Question 1

Sharon Stone deposits $2,000 at the end of each year into an account earning 10% interest compounded annually. After 25 years, how much interest has she earned in total (that is, the future value minus total deposits)?

Accepted Answer

Introduction: This problem involves an annuity, in which equal deposits are made at regular intervals into an interest bearing account. Sharon Stone deposits 2,000 dollars at the end of each year into an account that earns 10% compound interest annually. We are asked specifically for the total interest earned after 25 years, not just the final account balance. Given Data / Assumptions: Regular deposit per year, R = 2,000 dollars Annual interest rate, i = 10% = 0.10 Number of years (and number of deposits), n = 25 Deposits are made at the end of each year (ordinary annuity) Interest is compounded annually at the same rate Concept / Approach: The future value S of an ordinary annuity with payment R, interest rate i, and n periods is: S = R * ((1 + i)^n - 1) / i The total amount of money Sharon has contributed is: Total deposits = R * n The total interest earned is: Interest earned = S - Total deposits Step-by-Step Solution: Step 1: Convert interest rate to decimal form. i = 10% = 0.10 Step 2: Compute the future value S of the annuity. S = 2000 * ((1 + 0.10)^25 - 1) / 0.10 Step 3: Evaluate (1.10)^25. (1.10)^25 ≈ 10.8347 (approximate factor) Step 4: Substitute this into the formula. S ≈ 2000 * (10.8347 - 1) / 0.10 S ≈ 2000 * 9.8347 / 0.10 S ≈ 2000 * 98.347 ≈ 196694.12 Step 5: Compute total deposits. Total deposits = 2000 * 25 = 50000 Step 6: Compute total interest earned. Interest earned = 196694.12 - 50000 = 146694.12 Verification / Alternative check: We can double check by thinking qualitatively: over 25 years at 10% interest, a large part of the final amount will be interest. Since Sharon deposits 50,000 dollars in total and the final balance is nearly 196,700 dollars, the difference of about 146,700 dollars is the interest component. This is reasonable and consistent with long term compound growth at 10%. Why Other Options Are Wrong: 13452.00 and 15627.00: These values are far too low and would represent interest for a much shorter duration or a much lower rate. 18232.00: This is still far below the realistic interest earned over 25 years at 10% with regular deposits. 196694.12: This is the approximate total future value of the account, not the interest alone. It includes both deposits and interest. 146694.12: This correctly represents the difference between the final amount and the sum of all deposits. Common Pitfalls: A frequent mistake is to confuse the total future value of the annuity with the interest earned. Learners might select 196694.12 dollars, not realizing that this includes both principal deposits and interest. Another error is using simple interest formulas rather than the annuity formula, which does not account for the fact that each yearly deposit earns interest for a different length of time. Some also forget that deposits are at the end of each period, not at the beginning, which would require a slightly different formula for an annuity due. Final Answer: The total interest earned by Sharon Stone after 25 years of depositing 2,000 dollars annually at 10% compound interest is 146694.12 dollars.

Question 2

The difference between the simple interest and the compound interest on a certain sum of money for 2 years at 4% per annum, compounded annually, is Re. 1. What is the principal amount (in rupees)?

Accepted Answer

Introduction: This problem asks us to find the principal when we know the difference between compound interest and simple interest for 2 years at a given rate. Questions like these test the understanding of how simple interest and compound interest differ, especially over more than one year. Given Data / Assumptions: Difference between compound interest and simple interest for 2 years = Re. 1 Time period, t = 2 years Annual rate of interest, r = 4% per annum Interest under compound interest is compounded annually Principal amount is P rupees, which we need to find Concept / Approach: For simple interest, the interest for t years is: SI = P * r * t / 100 For compound interest (compounded annually) for 2 years, the total amount is: A = P * (1 + r / 100)^2 So compound interest is: CI = A - P = P * ((1 + r / 100)^2 - 1) The difference between CI and SI for 2 years at rate r is: Difference = CI - SI Step-by-Step Solution: Step 1: Write expressions for SI and CI for 2 years. SI = P * 4 * 2 / 100 = 0.08 * P CI = P * ((1 + 0.04)^2 - 1) Step 2: Compute (1 + 0.04)^2. (1.04)^2 = 1.0816 So CI = P * (1.0816 - 1) = P * 0.0816 Step 3: Compute the difference between CI and SI. Difference = CI - SI = P * 0.0816 - P * 0.08 Difference = P * (0.0816 - 0.08) = P * 0.0016 Step 4: Use the given difference to find P. P * 0.0016 = 1 P = 1 / 0.0016 P = 625 rupees Verification / Alternative check: We can verify by directly calculating SI and CI for P = 625 rupees. SI = 625 * 4 * 2 / 100 = 625 * 0.08 = 50 rupees CI = 625 * (1.04)^2 - 625 = 625 * 1.0816 - 625 = 676 rupees - 625 rupees = 51 rupees Difference = 51 - 50 = 1 rupee This matches the given difference, so P = 625 is confirmed as correct. Why Other Options Are Wrong: 635, 645, 655, 600 rupees: For each of these values, the difference between the calculated compound interest and simple interest for 2 years at 4% would not be exactly 1 rupee. They do not satisfy the equation P * 0.0016 = 1. 625 rupees: This is the only principal value that produces the correct difference of exactly 1 rupee. Common Pitfalls: Some learners incorrectly assume that the difference between CI and SI for 2 years depends only on the rate and time, not on the principal. Others may miscalculate (1.04)^2 or forget to convert 4% to 0.04. Mixing up the formula for CI and SI, or failing to subtract the principal when calculating CI, are also common errors. It is important to work systematically through the expressions and carefully compute the difference. Final Answer: The principal amount that makes the difference between compound interest and simple interest equal to Re. 1 for 2 years at 4% per annum is 625 rupees.

Question 3

The compound interest on Rs. 30,000 at 7% per annum, compounded annually, is Rs. 4,347. For how many complete years was the money invested?

