Simple Interest & Compound Interest PDF

Simple Interest ( *Iѕ )* ======================== **Lender or creditor** -- person (or institution) who invests the money or makes the funds available **Borrower or debtor --** person (or institution) who owes the money or avails of the funds from the lender **Origin or loan date** -- date on wh...

Simple Interest ( *Iѕ )* ======================== **Lender or creditor** -- person (or institution) who invests the money or makes the funds available **Borrower or debtor --** person (or institution) who owes the money or avails of the funds from the lender **Origin or loan date** -- date on which money is received by the borrower **Repayment date or maturity date --** date on which the money borrowed or loan is to be completely repaid **Time or term (t) --** amount of time in years the money is borrowed or invested; length of time between the origin and maturity dates **Principal (P) --** amount of money borrowed or invested on the origin date **Rate (r) --** annual rate, usually in percent, charged by the lender, or rate of increase of the investment **Interest (I) --** amount paid or earned for the use of money **Maturity value or future value (F) --** amount after t years that the lender receives from the borrower on the maturity date Compound Interest ( *Ic )* ========================== **Simple Interest Formula** - *Is = Prt* - *F = P + Is or F = P ( 1 + rt)* - *P =* 𝑟𝑡^[𝐼𝑠]^ *or P = F - Is* - *~t\ =~* 𝑃𝑟𝐼𝑠𝑃𝑡𝐼𝑠 - *r =* where; *Is* - simple interest P - principal r - simple interest rate t - term of time in years F - future value or maturity value **Compound Interest Formula** - *Ic = F - P ^^* 𝐼𝑐 = 𝑃\[(1 + 𝑖)^𝑛^ − 1\] **Present Value** - 𝑃 = 𝐹 ( 1 + 𝑖 )^𝑛^ **^^~~** 𝑃 = 𝐹 ( 1 +^^ 𝑚^𝑟^ )^𝑚𝑡^ - 𝐥𝐨𝐠( 𝑃^𝐹^ ) 𝑚\[𝐥𝐨𝐠(1+𝑖)\] where: 𝑰𝒄 − compound interest 𝑷 − present value of F 𝒓 − annual interest rate 𝒕 − time (per year) 𝑭 − compound amount or maturity value 𝒎 − conversion period annually : m = 1 semi-annually : m = 2 quarterly : m = 4 monthly : m = 12 𝒏 − total number of conversion periods (𝑛 = 𝑚𝑡) 𝒊 − periodic rate (𝑖 = 𝑚^[𝑟]^ ) Annuities ========= **Simple Annuity** - payment interval is also the same as the interest period. **General Annuity -** annuity where the length of the payment interval is not the same as the length of the interest compounding period. **Annuity Payment** - the payment for each period is fixed and the compound interest rate is fixed over a specified time. **Regular / Periodic Payment (R)** - each payment in an annuity. **Payment Interval** - the time between the successive payments dates of an annuity. **Future Value or Amount of an Annuity (F)** sum of the future values of all the payments to be made during the entire term of the annuity. **Present Value of an Annuity (P)** - The sum of the present values of all payments to be made during the entire term of the annuity. **Examples of Annuity** - Rental payment - Monthly pensions - Monthly payment for car loan - Educational plan **Annuities** +-----------------------+-----------------------+-----------------------+ | According to payment | | \- not the same as | | interval and interest | | the | | period | | | +=======================+=======================+=======================+ | | Annuity payments are | Annuity Due made at | | | | the | | | | | | | | each payment | | | | interval. | +-----------------------+-----------------------+-----------------------+ | | | Annuity payments | | | | extend | | | | | | | | over an indefinite | | | | length of time. | +-----------------------+-----------------------+-----------------------+ **Future and Present Values of a Simple Annuity** - **Present Value ( P )** ^○^ 𝑃 = 𝑅 ⎡⎢⎣ [1−( 1 +]𝑖 [𝑖 )] ^−𝑛^ ⎤⎥⎦ - **Future Value ( F ) ^○^** 𝐹 = 𝑅 ⎡⎢⎣ [(1 + 𝑖]𝑖 [)]^𝑛^[−1] ⎤⎥⎦ - where: **Cash Value or Cash Price (CV)** - equal to the down payment plus the present value of the installment payments. **Future and Present Values of a General Annuity** - **Present Value ( P )** ^○^ 𝑃 = 𝑅 ⎡^⎢^⎣ 1(−1+(1𝑖+)𝑏𝑖−)^−^1^𝑛^ ⎤^⎥^⎦ - Future Value ( F ) ^○^ 𝑃 = 𝑅 ^⎢^⎣⎡ ((11++𝑖𝑖))^𝑛^𝑏−−11 ^⎤⎥^⎦ - where: **Time Diagram** - ordinary annuity **Cash flow** - payments received (cash inflows) or payments or deposits made (cash outflows) **Cash inflows** - positive numbers and cash outflows **Fair Market Value or Economic Value** - amount that is equivalent to the value of the payment stream at that date **Deferred Annuity** - are a series of payments that will start on a later date. - annuity which payments (or deposits) starts in more than one period from the present **Period of Deferral** - time between the purchase of an annuity and the start of the payments **Time Diagram** - the period of deferral is k because the regular payments **Present Value (P) of a Deferred Annuity** where: **P** = present value **R** = periodic payment **i** = interest rate per period **n** = total number of conversion periods **k** = number of conversion periods in the deferral or number of artificial payments. **Stocks and Bonds** **Stocks** - share of ownership in a business or company. Two Types of Stocks: **Common stock and preferred stocks** **Bonds** - debt of the firm +-----------------------------------+-----------------------------------+ | STOCKS | BONDS | +===================================+===================================+ | Form of equity financing or | form of debt financing, or | | raising money by allowing | raising money by borrowing from | | investors to be part owners of | investors. | | the