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Financial market interest
rates
4. Financial market interest rates
Interest rates – are promised rates of
return on fixed-income instruments that are a
contractual obligation
Factors impacting on interest
rates
Unit of account
Maturity
Default risk
The general form of the formula for finding the
future value at a simple interest rate is:
𝑹𝒇𝒖𝒕𝒖𝒓𝒆 = 𝑹 𝟏 + 𝒊 ∗ 𝒏
The general form of the formula for finding the
future value at a compound interest rate is:
𝑹𝒕 = 𝑹(𝟏 + 𝒊)𝒕
where: Rt – the increased value for time t
𝑟
𝑖=
– interest rate in shares;
100
R – present value of capital;
Interest rate examples
If the interest rate is simple, the interest accrued
on the deposit or loan is calculated as the product of
the number of years t (or their respective shares)
and the amount of the deposit or loan R before the i
interest rate matures.
For example, if $1000 was placed by 8% per
annum for 9 months, then the amount calculated with
simple interest rate is:
9
1000 ∗ 0,08 ∗
= $60
12
Future value of deposit is:
9
1000 1 + 0,08 ∗
= $1060
12
The graph below shows the time dependence
of the increased value of capital for fixed interest
rates for the period t.
The expression needs to be changed to
formulate the percentage increase more than
once a year. The annual interest rate is divided
by the number of accounting periods in a year,
and the t level is multiplied by the number of
accounting periods in a year:
𝑹𝒕 = 𝑹(𝟏 +
𝒊 𝒎∗𝒕
)
𝒎
where m is the number of computational
cycles per year.
So far, we have considered the exact
percentage of interest. Determining the
percentage as a result of continuous calculation
is also of particular interest.
We have an R deposit in the initial period.
We set a goal to increase this amount to
the maximum by the end of the year. If the
bank pays r% per annum, then the deposit
is increased by r% for the year, the deposit
for any short period increases in proportion
to this period, e.g., an increase of r / 12%
over a month and a r / 365% for the day.
Assuming that the opening and closing of
the deposit will be continuous (only in
theory), then we can consider the following
general issue.
The amount R deposited in the bank at r% per
annum is kept for t years. [0; t] dividing the section
into equal periods, we obtain the theoretically
possible final sum:
𝑹𝒕 = 𝒍𝒊𝒎 𝑹 𝟏 +
𝒏→∞
𝒏
𝒓
𝒕
𝟏𝟎𝟎𝒏
= 𝑹 𝒍𝒊𝒎 (𝟏 +
𝒏→∞
𝒓𝒕 𝟏𝟎𝟎𝒏
) 𝒓𝒕
𝟏𝟎𝟎𝒏
𝒓𝒕
𝟏𝟎𝟎
= 𝑹∗𝒆
𝒓𝒕
𝟏𝟎𝟎
Thus, we create a continuous interest formula
for the final amount of the deposit:
𝑹𝒕 = 𝑹 ∗ 𝒆
where 𝑖 =
𝑟
.
100
𝒓𝒕
𝟏𝟎𝟎
𝒊𝒕
= 𝑹 ∗ 𝒆 , (6.3)
Example. Find the difference between the
values ​increased for two years in the calculation
of the amount at a rate of 10% continuous and
monthly interest. Principal value is $300K
The amount added in the continuous
calculation of interest is as follows:
𝑅𝑡 = 300,0 ∗ 𝑒 0,1∗2 = $366,421
The amount increased in the calculation of
monthly interest is as follows:
0,1 12∗2
𝑅𝑡 = 300(1 + )
= $366,117
12
Difference is:
366421 – 366117 = 304 sh.b.
Effect of maturity on interest rates
U.S Treasury Yield Curve
Yield comparisons (%)
Treasury 1 – 10 years
5,70
Treasury more than 10 years
6,21
Corporate bonds 1 – 10 years:
High quality corporate bonds
6,45
Low quality corporate bonds
6,94
Corporate bonds more than 10 years:
High quality
Low quality
7,09
7,56
Return comes from two sources in
common stocks
Dividend
Gain in market
price of the stock
over the period it
is held
Example. To illustrate how returns are
measured, suppose you buy shares of stock at a
price of $100 per share. One day later the price
is $101 per share and you sell. Your rate of
return for the day is 1 % - capital gain of $1 per
share divided by the purchase price of $100.