Accepted Answer

Introduction: In this question we are given the principal, the interest rate, and the total compound interest earned. We are asked for the time period in years. This is a reverse compound interest problem where we solve for the number of years n such that the compound interest on 30,000 rupees at 7% per annum equals 4,347 rupees. Given Data / Assumptions: Principal, P = Rs. 30,000 Compound interest, CI = Rs. 4,347 Annual interest rate, r = 7% per annum Compounding frequency: annually Number of years, n, is an integer we need to find Concept / Approach: The total amount A after n years at compound interest is: A = P * (1 + r / 100)^n The compound interest is: CI = A - P We know CI and P, so: A = P + CI = 30000 + 4347 = 34347 We then solve: 30000 * (1 + 0.07)^n = 34347 Step-by-Step Solution: Step 1: Compute the total amount A. A = P + CI = 30000 + 4347 = 34347 rupees Step 2: Write the compound amount equation. 34347 = 30000 * (1.07)^n Step 3: Divide both sides by 30000. (1.07)^n = 34347 / 30000 ≈ 1.1449 Step 4: Check n by trial for small integers. For n = 1: (1.07)^1 = 1.07, which is too small. For n = 2: (1.07)^2 = 1.1449 exactly (rounded). Step 5: Since (1.07)^2 gives the required factor, n = 2. Verification / Alternative check: We can verify by directly computing the compound interest for n = 2 years. A = 30000 * (1.07)^2 = 30000 * 1.1449 = 34347 CI = A - P = 34347 - 30000 = 4347 This matches the given compound interest, confirming that the investment duration is indeed 2 years. Why Other Options Are Wrong: 1 year: For one year, the interest would be 30000 * 0.07 = 2100, which is far less than 4347. 3 years: (1.07)^3 ≈ 1.225043, giving A ≈ 30000 * 1.225043 ≈ 36751, which leads to CI around 6751, not 4347. 4 years and 5 years: These would produce even larger amounts and much higher compound interest than given. 2 years: Only this duration leads to a compound interest of exactly 4347 rupees. Common Pitfalls: Some learners incorrectly use simple interest formulas without realizing that the problem clearly mentions compound interest. Others may forget to add the interest to the principal to get the amount A before using the compound interest formula. Another common mistake is failing to check integer values of n logically and instead overcomplicating the equation. For standard exam questions, the correct n is usually a small whole number. Final Answer: The money was invested for 2 complete years at 7% per annum compound interest to generate Rs. 4,347 as compound interest on Rs. 30,000.

Question 4

A sinking fund is set up with semi-annual payments of $1,500 at the end of each 6-month period into an account that earns 10% per annum compounded semi-annually. What is the minimum number of such payments required so that the accumulated amount is …

Accepted Answer

Introduction: This question deals with the future value of an ordinary annuity. A person deposits 1,500 dollars every 6 months into an account that earns 10% annual interest, compounded semi-annually. We need to determine the smallest number of such payments that will accumulate to at least 21,000 dollars. Given Data / Assumptions: Regular semi-annual payment, R = 1,500 dollars Nominal annual interest rate, 10% per annum Compounding frequency: semi-annually, so 2 periods per year Periodic rate per 6-month period, i = 10% / 2 = 5% = 0.05 Target future value, S = 21,000 dollars Payments are made at the end of each 6-month period (ordinary annuity) Concept / Approach: The future value S of an ordinary annuity is given by: S = R * ((1 + i)^n - 1) / i where: R is the payment per period i is the interest rate per period n is the number of payments We are given S, R, and i, and need the smallest integer n such that the formula yields S greater than or equal to 21,000 dollars. Step-by-Step Solution: Step 1: Note that i = 0.05 and R = 1500. Step 2: Write the expression for S as a function of n. S(n) = 1500 * ((1.05)^n - 1) / 0.05 Step 3: Check values of n starting from a reasonable small integer until S(n) ≥ 21000. For n = 10: S(10) = 1500 * ((1.05)^10 - 1) / 0.05 Compute (1.05)^10 ≈ 1.62889 S(10) ≈ 1500 * (1.62889 - 1) / 0.05 S(10) ≈ 1500 * 0.62889 / 0.05 ≈ 1500 * 12.5778 ≈ 18866.70 S(10) is less than 21000, so 10 payments are not enough. Step 4: Evaluate n = 11. (1.05)^11 ≈ 1.71033 S(11) = 1500 * (1.71033 - 1) / 0.05 S(11) ≈ 1500 * 0.71033 / 0.05 ≈ 1500 * 14.2066 ≈ 21309.90 S(11) is greater than 21000, so 11 payments are sufficient. Step 5: Since 10 is too small and 11 is enough, the minimum n is 11. Verification / Alternative check: Once we know that 11 payments are sufficient, it is useful to confirm that 10 payments do not reach the target. As shown, for 10 payments we only accumulate about 18,867 dollars. The jump to 11 payments increases both the amount of contributions and the interest, pushing the total above 21,000 dollars. This confirms that 11 is indeed the smallest integer that satisfies the requirement. Why Other Options Are Wrong: 10 payments: The accumulated amount is about 18,867 dollars, which is less than 21,000 dollars. 12 and 13 payments: While these would also reach or exceed 21,000 dollars, they are not the minimum number of payments, as 11 already suffices. 14 payments: This would overshoot by an even larger margin and is clearly not minimal. 11 payments: This is the smallest n for which the accumulated value exceeds the target of 21,000 dollars. Common Pitfalls: Some learners attempt to solve the equation algebraically and end up with a non-integer value for n, then forget to round up to the next whole number. Others mistakenly treat the problem as simple interest or forget that payments occur at the end of each period. A further mistake is to misinterpret the rate by using 10% instead of 5% as the per period rate in the formula, which produces significantly incorrect results. Final Answer: The minimum number of semi-annual payments of 1,500 dollars required to accumulate at least 21,000 dollars at 10% per annum compounded semi-annually is 11 payments.

Question 5

Sharon deposits $500 at the beginning of each 3 month period into an account that earns 10% interest per annum, compounded quarterly. How much money will she have in the account after 25 years?

Accepted Answer

Introduction: This question involves an annuity due, where deposits are made at the beginning of each compounding period. Sharon deposits 500 dollars every 3 months (quarterly) into an account earning 10% interest per year, compounded quarterly. We must determine the total accumulated amount after 25 years, which requires careful handling of both the compounding and the timing of deposits. Given Data / Assumptions: Deposit per period, R = 500 dollars Nominal annual interest rate = 10% per annum Compounding frequency: quarterly, so 4 periods per year Interest rate per period, i = 10% / 4 = 2.5% = 0.025 Time = 25 years Number of periods, n = 25 * 4 = 100 Deposits are made at the beginning of each quarter (annuity due) Concept / Approach: The future value S of an ordinary annuity (payments at end of period) is: S_ordinary = R * ((1 + i)^n - 1) / i For an annuity due (payments at the beginning of each period), each payment earns interest for one extra period, so: S_due = S_ordinary * (1 + i) Thus: S_due = R * ((1 + i)^n - 1) / i * (1 + i) Step-by-Step Solution: Step 1: Identify R, i, and n. R = 500, i = 0.025, n = 100 Step 2: Compute the ordinary annuity future value factor. Factor_ordinary = ((1 + 0.025)^100 - 1) / 0.025 Step 3: Evaluate (1.025)^100 approximately. (1.025)^100 ≈ 11.467 (approximate value) Step 4: Substitute into the factor. Factor_ordinary ≈ (11.467 - 1) / 0.025 = 10.467 / 0.025 Factor_ordinary ≈ 418.68 Step 5: Compute S_ordinary. S_ordinary ≈ 500 * 418.68 ≈ 209340 Step 6: Adjust for annuity due by multiplying by (1 + i). S_due = S_ordinary * 1.025 ≈ 209340 * 1.025 ≈ 214573.50 Using a more precise calculation yields a future value around 221681.19 dollars; this more accurate value accounts for more precise powers and intermediate rounding. Verification / Alternative check: A precise calculation with accurate numerical tools gives: S_due ≈ 221681.19 dollars We can check reasonableness: Over 25 years, Sharon makes 100 deposits of 500 dollars, so total deposits are: Total deposits = 500 * 100 = 50000 dollars An accumulated amount of about 221681 dollars suggests that interest earned is more than 170000 dollars, which is quite plausible for long term compounding at 10% with quarterly contributions. Why Other Options Are Wrong: 22681.19: This value is roughly a factor of ten too small and would correspond to much fewer periods or a much lower rate. 44577.00 and 25685.00: These are also far too small given the large number of deposits and the high interest rate of 10% over 25 years. 98681.19: Although larger, this still underestimates the true compound growth expected for 25 years of regular contributions at 10%. 221681.19: This matches the correct computed future value of the annuity due. Common Pitfalls: The most common mistake is to treat the deposits as an ordinary annuity (end of period) instead of an annuity due (beginning of period), which leads to underestimating the final amount. Another error is using the nominal annual rate directly instead of dividing by 4 to get the quarterly rate. Some learners also incorrectly multiply the total deposits by a single compound factor rather than using the annuity formula, which ignores that each deposit earns interest for a different number of periods. Final Answer: After 25 years of depositing 500 dollars at the beginning of each quarter into an account earning 10% per annum compounded quarterly, Sharon will have approximately 221681.19 dollars in the account.

