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**Sections 1.7, 1.8**Recall: The effective rate of interesti is the amount of money that one unit (one dollar) invested at the beginning of a (the first) period will earn during the period, with interest being paid at the end of the (first) period. That is, i = a(1) –a(0) = a(1) – 1 . We can also say that the effective rate of interesti is the ratio of the amount of interest earned during the (first) period to the amount invested at the beginning of the period. That is, A(1) –A(0) i = ————— . A(0) The effective rate of discountd is the ratio of the amount of interest (discount) earned during the (first) period to the amount invested at the end of the period. That is, A(1) –A(0) d = ————— . A(1)**The effective rate of discount during the nth period from**the date of investment is a(n) –a(n – 1) A(n) –A(n – 1) dn = —————— = —————— a(n) A(n) (The effective rate of interest can be considered interest paid at the end of the period on the balance at the beginning of the period, whereas the effective rate of discount can be considered interest paid at the beginning of the period on the balance at the end of the period.) Recall that compound interest implies a constant rate of effective interest; it will be proven in the exercises that compound interest implies a constant rate of effective discount which can be called “compound discount”. t2 If a(t) = 1 + — , then d1 = , d2 = , and d3 = . 25 1 — 26 3 — 29 5 — 34**In = A(n) –A(n – 1) can be called the “amount of**interest” or the “amount of discount” for the nth period. A(1) –A(0) i = ————— A(0) A(1) –A(0) d = ————— A(1) iA(0) = dA(1) = A(1) – A(0) –1 A(1) –A(0) d = ————— = A(1) A(0) 1 –—— = A(1) A(1) 1 –—— = A(0) –1 A(1) – A(0) + A(0) 1 –———————– = A(0) 1 1 – —— = 1 + i i —— 1 + i 1 – (i + 1)–1 =**The amount of interest earned for one year when X is**invested is $108. The amount of discount earned when an investment grows to value X at the end of one year is $100. Find X, i, and d. i = d = X = 0.08 108 —— = 100 1 + i iX = 108 dX = 100 i —— X = 100 1 + i 2 / 27 $1350**a(t) = amount to which 1 unit accumulates in t periods**[a(t)]–1 = amount which must be invested to accumulate 1 unit in t periods The accumulation function for simple interest is one where the amount of interest earned in t periods is the effective rate of interest times t: a(t) = 1 + it The accumulation function for simple discount is defined to be one where the amount of discount earned in t periods is the effective rate of discount times t: [a(t)]–1 = 1 –dt for 0 t< 1/d (Note that the condition t< 1/d is required to insure that a(t) is defined and positive.) 1 a(t) = ——– for 0 t< 1/d 1 –dt Are simple interest and simple discount the same?**1**With a(t) = ——– for 0 t< 1/d , 1 –dt the effective rate of discount is a(1) –a(0) 1/(1 –d) – 1 ———— = ————— = a(1) 1/(1 –d) (as expected), and the effective rate d a(1) –a(0) 1/(1–d) – 1 ———— = ————— = a(0) 1 d —— . 1 –d of interest is i = d 1 + ——– t 1 –d 1 ——– 1 –dt Are 1 + it = and the same for all t? No, so simple interest and simple discount are not the same.**The accumulation function for compound interest is one where**the effective rate of interest earned each period is constant: a(t) = (1 + i)t The accumulation function for compound discount is defined to be one where the effective rate of discount earned each period is constant: We see that the effective rate of discount earned in period n is 1 a(t) = ——– (1 –d)t a(n) –a(n–1) 1/(1 –d)n– 1/(1 – d)n–1 —————– = ————————— = a(n) 1/(1 –d)n d which is indeed constant. (This is the proof needed for Text Exercise #24(a).) Are compound interest and compound discount the same? t d 1 + —— 1 –d 1 ——— (1 –d)t Since (1 + i)t = = for all t, compound interest is the same as compound discount.**We say that a rate of interest and a rate of discount are**equivalent, if a given amount of principal invested for the same length of time at each of the rates produces the same accumulated value (i.e., the accumulation functions are identical). We have previously seen that an effective rate of simple interest can never be equivalent to an effective rate of simple discount. We have previously seen that an effective rate of compound discount d is equivalent to an effective rate of compound interest i = d / (1 –d). Observe that several identities can be derived from this equivalency: d i = —— 1 –d i d = —— 1 + i Interpretation: the ratio of amount of interest to beginning principal. Interpretation: the ratio of amount of interest to ending principal. d = 1 – v or v = 1 – d d = iv Interpretation: v and 1–d are each the present value of 1 paid at the end of the period. Interpretation: discount i from the end of a period to the beginning of a period.