Part 1Interest rate. Upstate Bank is offering long-term certificates of deposit (CDs) with a face value of $100,000100,000 (future value). Bank customers can buy these CDs today for $50,00050,000 and will receive the $100,000100,000 in 2020 years. What interest rate is the bank paying on these CDs? Part 1What interest rate is the bank paying on these CDs? (Round to two decimal places.)[input]enter your response here %

Question

Part 1Interest rate. Upstate Bank is offering​ long-term certificates of deposit​ (CDs) with a face value of ​$100,000100,000 ​(future value). Bank customers can buy these CDs today for ​$50,00050,000 and will receive the ​$100,000100,000 in 2020 years. What interest rate is the bank paying on these​ CDs? Part 1What interest rate is the bank paying on these​ CDs? ​(Round to two decimal​ places.)[input]enter your response here ​%

BlackTom AI · Accepted Answer

We are told that a CD has a face value (future value) of $100,000 but can be bought today for $50,000 and will mature in 20 years. This means the present value (PV) is $50,000, the future value (FV) is $100,000, and the number of periods (n) is 20 years, with a single compounding period per year.

The relationship between PV, FV, rate r, and time n for compounding annually is: FV = PV × (1 + r)^n.

To solve for r, rearrange the equation: (1 + r)^n = FV / PV. Here FV / PV = 100{,}000 / 50{,}000 = 2.

Thus, (1 + r)^{20} = 2. To isolate r, take the 20th root: 1 + r = 2^{1/20}.

Compute the 20th root of 2. Using natural logarithms or a calculator, 2^{1/20} ≈ exp(ln(2)/20) ≈ exp(0.6931 / 20) ≈ exp(0.034655) ≈ 1.035265.

Subtract 1 to solve for r: r ≈ 1.035265 − 1 ≈ 0.035265, which is about 3.5265%.

When rounding to two decimal places, this rate is approximately 3.53%.

Test:FINAL

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