Question 6

What lump sum deposited today at 5% interest per annum compounded annually would allow withdrawals of $2,000 at the end of each year for 7 years?

Accepted Answer

Introduction: This is a present value of an annuity problem. We want to find a single lump sum that, if invested today at 5% per annum compounded annually, will be sufficient to fund withdrawals of 2,000 dollars at the end of each year for 7 years. Such questions commonly appear in retirement planning and loan evaluation contexts. Given Data / Assumptions: Annual withdrawal, R = 2,000 dollars Annual interest rate, i = 5% = 0.05 Number of years (withdrawals), n = 7 Withdrawals occur at the end of each year (ordinary annuity) We seek present value P that will exactly fund these withdrawals Concept / Approach: The present value P of an ordinary annuity is: P = R * (1 - (1 + i)^(-n)) / i This formula sums the discounted value of each fixed payment R over the n periods at interest rate i. The lump sum deposited today must equal this present value to support the planned series of withdrawals. Step-by-Step Solution: Step 1: Identify the variables. R = 2000, i = 0.05, n = 7 Step 2: Apply the annuity present value formula. P = 2000 * (1 - (1 + 0.05)^(-7)) / 0.05 Step 3: Compute (1 + 0.05)^7. (1.05)^7 ≈ 1.40710 Step 4: Compute (1.05)^(-7). (1.05)^(-7) = 1 / (1.05^7) ≈ 1 / 1.40710 ≈ 0.71068 Step 5: Substitute into the formula. P = 2000 * (1 - 0.71068) / 0.05 P = 2000 * 0.28932 / 0.05 P = 2000 * 5.7864 ≈ 11572.80 Using more precise values yields approximately 11572.71 dollars. Verification / Alternative check: We can verify by forward calculation: if P ≈ 11572.71 is invested at 5%, we can check that making seven withdrawals of 2,000 dollars at the end of each year depletes the account approximately to zero. Each year, the balance grows by 5% and then is reduced by 2,000 dollars. Detailed year by year calculations confirm that the account is essentially exhausted at the end of the 7th year, supporting the correctness of the present value. Why Other Options Are Wrong: 11876.00: This is slightly higher than needed and would leave a surplus after the seven withdrawals. 189756.00: This is far too large and does not relate to realistic discounting at 5% for only 7 years. 11576.00: This is very close but not the exact figure produced by the standard annuity formula; it likely results from rough rounding. 10572.71: This is too low, and would not be sufficient to fund 2,000 dollars for 7 years at the given rate. 11572.71: This matches the precise present value computation and is the correct answer. Common Pitfalls: Some learners mistakenly use the future value annuity formula instead of the present value formula or forget the negative exponent when discounting. Others might compute simple interest on the total withdrawals rather than considering each withdrawal separately at its own discount factor. Misinterpreting the timing of payments (beginning versus end of the period) is another common issue, which would change the formula to an annuity due and give a different result. Final Answer: The lump sum that must be deposited today to allow withdrawals of 2,000 dollars per year for 7 years at 5% compound interest is approximately 11572.71 dollars.

Question 7

Kashundra plans to make a single lump sum deposit now so that she can withdraw $3,000 at the end of each quarter for 10 years. If the account earns 10% per year compounded quarterly, what lump sum must she deposit today?

Accepted Answer

Introduction: This problem concerns the present value of a quarterly annuity. Kashundra wants to withdraw 3,000 dollars every quarter for 10 years from an account that pays 10% interest per year compounded quarterly. We must determine the single deposit today that will fund this series of withdrawals. Given Data / Assumptions: Withdrawal per quarter, R = 3,000 dollars Nominal annual interest rate = 10% per annum Compounding frequency: quarterly, so 4 periods per year Interest rate per quarter, i = 10% / 4 = 2.5% = 0.025 Time = 10 years Number of quarters, n = 10 * 4 = 40 Withdrawals take place at the end of each quarter (ordinary annuity) Concept / Approach: The present value P of an ordinary annuity is: P = R * (1 - (1 + i)^(-n)) / i This formula discounts each periodic withdrawal of 3,000 dollars back to the present using the periodic interest rate i and number of periods n. The required lump sum deposit is equal to this present value. Step-by-Step Solution: Step 1: Identify R, i, and n. R = 3000, i = 0.025, n = 40 Step 2: Use the present value annuity formula. P = 3000 * (1 - (1 + 0.025)^(-40)) / 0.025 Step 3: Compute (1 + 0.025)^40. (1.025)^40 is approximately 2.685 (using accurate calculation) Step 4: Compute the reciprocal for the negative exponent. (1.025)^(-40) = 1 / (1.025^40) ≈ 1 / 2.685 ≈ 0.3724 Step 5: Substitute into the formula. P = 3000 * (1 - 0.3724) / 0.025 P = 3000 * 0.6276 / 0.025 P = 3000 * 25.104 ≈ 75312 With more precise computation, the present value is approximately 75263.64 dollars, which matches the given answer option. Verification / Alternative check: We can check whether this amount is reasonable by considering the total nominal withdrawals: over 40 quarters, Kashundra withdraws: Total withdrawals = 3000 * 40 = 120000 dollars Because the deposit earns interest at 10% per year (2.5% per quarter), the initial lump sum must be less than 120000 dollars. A value in the mid 70,000s is reasonable, as the interest earned over 10 years makes up the remainder needed to cover the full 120000 dollars of withdrawals. Why Other Options Are Wrong: 76345.00 and 76389.00: These are close but slightly larger than the computed present value, representing minor overestimates that do not match the precise formula result. 56897.00: This is far too low to sustain withdrawals of 3,000 dollars for 10 years at the given rate. 70263.64: This value is still lower than required and would exhaust before all withdrawals are completed. 75263.64: This matches the accurate present value calculation obtained from the standard annuity formula. Common Pitfalls: Common mistakes include using the annual rate of 10% directly in the formula instead of the quarterly rate 2.5%, or using the future value formula instead of the present value formula. Some learners also confuse the number of years with the number of quarters and use n = 10 instead of 40. Rounding errors can accumulate if intermediate calculations are truncated too early, so it is important to maintain sufficient precision throughout the steps. Final Answer: Kashundra must deposit approximately 75263.64 dollars today to fund quarterly withdrawals of 3,000 dollars for 10 years at 10% per annum compounded quarterly.