company. | | +-----------------------------------+-----------------------------------+ | Stock prices vary every day. | Investors are guarantee interest | | | payments and a return of their | | | money the maturity date. | +-----------------------------------+-----------------------------------+ | Investing in stock involves some | issuer to pay the bondholders. | | uncertainty. | | | | | | Investors can earn if the stock | | | prices increase, but they can | | | lose money if the stock prices | | | decrease or worse, if the company | | | goes bankrupt. | | +-----------------------------------+-----------------------------------+ | Higher risk but with possibility | Lower risk but lower yield. | | of higher | | | | | | returns | | +-----------------------------------+-----------------------------------+ | Can be appropriate if the | Can be appropriate for retirees | | investment is for the long term | or for those who need the money | | | soon | +-----------------------------------+-----------------------------------+ **Relation to Stocks** **Term of an Annuity (t)** -- the time between the first payment interval and the last payment interval. **Dividend** -- share in the company's profit **Dividend per share** -- ration of the dividends to the number of shares **Stock Market** -- a place where stocks can be bought or sold. **Market Value** -- the current price of a stock at which it can be sold **Stock Yield Ratio** -- ratio of the annual dividend per share and the market value per share. **Par Value** -- the per share amount as stated on the company certificate. **Relation to Bonds** **Bond** -- interest bearing security which promises to pay a stated amount of money on the maturity date, and regular interest payments called coupons **Coupon** -- periodic interest payment that the bondholder receives during the time between purchase date and maturity date **Coupon Rate** -- the rate per coupon payment period **Price of a Bond** -- the price of the bond at a purchase time **Par Value or Face Value** -- the amount payable on the maturity date **Term (or Tenor) of a Bond** -- fixed period of time (in years) at which the bond is redeemable as stated in the bond certificate **Fair Price of a Bond** -- present value of all cash inflows to the bondholder. **Stock Index or Stock Market Index** - measure of the value of a section of the stock market and is computed from the price of selected stocks. **Business Loan** -- money lent specifically for a business purpose. **Consumer Loan** -- money lent to an individual for personal or family purpose **Collateral** -- assets used to secure the loan. It may be real-estate or other investments **Term of the Loan** -- time to pay the entire loan **Amortization Method** -- method of paying a loan (principal and interest) on installment basis, usually of equal amounts at regular intervals **Mortgage** -- a loan, secured by a collateral, that the borrower is obliged to pay at specified terms. **Chattel Mortgage** -- a mortgage on a movable property **Collateral** -- assets used to secure the loan. It may be a real-estate or other investments **Outstanding Balance** -- any remaining debt at a specified time PROPOSITIONS ============ **Notation**: Variables are used to represent propositions. The most common variables used are p, q, and r. If a proposition is true, then its truth value is true, which is denoted by T; otherwise, its true value is false, which is denoted by F. **Simple Proposition** -- conveys one thought with no connecting words. **Compound Proposition** -- contains two or more simple propositions that are put together using connective words. **BASIC LOGICAL CONNECTIVES** - proposition is compound, then it must be one of the following: conjunction, disjunction, conditional, biconditional, or negation. **Conjunction** - p and q are connected by the word "**and**\", then the compound proposition - is written in symbolic form as \"p \^ q\". **Disjunction** - If two simple propositions p and q are connected by the word **\'or**\", then the compound proposition \"p or q\" - written in symbolic form as \"pVq\". **Conditional** - p and q are joined by a connectivity "**if then**", then the resulting compound proposition \"if p then q\" - is written in symbolic form as \"p→q\" or \"p q\". \- p. is called hypothesis (or antecedent) q. is called the conclusion (or consequent) **Biconditional** - p and q are connected by the connective "**if and only if**" then the resulting compound proposition \"p if and only if q\" - is written in symbolic form as p↔q. The proposition may also be written as \"p iff q\". **Negation** - An assertion that a statement fails, or denial of a statement - The negative of a statement is generally formed by introducing the word \"**not**\" at some proper place in the statement or by prefixing the statement with \"**it is not the case that**\" or \"**It is false that**\". - p in symbolic form is written as \"\~p" **Truth Value** - Displays the relationship between the possible truth values **Conjunction** 𝑝 𝑞 --- --- -- T T T F F T F F **Disjunction** ---- -- -- T𝑝 T F F ---- -- -- **Conditional** 𝑝 𝑞 --- --- -- T T T F F T F F **Biconditional** ---- -- -- T𝑝 T F F ---- -- -- **Negation** ---- -------- T𝑝 \~~F~𝑝 F T ---- -------- **Forms of Conditional Proposition** **Converse** - q→p Conditional Statement - if p, then q Converse - if q, then p Biconditional statement - p if and only if q **Inverse and its contrapositive** Conditional Statement - if p, then q Inverse - if not p, then not q Contrapositive - if not q, then not p