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Financial market interest rates 4. Financial market interest rates Interest rates β 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: πΉππππππ = πΉ π + π β π The general form of the formula for finding the future value at a compound interest rate is: πΉπ = πΉ(π + π)π where: Rt β the increased value for time t π π= β interest rate in shares; 100 R β 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 β 0,08 β = $60 12 Future value of deposit is: 9 1000 1 + 0,08 β = $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: πΉπ = πΉ(π + π πβπ ) π 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: πΉπ = πππ πΉ π + πββ π π π ππππ = πΉ πππ (π + πββ ππ ππππ ) ππ ππππ ππ πππ = πΉβπ ππ πππ Thus, we create a continuous interest formula for the final amount of the deposit: πΉπ = πΉ β π where π = π . 100 ππ πππ ππ = πΉ β π , (6.3) Example. Find the difference between the values βincreased 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: π π‘ = 300,0 β π 0,1β2 = $366,421 The amount increased in the calculation of monthly interest is as follows: 0,1 12β2 π π‘ = 300(1 + ) = $366,117 12 Difference is: 366421 β 366117 = 304 sh.b. Effect of maturity on interest rates U.S Treasury Yield Curve Yield comparisons (%) Treasury 1 β 10 years 5,70 Treasury more than 10 years 6,21 Corporate bonds 1 β 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= π¬ππ πππ πππππ ππ π πππππ βπ©ππππππππ π·ππππ+πͺπππ π«ππππ πππ π©ππππππππ π·ππππ In the example we have: $πππβ$πππ+$π r= $πππ = π. ππ ππ ππ% Note that we can present the total rate of return as the sum of the dividend income component and the price change component: π= ππππ π ππππ πππ πππ πππ πππππ ππ π πππππ β πππππππππ πππππ + πππππππππ πππππ πππππππππ πππππ π = π ππππ πππ ππππππ πππππππππ + πππππ ππππππ πππππππππ 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 + πππππππ πππ‘ππππ π‘ πππ‘π 1 + ππππ πππ‘π ππ πππ‘π’ππ = 1 + πππ‘π ππ ππππππ‘πππ or equivalently, π΅ππππππ ππππππππ ππππ β πΉπππ ππ πππππππππ πΉπππ πΉπππ = π + πΉπππ ππ πππππππππ 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