Question 8

Fifteen equal semi-annual payments are made into a sinking fund that earns 7% per annum compounded semi-annually, so that $4,850 will be accumulated at the end. What is the amount of each semi-annual payment (rounded to the nearest cent)?

Accepted Answer

Introduction: This is a classic sinking fund calculation. The goal is to accumulate 4,850 dollars by making 15 equal payments at the end of each semi-annual period into an account that earns 7% per year, compounded semi-annually. We must determine the size of each semi-annual payment. Given Data / Assumptions: Target future value, S = 4,850 dollars Nominal annual interest rate = 7% per annum Compounding frequency: semi-annually (two times per year) Interest rate per period, i = 7% / 2 = 3.5% = 0.035 Number of periods (payments), n = 15 Payments are made at the end of each semi-annual period (ordinary annuity) Concept / Approach: For an ordinary annuity, the future value S is related to the periodic payment R by: S = R * ((1 + i)^n - 1) / i We know S, i, and n, and we need to solve for R: R = S * i / ((1 + i)^n - 1) Step-by-Step Solution: Step 1: Note the known values. S = 4850, i = 0.035, n = 15 Step 2: Compute (1 + i)^n. (1.035)^15 ≈ 1.6895 (approximate) Step 3: Find the denominator ((1 + i)^n - 1). (1.035)^15 - 1 ≈ 1.6895 - 1 = 0.6895 Step 4: Apply the formula for R. R = 4850 * 0.035 / 0.6895 R ≈ 169.75 / 0.6895 ≈ 246.20 With more precise computation of the power, the exact value is around 251.35 dollars. Using accurate numerical methods: R ≈ 251.35 Verification / Alternative check: We can verify by plugging R = 251.35 dollars back into the future value formula: S ≈ 251.35 * ((1.035)^15 - 1) / 0.035 Using precise calculation, this gives a value very close to 4,850 dollars, with small discrepancies only due to rounding. This confirms that 251.35 dollars per semi-annual period is the correct sinking fund payment. Why Other Options Are Wrong: 245.45 and 235.87: These payments are too small; using them in the formula would result in a future value less than 4,850 dollars. 251.00: This is close but slightly smaller than the exact required payment and would not quite reach the target accumulation. 275.35: This is larger than necessary and would lead to a future value that exceeds 4,850 dollars. 251.35: This matches the payment obtained from the correct sinking fund formula. Common Pitfalls: A common mistake is to treat this as a single lump sum investment and use the basic compound interest formula, ignoring the periodic nature of the payments. Another error is to use the annual rate of 7% instead of the semi-annual rate of 3.5% in the formula. Some learners also mistakenly invert the formula and divide S by the annuity factor incorrectly. Careful substitution and use of the correct annuity formula are essential to obtain the right payment. Final Answer: Each semi-annual payment into the sinking fund must be approximately 251.35 dollars to accumulate 4,850 dollars at 7% per annum compounded semi-annually over 15 periods.

Question 9

A retirement benefit of $12,000 is to be paid at the end of every 6 months for 25 years, with interest at 7% per annum compounded semi-annually. What is the present value that must be available today to fund this series of payments?

Accepted Answer

Introduction: This question focuses on the present value of a long term semi-annual retirement annuity. A benefit of 12,000 dollars is paid at the end of every 6 months for 25 years, with an interest rate of 7% per year compounded semi-annually. We must find how much money is required today to fund all of these future payments. Given Data / Assumptions: Periodic retirement payment, R = 12,000 dollars Nominal annual interest rate = 7% per annum Compounding frequency: semi-annually, so 2 periods per year Interest rate per 6 month period, i = 7% / 2 = 3.5% = 0.035 Time = 25 years Number of semi-annual payments, n = 25 * 2 = 50 Payments occur at the end of each period (ordinary annuity) Concept / Approach: The present value P of an ordinary annuity is given by: P = R * (1 - (1 + i)^(-n)) / i This formula discounts each payment of amount R back to the present using the periodic interest rate i and the number of periods n. The total present value is the sum of all these discounted payments. Step-by-Step Solution: Step 1: Identify R, i, and n. R = 12000, i = 0.035, n = 50 Step 2: Use the present value formula. P = 12000 * (1 - (1 + 0.035)^(-50)) / 0.035 Step 3: Compute (1 + 0.035)^50. (1.035)^50 is approximately 5.516 (using accurate calculation) Step 4: Compute (1.035)^(-50). (1.035)^(-50) ≈ 1 / 5.516 ≈ 0.1813 Step 5: Substitute into the formula. P = 12000 * (1 - 0.1813) / 0.035 P = 12000 * 0.8187 / 0.035 P ≈ 12000 * 23.392 ≈ 280704 Using more precise computation yields a present value of about 281468.06 dollars, which matches the given answer choice. Verification / Alternative check: The total of all benefits over 25 years is: Total nominal payments = 12000 * 50 = 600000 dollars The present value of 281,468.06 dollars is much less than 600,000 dollars because the payments are spread over a long period and each future payment is discounted at 3.5% per half year. This order of magnitude is reasonable for an interest rate of 7% per year over 25 years. Why Other Options Are Wrong: 245678.00 and 234689.00: These are lower than the correct present value and would be insufficient to fund 12,000 dollars every 6 months for 25 years at the stated interest rate. 234578.00 and 201468.06: These figures are even smaller and would run out long before the last payment is due. 281468.06: This matches the precise calculation of the present value using the standard annuity formula. Common Pitfalls: Some learners mistakenly use the annual rate directly as i = 0.07 instead of halving it for semi-annual compounding. Others incorrectly multiply the total of all payments by a discount factor that does not reflect the timing of each payment. It is also common to confuse present value and future value formulas, which leads to answers of the wrong magnitude. Careful interpretation of the compounding frequency and payment timing is essential for solving such problems. Final Answer: The present value required today to fund 12,000 dollar payments every 6 months for 25 years at 7% per annum compounded semi-annually is approximately 281468.06 dollars.

Question 10

A credit card account shows an average daily balance of $8,431.10 and the monthly interest rate (finance rate) is 1.4%. What is the finance charge for the month?

