You place a deposit for 91 days at 11.5% on an ACT/360 basis. What would the rate have been if it had been quoted on an ACT/365 basis? What is the effective yield on an ACT/365 basis and on an ACT/360 basis?
The correct answer and explanation is:
Answer:
To convert the interest rate from an ACT/360 basis to an ACT/365 basis, we use the following formula: r365=r360×360365r_{365} = r_{360} \times \frac{360}{365}
Given that r360=11.5%r_{360} = 11.5\%: r365=11.5%×360365r_{365} = 11.5\% \times \frac{360}{365} r365=11.5%×0.9863r_{365} = 11.5\% \times 0.9863 r365≈11.34%r_{365} \approx 11.34\%
Effective Yield Calculation:
The effective yield accounts for compounding and is calculated using the formula: Effective Yield=(1+r×tdenominator)−1\text{Effective Yield} = \left(1 + \frac{r \times t}{\text{denominator}}\right) – 1
For ACT/360: Effective Yield360=(1+0.115×91360)−1\text{Effective Yield}_{360} = \left(1 + \frac{0.115 \times 91}{360}\right) – 1 =(1+0.02906)−1= \left(1 + 0.02906\right) – 1 =2.91%= 2.91\%
For ACT/365: Effective Yield365=(1+0.1134×91365)−1\text{Effective Yield}_{365} = \left(1 + \frac{0.1134 \times 91}{365}\right) – 1 =(1+0.02828)−1= \left(1 + 0.02828\right) – 1 =2.83%= 2.83\%
Explanation:
Interest rates in finance are quoted based on different day count conventions. The ACT/360 method assumes a 360-day year, which is common in money markets and banking, whereas the ACT/365 method considers the actual number of days in a year, typically used in government securities and bonds.
Since ACT/360 uses fewer days in the denominator, it results in a higher annualized interest rate compared to ACT/365. To convert from ACT/360 to ACT/365, we adjust the rate by multiplying by 360365\frac{360}{365}, leading to a slightly lower rate.
The effective yield represents the actual return over the period. Since ACT/360 assumes a shorter year, it results in a slightly higher effective yield compared to ACT/365. In this case, the effective yield on an ACT/360 basis is 2.91%, while on an ACT/365 basis, it is 2.83%.
This distinction is crucial in financial calculations as it impacts interest earnings, loan costs, and return on investments.
I’ve generated a financial chart comparing ACT/360 and ACT/365 interest rate calculations. The image visually represents the formulas, effective yields, and the differences between both methods. Let me know if you need any modifications or further explanations!
