Interest, Compound Interest, Simple Interest, Present Value, Interest in Advance, and Nominal Interest A Primer for MATH 393 1 What is Interest • When lenders invest money, they usually do so with the expectation (hope) of a financial gain. If an investment K grows to amount S, then the difference S − K is interest. • The interest may be thought of as rent paid by the borrower for the use of the K. Historically/Culturally Speaking... • Western society • The Catholic Church of the Middle Ages • The Islamic World Why do we charge interest? • Investment opportunities theory • Time preference theory • Default risk 2 Accumulation and Amount Functions • The amount K of money the investor loans is called the principal. From the borrower’s perspective, the amount of money borrowed is the principal. • We’ll call time 0 the time of our initial transaction. Thus we suppose that K is invested at time t = 0. Definition. AK (t) is a real-valued function with domain {t|t ≥ 0} such that AK (t) equals the balance at time t. The function AK (t) is called the amount function for principal K. We write a(t) for A1 (t) and call a(t) the accumulation function. FACT: K invested at time 0 grows to AK (t) at time t, and 1 invested at time 0 grow to a(t) at time t. Definition. If t2 > t1 ≥ 0, then AK (t2 ) − AK (t1 ) gives the amount of interest earned on an investment of K made at time 0 between time t1 and t2 . We define the effective interest rate for the interval [t1 , t2 ] to be i[t1 ,t2 ] = a(t2 ) − a(t1 ) a(t2 ) 1 Note that whenever AK (t) = Ka(t), we also have i[t1 ,t2 ] = A(t2 ) − A(t1 ) A(t2 ) Observation 1: The effective annual rate of interest earned by an investment during a one-year period is the percentage change in the value of the investment from the beginning to the end of the year, without regard to the investment behavior at tinermediate points in the year. Observation 2: We can calculate the effective interest rate for any time period [t1 , t2 ], t2 > t1 ≥ 0. 3 Simple Interest and Linear Accumulation Functions Definition. AK (t) = K(1 + st) is called the amount function for K invested by simple interest at rate s. The function a(t) = 1 + st is the simple interest accumulation function at rate s. Observation: When the growth of money is governed by simple interest at rate s, in = (1 + sn) − [1 + s(n − 1)] s a(n) − a(n − 1) = = a(n − 1) a + s(n − 1) 1 + s(n − 1) Hence, {in } is a decreasing sequence that converges to 0. This is a reason why simple interest is not often used for long period loans. Ways to Calculate Simple Interest • Exact simple interest (actual/actual) • Ordinary simple interest (30/360) • Banker’s rule (actual/360) 4 Compound Interest Definition. We will use a(t) = (1 + i)t for all t ≥ 0 and will call this the compound interest accumulation function at interest rate i. Observations • For large t > 1, the graph of compound interest accumulation function lies above the simple interst accumulation function. • For a tiered investment account, the function AK (t) 6= Ka(t). 5 Interest in Advance and the Effective Discount Rate Definition. When money is borrowed with interest due before the money is released, we describe the relationship using discount rates. If an investor lends K for one basic period at 2 discount rate, D, then the borrower will have to pay KD in order to received the use of K. Therefore, instead of having the use of an extra K, the borrower only has the use of an extra K − KD = (1 − D)K. The quantity KD is called the amount of discount for the loan. Observation: In any cashflow with a beginning and ending balance and no withdrawals or deposits, the amount of interest and the amount of discount are the same; they are both equal to the change in the balance. Definition. We define the effective discount rate for the interval [t1 , t2 ] to be d[t1 ,t2 ] = a(t2 ) − a(t1 ) a(t2 ) d[t1 ,t2 ] = A(t2 ) − A(t1 ) A(t2 ) and if AK (t) = Ka(t), then Usually i[t2 ,t1 ] and d[t2 ,t1 ] are not equal. They are, however, related. Definition. A rate of interest and a rate of discount are said to be equivalent for an interval [t1 , t2 ] if for each 1 invested at t1 , the two rates produce the same accumulated value at time t2 . More generally, two different methods of specifying an investment’s growth (over a given time period) are called equivalent if they correspond to the same accumulation function. Observation: On the intervale [t1 , t2 ], an interest rate of it1 ,t2 ] is equivalent to a discount rate of d[t1 ,t2 ] precisely when 1 = (1 + i[t1 ,t2 ] )(1 − d[t1 ,t2 ] ) 6 Discount Functions and the Time Value of Money Definition. We define the discount function by v(t) = 1 a(t) Observation: In general, the accumulation of money over time depends not only on the length of the time interval but also on where in time the interval lies. Observation: Fortunately, for compound interest, we only have to worry about the length of the interval. FACT: If we wish to invest money t1 years from now in order to have S in t2 years from v(t2 ) 1) now, we should invest Sv(t2 )a(t1 ) = S a(t = S v(t . a(t2 ) 1) Definition. We define the discount factor to be v= 1 1+1 3 Then, v(t) = 1 1 = = vt. a(t) (1 + 1)t Definition. We defined present value with respect to a(t) of L to be received at time t0 to be Lv(t0 ). P Va(t) (L at t0 ) = Lv(t0 ) How to compare investments • NPV • Effective rate of interest 7 Nominal Interest Banks commonly credit interest more than once per year, say m times per year. They advertise a nominal (annual) interest rate of i(m) convertible m times per year. FACT: If an account is governed by a nominal interest rate of i(m) payable m times per 1 (m) year, the bank pays interest at a rate of i m = [(1 + i) m − 1] per m-th of a year. Other ways to phrase nominal interest • nominal interest rate convertible m times per year • nominal interest rate compounded m times per year • nominal interest rate payable m times per year • annual percentage yield (APR) Observation: With a given nominal annual interest rate, the more often compounding takes place in a year, the larger the year-end accumulation value will be and therefore also the effective annual interest rate. 4