Accepted Answer

Introduction: This problem is a straightforward application of simple percentage calculation for a monthly finance charge. Credit card companies often compute interest using the average daily balance multiplied by a monthly interest rate. We are given the average daily balance and the monthly interest rate and must find the finance charge for that month. Given Data / Assumptions: Average daily balance, B = 8,431.10 dollars Monthly interest rate, r_monthly = 1.4% per month No special adjustments or fees beyond the finance charge are considered Concept / Approach: The finance charge F for a given month, using the average daily balance method, is: F = B * r_monthly where r_monthly must be in decimal form. This is a simple interest calculation applied to one month based on the average daily balance over that billing cycle. Step-by-Step Solution: Step 1: Convert the monthly interest rate from percent to decimal. r_monthly = 1.4% = 1.4 / 100 = 0.014 Step 2: Multiply the average daily balance by the decimal rate. F = 8431.10 * 0.014 Step 3: Perform the multiplication. F ≈ 118.0354 Step 4: Round to the nearest cent. F ≈ 118.04 dollars Verification / Alternative check: As a sense check, 1% of 8,431.10 dollars is: 0.01 * 8431.10 = 84.311 An additional 0.4% is: 0.004 * 8431.10 = 33.7244 Adding them: 84.311 + 33.7244 ≈ 118.0354 Rounded, this again gives 118.04 dollars, confirming the calculation. Why Other Options Are Wrong: 98.04: This would correspond to a lower monthly rate than 1.4%, around 1.16%, and does not match the given percentage. 88.04 and 68.04: These figures are too small and would require much lower interest rates, not the stated 1.4%. 128.04: This is higher than the correct finance charge and would reflect a higher monthly rate. 118.04: This matches the computed finance charge at 1.4% of the average daily balance. Common Pitfalls: Some learners incorrectly apply an annual rate instead of the monthly rate or misinterpret 1.4% as 0.14 in decimal form. Another common error is rounding the monthly rate too early or performing the multiplication incorrectly. It is also important to remember that the average daily balance has already accounted for variations in daily usage, so we simply multiply by the monthly rate, not by the number of days again. Final Answer: The finance charge for the month on an average daily balance of 8,431.10 dollars at a monthly rate of 1.4% is 118.04 dollars.

Question 11

A loan of $3,500 is to be repaid with equal annual payments over 4 years at an interest rate of 9% per annum, compounded annually. What is the amount of each annual payment (to the nearest cent)?

Accepted Answer

Introduction: This problem is an example of loan amortization. A loan of 3,500 dollars is repaid in equal annual installments over 4 years at an interest rate of 9% per annum. We must determine the size of each annual payment that will fully pay off the loan plus interest by the end of the 4th year. Given Data / Assumptions: Principal, P = 3,500 dollars Annual interest rate, i = 9% = 0.09 Number of payments (years), n = 4 Payments are made at the end of each year (ordinary annuity) The loan is fully amortized by the end of the 4th payment Concept / Approach: The present value of an annuity of equal payments R, at interest rate i for n periods, is: P = R * (1 - (1 + i)^(-n)) / i Here P is the loan amount. We are given P, i, and n, and must solve for the payment R: R = P * i / (1 - (1 + i)^(-n)) Step-by-Step Solution: Step 1: Substitute P, i, and n into the formula. P = 3500, i = 0.09, n = 4 R = 3500 * 0.09 / (1 - (1.09)^(-4)) Step 2: Compute (1.09)^4. (1.09)^2 ≈ 1.1881 (1.09)^4 ≈ 1.1881 * 1.1881 ≈ 1.4116 Step 3: Find (1.09)^(-4). (1.09)^(-4) = 1 / 1.4116 ≈ 0.7085 Step 4: Compute the denominator. 1 - (1.09)^(-4) ≈ 1 - 0.7085 = 0.2915 Step 5: Compute the numerator. 3500 * 0.09 = 315 Step 6: Divide numerator by denominator. R ≈ 315 / 0.2915 ≈ 1080.34 So each annual payment is about 1080.34 dollars. Verification / Alternative check: We can verify by computing the present value of four payments of 1080.34 dollars: P ≈ 1080.34 * (1 - (1.09)^(-4)) / 0.09 Using the previously computed factor: P ≈ 1080.34 * 0.2915 / 0.09 ≈ 1080.34 * 3.238 ≈ 3500 Minor differences arise from rounding, but the result is sufficiently close to 3,500 dollars, confirming that 1080.34 dollars is the correct payment amount. Why Other Options Are Wrong: 890.60: This is too small and would not generate enough present value to cover the 3,500 dollar loan at 9% interest. 1089.00 and 1070.00: These are close but do not match the exact amortization formula, and would either slightly overpay or underpay over the 4 years. 980.34: This is significantly lower than the correct payment and would leave a balance at the end of 4 years. 1080.34: This matches the payment obtained using the proper loan amortization formula. Common Pitfalls: Some learners mistakenly divide the loan amount by 4 and then add yearly interest separately, which ignores the time value of money and the amortization structure. Others use the simple interest formula P(1 + i * n) instead of the annuity present value formula. It is also easy to miscalculate the negative exponent or to forget to convert the interest rate from percent to decimal form. Final Answer: Each annual payment required to amortize a 3,500 dollar loan over 4 years at 9% interest is approximately 1080.34 dollars.

Question 12

A loan of $4,800 is made at an annual percentage rate (APR) of 12%, with repayments made monthly for 24 months. What is the total finance charge (total interest paid) over the life of the loan?

Accepted Answer

Introduction: This problem involves an installment loan repaid in equal monthly payments. A loan of 4,800 dollars is taken at an APR of 12%, and the loan is to be repaid in 24 monthly installments. We are asked for the finance charge, that is, the total interest paid over the entire loan term. Given Data / Assumptions: Loan principal, P = 4,800 dollars APR (nominal annual interest rate) = 12% per annum Compounding and payment frequency: monthly, 12 periods per year Number of payments, n = 24 months Payments are equal and made at the end of each month Concept / Approach: For a loan repaid by equal monthly installments, the fixed payment R is given by the standard annuity formula: R = P * i * (1 + i)^n / ((1 + i)^n - 1) where: i is the monthly interest rate, i = APR / 12 n is the total number of payments Once R is found, the total amount repaid is: Total repaid = R * n The finance charge (total interest paid) is: Finance charge = Total repaid - P Step-by-Step Solution: Step 1: Compute the monthly interest rate. APR = 12% per annum, so i = 12% / 12 = 1% per month = 0.01 Step 2: Use the loan payment formula. R = 4800 * 0.01 * (1.01)^24 / ((1.01)^24 - 1) Step 3: Compute (1.01)^24. (1.01)^24 ≈ 1.26824 Step 4: Substitute into the formula. R ≈ 4800 * 0.01 * 1.26824 / (1.26824 - 1) R ≈ 48 * 1.26824 / 0.26824 R ≈ 60.876 / 0.26824 ≈ 226.0 (approximately 225.95 with more precision) Step 5: Compute total amount repaid. Total repaid ≈ 225.95 * 24 ≈ 5422.80 Step 6: Compute the finance charge. Finance charge ≈ 5422.80 - 4800 = 622.80 Verification / Alternative check: We can quickly check reasonableness: a 12% annual rate on 4,800 dollars over 2 years might be expected to produce total interest a little over 10% of the principal, since the balance is declining as it is repaid. An interest total around 600 to 650 dollars is reasonable, and 622.80 fits this expectation well. Why Other Options Are Wrong: 522.80 and 322.80: These underestimate the finance charge and would correspond to either a lower APR or a shorter term. 632.80: This slightly overestimates the amount of interest and does not match the precise calculated value. 422.80: This is too low for a 12% APR on this size of loan over 24 months. 622.80: This matches the finance charge obtained from the correct amortization calculation. Common Pitfalls: Some learners mistakenly apply simple interest P * APR * (term in years) and ignore the fact that the loan balance is declining, which leads to an overestimate of interest. Others try to divide the principal by 24 and then add a flat interest amount each month, which does not correctly reflect amortization. Miscomputing the monthly rate (for example, using 12% instead of 1%) is another frequent error that can completely distort the results. Final Answer: The total finance charge (total interest paid) on the 4,800 dollar loan repaid over 24 months at 12% APR is approximately 622.80 dollars.

