o True or False INSTRUCTIONS: If you think the statement is true, then show that it is true. On the other hand, if you think the statement is false, then give an example that disproves the statement. 1. True or false. When simple interest is used, the accumulated amount is a linear function of time. TRUE. This is because the rate of increase in your interest is dependent on how much time has elapsed. The rate will never increase nor will your initial lump sum until the interest date has been reached. So yes, it is linear because the amount of money and the interest rate never change over the given amount of time. The formula for simple interest is; I=P*r*t, so the relationship itself says that I is directly proportional to t and t is of order 1, so the interest is linearly dependent. 2. True or false. Compound interest that is converted once a year is the same as simple interest. FALSE. The statement is not true because in compound interest the interest part also gets added to the principal and the next years interest is computed on the total amount, while in simple interest, the interest part is not included in the principal to compute the interest. 3. True or false. If interest is compounded annually, then the effective rate of interest is the same as the nominal rate of interest. TRUE. The effective interest rate is given by r=(1+i/n)^n When n=1 (annual compounding) then we have r=(1+i)^1 Now this is exactly same as the nominal rate of interest. 4. True or false. The present value is always smaller than the future value. TRUE. Future value is a result of regular investment over time and therefore it has interest accumulated in it while present value of a future amount will always be less because it is the estimate of amount as of today, so it's without any interest. 5. True or false. The future value of an annuity can be found by adding together all the payments that are paid into an account. FALSE. The future value of an annuity with payments at the beginning of each period: m is the amount, r is the interest, n is the number of periods per year, and t is the number of years. The Future Value of an Annuity due m [ ( 1+r /n ) nt +11 ] m r /n 6. True or false. The periodic payment R where and P is the loan amount and i is the interest per period that will amortize the loan at the end of the term comprising n periods. TRUE. Let where P=principal, i = interest rate per period, n = number of periods, R=monthly payment, then R=P x r/ [1-(1 + r)^-n] The calculation used to arrive at the periodic payment amount assumes that the first payment is not due on the first day of the loan, but rather one full payment period into the loan. While normally used to solve for A, (the payment, given the terms) it can be used to solve for any single variable in the equation provided that all other variables are known. 7. True or false. A sinking fund is the accumulated amount to be realized at some future date (the end of the term) when a fixed number of periodic payments are paid into an account earning interest at the rate of i per period. TRUE. It is a fund formed by periodically setting aside money for the gradual repayment of a debt or replacement of a wasting asset. The amount at period n is the amount at period (n-1) plus interest earned on that plus the periodic payment. Answer 1: True If I is interest on a principal P at an interest rate r per year for t year, then I Prt The accumulated amount A (sum of principal and interest) after t year is A I P Prt P P 1 rt Here A is linear function of time. Thus the given statement is true. Answer 2: False If $2000 are invested over a terms of 3 years earning interest at the rate of 8% per year compounded annually, so Compound interest 2000 1 0.08 2000 3 CI P 1 r t P $519.424 And Simple interest 2000 0.08 3 SI Prt $480 Thus, Compound interest that is converted once a year is the same as simple interest are not equal. Answer 3: True Effective rate of interest is given by the formula m r reff 1 1 m Here r is nominal interest rate and m is number of conversion period per year. Since interest is compounded annually, so m 1 . 1 r reff 1 1 1 1 r 1 1 r 1 r This is same as nominal interest rate. Thus the given statement is true. Answer 4: True The future value of compound interest is calculated by formula FV PV 1 i n Future value is a result of regular investment over time and therefore it has interest accumulated in it while present value of a future amount will always be less because it is the estimate of amount as of today, so it's without any interest. Thus, present value is always either less than or equal to future value. Answer 5: False The future value of an annuity is the sum of the future values of all of the payments in the annuity. The formula for annuity is 1 r n 1 A P r Here P is the periodic payment, r is the rate per period and n is the number of periods. The future value of an annuity formula is used to calculate what the value at a future date would be for a series of periodic payments. So, the future value of an annuity is not the sum of payments that are paid into an account, it is the sum of the future values of all of the payments in the annuity. Answer 6: True The present value of annuity is given by 1 1 i n P R i Here P is the loan amount, R is periodic payment and i is the interest per period that will amortize the loan at the end of the term comprising n periods. Solve equation for R as n Pi R 1 1 i Pi R n 1 1 i Thus, the statement is true. Answer 7: True The interest bearing fund which payments are made at periodic time intervals to provide a desired sum of money at a specified future point in time is called sinking fund. This is a type of annuity in which the goal is to save a specific amount of money in a specific amount of time. p r A n nt r 1 1 n In the formula, p is the payment needed to reach the accumulated amount, A. Payments are made n times per year, for t years, into a sinking fund with interest rate r, compounded n times per year. Answer 1: True If I is interest on a principal P at an interest rate r per year for t year, then I Prt The accumulated amount A (sum of principal and interest) after t year is A I P Prt P P 1 rt Here A is linear function of time. Thus the given statement is true. Answer 2: False If $2000 are invested over a terms of 3 years earning interest at the rate of 8% per year compounded annually, so Compound interest 2000 1 0.08 2000 3 CI P 1 r t P $519.424 And Simple interest 2000 0.08 3 SI Prt $480 Thus, Compound interest that is converted once a year is the same as simple interest are not equal. Answer 3: True Effective rate of interest is given by the formula m r reff 1 1 m Here r is nominal interest rate and m is number of conversion period per year. Since interest is compounded annually, so m 1. 1 r reff 1 1 1 1 r 1 1 r 1 r This is same as nominal interest rate. Thus the given statement is true. Answer 4: True The future value of compound interest is calculated by formula FV PV 1 i n Future value is a result of regular investment over time and therefore it has interest accumulated in it while present value of a future amount will always be less because it is the estimate of amount as of today, so it's without any interest. Thus, present value is always either less than or equal to future value. Answer 5: False The future value of an annuity is the sum of the future values of all of the payments in the annuity. The formula for annuity is 1 r n 1 A P r Here P is the periodic payment, r is the rate per period and n is the number of periods. The future value of an annuity formula is used to calculate what the value at a future date would be for a series of periodic payments. So, the future value of an annuity is not the sum of payments that are paid into an account, it is the sum of the future values of all of the payments in the annuity. Answer 6: True The present value of annuity is given by 1 1 i n P R i Here P is the loan amount, R is periodic payment and i is the interest per period that will amortize the loan at the end of the term comprising n periods. Solve equation for R as n Pi R 1 1 i Pi R n 1 1 i Thus, the statement is true. Answer 7: True The interest bearing fund which payments are made at periodic time intervals to provide a desired sum of money at a specified future point in time is called sinking fund. This is a type of annuity in which the goal is to save a specific amount of money in a specific amount of time. r A n p nt r 1 1 n In the formula, p is the payment needed to reach the accumulated amount, A. Payments are made n times per year, for t years, into a sinking fund with interest rate r, compounded n times per year