if its beginning and ending have a money valua
at rate of i =10% 160 days before its maturity (the base annual number
is 360 days). At the execution of the transaction a commission fee was
retained from the note owner that equaled 0.5% of the prin