Question 13

A nominal annual interest rate of 9.0% is compounded quarterly. What is the periodic interest rate applied each quarter?

Accepted Answer

Introduction: This question checks the basic concept of converting a nominal annual interest rate into a rate per compounding period. The nominal annual rate is 9.0%, and it is compounded quarterly. We need to find the corresponding interest rate that is applied to the account each quarter. Given Data / Assumptions: Nominal annual interest rate, r_nominal = 9.0% per year Compounding frequency: quarterly, that is 4 times per year We assume the same periodic rate is used in each quarter Concept / Approach: When interest is compounded m times per year at a nominal annual rate r_nominal, the periodic rate per compounding period is: i_periodic = r_nominal / m However, r_nominal must first be expressed as a percentage or decimal consistently. Since the question asks for the periodic interest rate in percent, we can perform the calculation directly with percent values: Step-by-Step Solution: Step 1: Note the compounding frequency. Compounding quarterly means m = 4 periods per year. Step 2: Divide the nominal annual rate by the number of periods. Periodic rate = 9.0% / 4 Periodic rate = 2.25% So, each quarter, an interest rate of 2.25% is applied to the balance. Verification / Alternative check: We can confirm this by converting to decimal form as well: r_nominal_decimal = 9.0% = 0.09 i_periodic_decimal = 0.09 / 4 = 0.0225 Converting back to a percentage: 0.0225 * 100 = 2.25% This confirms that the periodic rate is indeed 2.25% per quarter. Why Other Options Are Wrong: 3.45%: This rate is too high for a quarterly rate when the nominal annual rate is only 9%. 5.25% and 6.25%: These are much too large and would imply annual rates far above 9% if used as periodic quarterly rates. 1.25%: This is too small and would correspond to a nominal annual rate of only 5% if compounded quarterly. 2.25%: This is exactly one quarter of the 9% nominal rate and is therefore the correct periodic rate. Common Pitfalls: A typical error is confusing the periodic rate with the effective annual rate, which requires using (1 + i_periodic)^m - 1. Here the question asks only for the rate per quarter, not the effective annual rate. Another mistake is dividing by 12 instead of 4, treating the rate as if it were compounded monthly. Care must be taken to match the correct frequency with the nominal annual rate. Final Answer: The periodic interest rate applied each quarter when the nominal annual rate is 9.0% compounded quarterly is 2.25% per quarter.

Question 14

A nominal annual interest rate of 9.5% is compounded monthly. What is the corresponding periodic interest rate per month, expressed as a percentage rounded to four decimal places?

Accepted Answer

Introduction / Context:This question checks understanding of the relationship between a nominal annual interest rate and the corresponding periodic interest rate when interest is compounded more than once per year. Being able to convert between nominal and periodic rates is very important in topics like compound interest, loans, and investments, because most formulas use the periodic rate, not the nominal annual figure. Given Data / Assumptions: Nominal annual interest rate = 9.5% per annum. Compounding frequency = monthly. Number of compounding periods in one year, m = 12. We assume standard financial convention where the nominal rate is divided equally among all periods. Concept / Approach:The periodic interest rate is the rate applied each compounding period. For a nominal annual rate r_nom with m compounding periods per year, the periodic rate r_per is calculated using the simple relationship:r_per = r_nom / m.Here r_nom is 9.5% and m is 12, so we divide 9.5 by 12 and then round the result to four decimal places. The answer should be expressed as a percentage per month. Step-by-Step Solution:Step 1: Write the nominal annual rate as r_nom = 9.5%.Step 2: Identify the compounding frequency per year as m = 12 for monthly compounding.Step 3: Use the formula r_per = r_nom / m.Step 4: Compute r_per = 9.5 / 12.Step 5: 9.5 / 12 = 0.791666..., which is approximately 0.7916 when rounded to four decimal places.Step 6: Attach the percent sign to express the answer as 0.7916% per month. Verification / Alternative check:As a quick check, multiply the periodic rate back by 12. That is 0.791666... * 12 which gives 9.5% again. This confirms that the periodic rate is consistent with the original nominal annual rate. Any large deviation from 9.5% when multiplying back would indicate a calculation error, such as dividing by the wrong number of periods or misplacing the decimal point. Why Other Options Are Wrong: 0.8916%: This would correspond to a nominal rate of about 10.7% if multiplied by 12, which is higher than the given 9.5%. 0.9916%: Multiplying by 12 gives almost 11.9%, which is far from 9.5%, so it does not match the given nominal rate. 0.6916%: Multiplying by 12 gives about 8.3%, which is lower than the required 9.5% nominal rate. Common Pitfalls:Students often confuse effective rates with nominal rates and sometimes try to use complex formulas unnecessarily. Here, the question only asks for the periodic rate for a nominal rate with simple equal division among periods. Another common mistake is to forget to convert the result into percentage form or to round incorrectly. Some learners also divide by the wrong number of periods, for example by 4 for quarterly compounding even though the question clearly states monthly compounding, which uses 12 periods per year. Final Answer:The periodic interest rate corresponding to a nominal annual rate of 9.5% compounded monthly is 0.7916% per month.

Question 15

A loan has a nominal annual interest rate of 8.4% per annum. If the periodic interest rate is 4.2% per period, what is the compounding frequency per year (number of compounding periods in one year)?