**Compound interest or discount will always be assumed, unless**specified otherwise. An interest (discount) rate may be stated in terms of one period, which we have called an effective rate of interest (discount), or in terms of some fraction of a period, which we shall call a nominal rate of interest (discount). When interest is paid (i.e., reinvested) more frequently than once per period, we say it is payable (convertible, compounded) each fraction of a period, and this fractional period is called the interest conversion period. For a positive integer m, we let i(m) represent a nominal rate of interest payable m times per period; that is, the rate of interest is i(m)/m for each mth of a period. The accumulation function with the nominal rate of interest i(m) is a(t) = mt i(m) 1 + — for t 0 m**Find the accumulated value of $3000 to be paid at the end of**8 years with a rate of compound interest of 5% (a) per annum. (b) convertible quarterly. (c) convertible monthly. 3000(1 + 0.05)8 = $4432.37 3000(1 + 0.05/4)4•8 = $4464.39 3000(1 + 0.05/12)12•8 = $4471.76**We say that two rates of interest are equivalent, if a given**amount of principal invested for the same length of time at each of the rates produces the same accumulated value. If two rates of compound interest i and i(m) are equivalent, then m mt i(m) 1 + — , m i(m) 1 + — for all t 0 . m 1 + i = which implies (1 + i)t = We find that with compound interest, the rates that are equivalent do not depend on the period of time chosen for the comparison, but this may not necessarily be true for another patterns of interest. Observe that nominal rates of interest are irrelevant with simple interest.**Find the compound yearly interest rate i which is equivalent**to a rate of compound interest of 8% convertible quarterly. Find the compound interest rate i(2) which is equivalent to a rate of compound yearly interest of 8%. Find the compound interest rate i(4) which is equivalent to a rate of compound yearly interest of 8%. 4 4 0.08 1 + —— 4 0.08 1 + —— – 1 4 1 + i = i = i = 0.08243216 2 i(2) 1 + — 2 i(2) 1 + — = 2 1 + 0.08 = (1.08)1/2 i(2) = 2[(1.08)1/2– 1] 0.07846 4 i(4) 1 + — 4 i(4) 1 + — = 4 1 + 0.08 = (1.08)1/4 i(4) = 4[(1.08)1/4– 1] 0.07771**For a positive integer m, we let d(m) represent a nominal**rate of discount payable at the beginning of mths of a period. The accumulation function with the nominal rate of discount d(m) is a(t) = –mt d(m) 1 –— for t 0 m Find the present value of $8000 to be paid at the end of 5 years at a rate of compound interest of 7% yearly (a) convertible semiannually. (b) payable in advance and convertible semiannually. 8000 —————— = $5671.35 (1 + 0.07/2)2•5 8000 —————— = 8000(1 – 0.07/2)2•5 = $5602.26 (1 – 0.07/2)–2•5**We say that two rates of discount are equivalent, if a given**amount of principal invested for the same length of time at each of the rates produces the same accumulated value. If two rates of compound discount d and d(m) are equivalent, then –m –mt d(m) 1 –— , m d(m) 1 –— for all t 0 . m (1 –d)–1 = which implies (1 –d)–t = We see that with compound discount, the rates that are equivalent do not depend on the period of time chosen for the comparison, but this may not necessarily be true for another patterns of interest. Observe that nominal rates of discount are irrelevant with simple discount. Equivalent interest and discount rates can be summarized as follows: –p m i(m) 1 + — = m d(p) 1 –— . p 1 + i = v–1 = (1 –d)–1 =**Find the compound yearly discount rate d which is equivalent**to a rate of compound discount of 8% convertible quarterly. Find the compound discount rate d(12) which is equivalent to a rate of compound yearly discount of 6%. Find the nominal discount rate convertible semiannually which is equivalent to a nominal interest rate 5% convertible quarterly. –4 4 0.08 1 –—— 4 0.08 1 – 1 –—— 4 (1 –d)–1 = d = d = 0.07763 –12 d(12) 1 –— = 12 (1 – 0.06)–1 d(12) = 12[1 – (1 – 0.06)1/12] d(12) = 0.06172 –2 4 –2 d(2) 1 –— = 2 0.05 1 + —— 4 0.05 1 + —— 4 d(12) = 2 1 – d(2) = 0.04908