Suppose you want to invest in a 3-year certificate of deposit (CD). You visit four banks to compare their CD rates.
Bank A offers annual interest compounded annually, Bank B offers compounded monthly, Bank C offers
compounded quarterly, and Bank D offers compounded daily. Determine which bank is offering the best
deal. The bank that offers the best deal is the one with the highest effective interest rate.
A

Ask by Medina Cook. in the United States

Mar 23,2025

Question

Suppose you want to invest in a 3-year certificate of deposit (CD). You visit four banks to compare their CD rates.
Bank A offers annual interest compounded annually, Bank B offers compounded monthly, Bank C offers
compounded quarterly, and Bank D offers compounded daily. Determine which bank is offering the best
deal. The bank that offers the best deal is the one with the highest effective interest rate.
A

Ask by Medina Cook. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

Let the nominal annual rate be $$ r $$ and the number of compounding periods per year be $$ n $$. The effective annual rate (EAR) is given by

$$
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1.
$$

### Bank A
- Nominal rate: $$ r_A = 0.0175 $$ (or $$ 1.75\% $$)
- Compounded annually: $$ n_A = 1 $$

$$
\text{EAR}_A = \left(1+\frac{0.0175}{1}\right)^1 - 1 = 0.0175 \quad \text{or } 1.75\%.
$$

### Bank B
- Nominal rate: $$ r_B = 0.0180 $$ (or $$ 1.80\% $$)
- Compounded monthly: $$ n_B = 12 $$

$$
\text{EAR}_B = \left(1 + \frac{0.0180}{12}\right)^{12} - 1.
$$

Calculate the monthly rate:

$$
\frac{0.0180}{12} = 0.0015.
$$

Then,

$$
\text{EAR}_B = \left(1.0015\right)^{12} - 1.
$$

Using a calculator or approximation,

$$
\left(1.0015\right)^{12} \approx 1.01816,
$$

so

$$
\text{EAR}_B \approx 0.01816 \quad \text{or } 1.816\%.
$$

### Bank C
- Nominal rate: $$ r_C = 0.0170 $$ (or $$ 1.70\% $$)
- Compounded quarterly: $$ n_C = 4 $$

$$
\text{EAR}_C = \left(1 + \frac{0.0170}{4}\right)^4 - 1.
$$

Compute the quarterly rate:

$$
\frac{0.0170}{4} = 0.00425.
$$

So,

$$
\text{EAR}_C = \left(1.00425\right)^4 - 1.
$$

Approximating,

$$
\left(1.00425\right)^4 \approx 1.0171,
$$

thus

$$
\text{EAR}_C \approx 0.0171 \quad \text{or } 1.71\%.
$$

### Bank D
- Nominal rate: $$ r_D = 0.0165 $$ (or $$ 1.65\% $$)
- Compounded daily: $$ n_D = 365 $$

$$
\text{EAR}_D = \left(1 + \frac{0.0165}{365}\right)^{365} - 1.
$$

Calculate the daily rate:

$$
\frac{0.0165}{365} \approx 0.00004521.
$$

Then, approximating using the exponential relation,

$$
\text{EAR}_D \approx e^{0.0165} - 1 \approx 0.016637 \quad \text{or } 1.66\%.
$$

### Comparison and Conclusion
- Bank A: $$ 1.75\% $$
- Bank B: $$ \approx 1.82\% $$
- Bank C: $$ \approx 1.71\% $$
- Bank D: $$ \approx 1.66\% $$

Thus, **Bank B** offers the highest effective annual rate, making it the best deal.
Bank B offers the best deal with an effective annual rate of approximately $$ 1.82\% $$.

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