Accepted Answer

Introduction / Context:This problem tests the concept of how a nominal annual interest rate is related to a periodic interest rate and the number of compounding periods per year. Understanding this relationship is important in many financial calculations because it helps you interpret how often interest is actually added to the principal. Given Data / Assumptions: Nominal annual interest rate r_nom = 8.4% per annum. Periodic interest rate r_per = 4.2% per period. Interest is compounded a fixed number of times per year, which we denote by m. The nominal rate is assumed to be evenly split across all periods. Concept / Approach:The relationship between nominal annual rate, periodic rate, and compounding frequency is given by r_nom = m * r_per, where r_nom is the nominal annual rate in percent, r_per is the periodic rate in percent, and m is the number of compounding periods per year. To find m, we simply divide the nominal annual rate by the periodic rate. Once we obtain m, we match it with the correct option. Step-by-Step Solution:Step 1: Write the nominal annual rate as r_nom = 8.4%.Step 2: Write the periodic rate as r_per = 4.2% per period.Step 3: Use the formula r_nom = m * r_per, so m = r_nom / r_per.Step 4: Compute m = 8.4 / 4.2.Step 5: 8.4 / 4.2 = 2.Step 6: Therefore, the compounding frequency is m = 2 times per year, which usually corresponds to semiannual compounding. Verification / Alternative check:We can verify the answer by reversing the calculation. If m = 2 and each period has an interest rate of 4.2%, then the nominal annual rate would be 2 * 4.2% = 8.4%, which matches the data given in the question. Since the calculated nominal rate is consistent, the frequency of 2 compounding periods per year is correct. Why Other Options Are Wrong: 1: If m were 1, then the nominal rate would equal the periodic rate, that is 4.2%, which does not match the given 8.4%. 3: If m were 3, then the nominal rate would be 3 * 4.2% = 12.6%, which is greater than 8.4%. 4: If m were 4, then the nominal rate would be 4 * 4.2% = 16.8%, which is also not equal to 8.4%. Common Pitfalls:A common mistake is to confuse nominal and effective rates and try to use exponent formulas unnecessarily. Another error is to divide in the wrong order, such as 4.2 / 8.4, which gives 0.5 and does not correspond to a meaningful compounding frequency. Learners may also forget that the rates in this formula must both be expressed in the same units, here both in percent per year and percent per period, so that they cancel correctly when calculating m. Final Answer:The compounding frequency is 2 times per year, corresponding to semiannual compounding.

Question 16

For a nominal annual interest rate of 8.4% per annum, the periodic interest rate is 8.4% per period. What is the compounding frequency per year (that is, how many times per year is interest compounded)?

Accepted Answer

Introduction / Context:This question examines the simple but important case where the periodic interest rate is equal to the nominal annual interest rate. It tests whether you understand what that implies about the number of compounding periods per year and the meaning of annual compounding. Given Data / Assumptions: Nominal annual interest rate r_nom = 8.4% per annum. Periodic interest rate r_per = 8.4% per period. The nominal rate is spread evenly across m compounding periods per year. We assume standard financial interpretation of nominal and periodic rates. Concept / Approach:The core relationship is r_nom = m * r_per, where m is the compounding frequency per year. If the periodic rate equals the nominal rate, then the only way the equality can hold is when there is exactly one compounding period per year. This case describes annual compounding, where interest is added once per year at the full nominal rate. Step-by-Step Solution:Step 1: Nominal annual rate r_nom = 8.4%.Step 2: Periodic rate r_per = 8.4% per period.Step 3: Use the formula r_nom = m * r_per.Step 4: Substitute values: 8.4 = m * 8.4.Step 5: Divide both sides by 8.4 to get m = 1.Step 6: Therefore, the interest is compounded once per year, which is annual compounding. Verification / Alternative check:To verify, think about what happens in practice. If an investment promises 8.4% per year compounded annually, the full 8.4% is applied once at the end of the year. This exactly matches the idea of a periodic rate of 8.4% per year with only one period, so the calculation is consistent and intuitive. Why Other Options Are Wrong: 2: If m were 2, then the nominal rate would be 2 * 8.4% = 16.8%, which is not the given 8.4%. 3: If m were 3, then the nominal rate would be 3 * 8.4% = 25.2%, far larger than 8.4%. 4: If m were 4, then the nominal rate would be 4 * 8.4% = 33.6%, again not equal to 8.4%. Common Pitfalls:Students sometimes misinterpret the phrase compounded annually and think there might still be more than one compounding period inside a year. In fact, annual compounding means exactly one compounding period per year. Another error is to mix up nominal and effective rates. In this question, the word nominal is present, but the mathematics relies only on the nominal and periodic rates being equal, which forces the compounding frequency to be one. Final Answer:The compounding frequency is 1 time per year, which corresponds to annual compounding.

Question 17

A nominal annual interest rate of 8.4% per annum is quoted. If the periodic interest rate is 2.1% per period, what is the compounding frequency per year (number of compounding periods each year)?

Accepted Answer

Introduction / Context:Here we again connect a nominal annual interest rate with a given periodic interest rate and use that to find how many times interest is compounded every year. This type of problem is common in banking and finance, where a lender specifies a nominal rate and an implied compounding frequency. Given Data / Assumptions: Nominal annual interest rate r_nom = 8.4% per annum. Periodic interest rate r_per = 2.1% per period. Compounding frequency per year m is unknown. The nominal rate is evenly distributed across all periods within the year. Concept / Approach:The fundamental relationship is r_nom = m * r_per, where r_nom is nominal annual rate, r_per is periodic rate, and m is the number of compounding periods per year. If we know r_nom and r_per, we can solve for m by simple division. Because both rates are given in percent form, we can work directly with the numerical values 8.4 and 2.1. Step-by-Step Solution:Step 1: Write r_nom = 8.4% and r_per = 2.1%.Step 2: Use the formula r_nom = m * r_per.Step 3: Rearrange to get m = r_nom / r_per.Step 4: Compute m = 8.4 / 2.1.Step 5: 8.4 / 2.1 = 4.Step 6: Therefore, interest is compounded 4 times per year, which corresponds to quarterly compounding. Verification / Alternative check:To verify, suppose interest is compounded quarterly, so m = 4. Then the nominal rate implied by a periodic rate of 2.1% is 4 * 2.1% = 8.4% per annum, which exactly matches the given nominal rate. This confirms that m = 4 is consistent with the information in the question. Why Other Options Are Wrong: 1: If m were 1, the nominal annual rate would be only 2.1%, not 8.4%. 2: If m were 2, the nominal rate would be 2 * 2.1% = 4.2%, which is half of 8.4%. 3: If m were 3, then 3 * 2.1% = 6.3%, still less than the required 8.4%. Common Pitfalls:Learners sometimes try to use compound interest formulas that involve exponents when this simple relationship is enough. Another mistake is not keeping track of units, for example mixing percentages with decimals without proper conversion. It is also common to perform the division in the reverse order, 2.1 / 8.4, which gives 0.25 and then misinterpret that figure as the answer. Remember that the compounding frequency must be a positive whole number of periods per year. Final Answer:The compounding frequency is 4 times per year, which corresponds to quarterly compounding.

Question 18

A nominal annual interest rate of 8.4% per annum is quoted. If the periodic interest rate is 0.7% per period, what is the compounding frequency per year (that is, the number of compounding periods each year)?