Now suppose you hold the stock for a year. At
the end of the year, the stock pays a cash
dividend of $5 per share and the price of a
share is $105 just after the dividend is paid.
The one-year rate of return, r, is:
r=
𝑬𝒏𝒅𝒊𝒏𝒈 𝒑𝒓𝒊𝒄𝒆 𝒐𝒇 𝒂 𝒔𝒉𝒂𝒓𝒆 −𝑩𝒆𝒈𝒊𝒏𝒏𝒊𝒏𝒈 𝑷𝒓𝒊𝒄𝒆+𝑪𝒂𝒔𝒉 𝑫𝒊𝒗𝒊𝒅𝒆𝒏𝒅
𝑩𝒆𝒈𝒊𝒏𝒏𝒊𝒏𝒈 𝑷𝒓𝒊𝒄𝒆
In the example we have:
$𝟏𝟎𝟓−$𝟏𝟎𝟎+$𝟓
r=
$𝟏𝟎𝟎
= 𝟎. 𝟏𝟎 𝒐𝒓 𝟏𝟎%
Note that we can present the total rate of return as the sum
of the dividend income component and the price change
component:
𝒓=
𝒄𝒂𝒔𝒉 𝒅𝒊𝒗𝒊𝒅𝒆𝒏𝒅
𝒆𝒏𝒅𝒊𝒏𝒈 𝒑𝒓𝒊𝒄𝒆 𝒐𝒇 𝒂 𝒔𝒉𝒂𝒓𝒆 − 𝒃𝒆𝒈𝒊𝒏𝒏𝒊𝒏𝒈 𝒑𝒓𝒊𝒄𝒆
+
𝒃𝒆𝒈𝒊𝒏𝒏𝒊𝒏𝒈 𝒑𝒓𝒊𝒄𝒆
𝒃𝒆𝒈𝒊𝒏𝒏𝒊𝒏𝒈 𝒑𝒓𝒊𝒄𝒆
𝒓 = 𝒅𝒊𝒗𝒊𝒅𝒆𝒏𝒅 𝒊𝒏𝒄𝒐𝒎𝒆 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕 + 𝒑𝒓𝒊𝒄𝒆 𝒄𝒉𝒂𝒏𝒈𝒆 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕
r = 5% +5% = 10%
The general formula relating the real rate of
return to the nominal rate of interest and the
rate of inflation is:
1 + 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
1 + 𝑟𝑒𝑎𝑙 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 =
1 + 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛
or equivalently,
𝑵𝒐𝒎𝒊𝒏𝒂𝒍 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝒓𝒂𝒕𝒆 − 𝑹𝒂𝒕𝒆 𝒐𝒇 𝒊𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏
𝑹𝒆𝒂𝒍 𝑹𝒂𝒕𝒆 =
𝟏 + 𝑹𝒂𝒕𝒆 𝒐𝒇 𝒊𝒏𝒇𝒍𝒂𝒕𝒊𝒐𝒏
To see why, let's compute the real rate of return
precisely. For every $100 you invest now, you will
receive $108 a year from now. But a basket of
consumption goods, which now costs $100, will cost
$105 a year from now. How much will your future
value of $108 be worth in terms of consumption
goods? To find the answer we must divide the $108 by
the future price of a consumption basket: $108/$105 =
1.02857 baskets. Thus, for every basket you give up
now, you will get the equivalent of 1.02857 baskets a
year from now. The real rate of return (baskets in the
future per basket invested today) is, therefore, 2.857%
per year.
There are four main factors that determine rates of
return in a market economy:
productivity of
capital goods
degree of
uncertainty
regarding the
productivity of
capital goods;
time
preferences
of people
risk aversion
Major stock indexes around the world
Country
USA
Japan
UK
Germany
France
Switzerland
Europe, Australia,
Far East
Indexes
Dow Jones Index,
Standard & Poor's 500
Nikkei, Topix
FT-30, FT-100
DAX
CAC 40
Credit Suisse
MSCI, EAFE