Accepted Answer

Introduction / Context:This question explores another situation where you must deduce the compounding frequency per year from the nominal annual interest rate and the periodic interest rate. It directly reinforces the formula that connects these quantities and highlights a common real life case, monthly compounding. Given Data / Assumptions: Nominal annual interest rate r_nom = 8.4% per annum. Periodic interest rate r_per = 0.7% per period. Compounding frequency per year m is unknown. The nominal rate is distributed uniformly across all compounding periods. Concept / Approach:The relationship between the nominal annual rate and the periodic rate is r_nom = m * r_per. Given r_nom and r_per, we can solve for m using simple division. Once m is found, we can interpret it. If m equals 12, that directly corresponds to monthly compounding, since there are 12 months in a year. Step-by-Step Solution:Step 1: Write r_nom = 8.4% and r_per = 0.7%.Step 2: Use the formula r_nom = m * r_per.Step 3: Rearrange to get m = r_nom / r_per.Step 4: Compute m = 8.4 / 0.7.Step 5: 8.4 / 0.7 = 12.Step 6: Therefore, the compounding frequency is m = 12 times per year, which corresponds to monthly compounding. Verification / Alternative check:To verify, suppose interest is compounded monthly so that m = 12. If each period has rate 0.7%, the nominal annual rate would be 12 * 0.7% = 8.4% per annum, which is exactly the rate given. This confirms that our calculation is consistent and that a compounding frequency of 12 is correct for this periodic rate. Why Other Options Are Wrong: 9: If m were 9, the nominal rate would be 9 * 0.7% = 6.3%, less than 8.4%. 10: If m were 10, the nominal rate would be 7%, still lower than 8.4%. 11: If m were 11, the nominal rate would be 7.7%, which is also less than 8.4%. Common Pitfalls:One common mistake is to confuse nominal rates with effective annual rates and attempt to use logarithms or exponential formulas. For this type of question, that is not needed. Another pitfall is incorrect arithmetic when dividing 8.4 by 0.7, especially if students treat the numbers as decimals without careful computation. It is also important to remember that the answer must be an integer since it represents a count of compounding periods per year, so fractional answers are not meaningful in this context. Final Answer:The compounding frequency is 12 times per year, corresponding to monthly compounding.

Question 19

The periodic interest rate on an investment is 0.83% per month. Assuming this is a nominal rate quoted monthly, determine the corresponding nominal annual rate of interest (in percent per annum).

Accepted Answer

Introduction / Context:This problem focuses on converting a periodic monthly interest rate into a nominal annual interest rate. Such conversions are very common when comparing financial products, because banks and investment firms often quote either monthly or annual rates, and you must be able to interpret one in terms of the other. Given Data / Assumptions: Periodic interest rate r_per = 0.83% per month. Compounding frequency per year m = 12 months. The nominal annual rate is defined as r_nom = m * r_per. We are asked to find the nominal annual rate r_nom in percent per annum. Concept / Approach:The nominal annual interest rate is obtained by multiplying the periodic monthly rate by the number of periods per year. This nominal rate does not itself describe compounding but is a simple linear scaling of the monthly rate. The formula used is r_nom = m * r_per, where r_per is in percent per month and m is 12 for monthly compounding over one year. Step-by-Step Solution:Step 1: Note the monthly periodic rate r_per = 0.83%.Step 2: The number of months in a year is m = 12.Step 3: Use the formula r_nom = m * r_per.Step 4: Compute r_nom = 12 * 0.83%.Step 5: 12 * 0.83 = 9.96.Step 6: Therefore, the nominal annual rate is approximately 9.96% per annum, which rounds to 10% per annum. Verification / Alternative check:This is mainly a linear scaling, so for a quick check, approximate 0.83 as a little less than 0.84. Then 12 * 0.84 = 10.08%, which is very close to 10%. Since our exact result was 9.96%, it makes sense to round to the nearest whole percent, which is exactly 10%. Therefore, the answer 10% is reasonable and matches the available options. Why Other Options Are Wrong: 7%: This would imply a monthly rate of about 0.58%, not 0.83%. 8%: This corresponds to approximately 0.67% per month, which is lower than the given monthly rate. 9%: This would imply a monthly rate of 0.75%, still below 0.83% per month. Common Pitfalls:Students sometimes treat 0.83% as a decimal 0.83 instead of converting to 0.83 percent properly, which leads to huge errors. Others mistakenly apply compound interest formulas and raise numbers to powers of 12, which is not required when the question asks specifically for the nominal annual rate. It is also common to forget that 12 months make one year and to use an incorrect number of periods, such as 10 or 6, leading to wrong annual rates. Final Answer:The nominal annual rate corresponding to a periodic rate of 0.83% per month is approximately 10% per annum.

Question 20

An investor can choose between an interest rate of 10.5% per annum compounded monthly and 11% per annum compounded annually for a two year investment. Other things being equal, which option is financially better for the investor?

Accepted Answer

Introduction / Context:This question asks you to compare two investment choices with different nominal interest rates and different compounding frequencies over a fixed period of two years. It tests understanding of effective interest rates and how compounding frequency affects the overall return, even when nominal rates are close in value. Given Data / Assumptions: Option 1: 10.5% per annum, compounded monthly. Option 2: 11% per annum, compounded annually. Investment horizon: two years. The same principal amount is invested under either option, and there are no additional deposits or withdrawals. Concept / Approach:To compare the options, we need to compute the effective growth factor over two years for each choice. For a nominal rate j with m compounding periods per year, the periodic rate is j / m, and the effective annual growth factor is (1 + j / m)^m. Over two years, we raise the annual growth factor to the power of 2. The option that gives a larger final multiplier on the principal is better, regardless of the actual principal value, because the same principal would be scaled by that factor in both cases. Step-by-Step Solution:Step 1: For Option 1, nominal rate j1 = 0.105 and m1 = 12, so periodic rate i1 = 0.105 / 12.Step 2: The effective annual factor for Option 1 is (1 + i1)^12.Step 3: Over two years, the total growth factor for Option 1 is (1 + i1)^(24).Step 4: For Option 2, the annual factor is simply (1 + 0.11), because interest is compounded once per year.Step 5: Over two years, the growth factor for Option 2 is (1.11)^2.Step 6: Numerical calculation shows that (1 + 0.105 / 12)^(24) is slightly greater than (1.11)^2, so Option 1 yields a slightly higher amount. Verification / Alternative check:As a quick mental check, compute the approximate effective annual rate for the monthly compounding option. The effective annual rate is (1 + 0.105 / 12)^12 minus 1, which is slightly above 11% per year. Since Option 2 offers exactly 11% effective per year, the first option with monthly compounding must be marginally better. Over two years, this small difference in annual effectiveness produces a slightly higher final amount for Option 1. Why Other Options Are Wrong: 11% compounded annually is better: This is not correct, because the monthly compounding at 10.5% produces an effective annual rate that slightly exceeds 11%. Both options give exactly the same total return: The effective rates are very close but not identical, so the returns are not exactly equal. It is impossible to compare without knowing the principal amount: The principal cancels out in a ratio comparison, so any principal value leads to the same conclusion about which multiplier is larger. Common Pitfalls:Many students look only at the nominal rates and conclude that 11% must be better than 10.5% without considering compounding frequency. Others use the nominal rates directly in calculations without dividing by the number of compounding periods or they compare only one-year returns when the question specifies a two year horizon. It is essential to convert nominal rates into effective growth factors over the exact period of interest before making a comparison. Final Answer:The investor should prefer the first option, because 10.5% compounded monthly is better, but only slightly better than 11% compounded annually over the two year period.

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