Actual/360 Interest Rate Convention

by | Aug 30, 2023 | finance

 

In this assignment, you will solve problems on Interest Rate Conventions.
Suppose an investment of $1 made today will be worth $1.03 in three months.
1. If the interest rate ℓ is expressed in the Actual/360 convention and the three-month horizon has 91 days in it, what is ℓ ?
If the interest rate r is expressed in the continuous-compounding convention and we treat three months as 1/4 years, what is r ?
Consider an investment of $1 over a horizon of one month.
If the interest rate ℓ expressed in the Actual/360 convention is 4% and the one-month horizon has 31 days in it, to what does the invested amount grow to?
If you had to express the same outcome using a continuous-compounding convention, and we treat one month as 1/12 of a year, what is the continuously-compounded rate r ?
Consider an investment of $1 over a horizon of one month.
If the interest rate r expressed in the continuously–compounded terms is 4% and we treat the one month horizon as 1/12 of a year, to what does the invested amount grow?
If you had to express the same outcome using an Actual/360 convention and the one month horizon has 31 days in it, what is the rate ℓ ?

 

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.


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God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.

Sample Answer

Sample Answer

Problem 1: Actual/360 Interest Rate Convention

To find the interest rate ℓ in the Actual/360 convention with a three-month horizon of 91 days, we can use the formula:

ℓ = (Future Value – Present Value) / (Present Value * Number of Days / 360Given that the Present Value (PV) is $1, the Future (F) is $1.03, and Number of Days is 91, we can substitute these values the formula:

ℓ = ($1.03 – $) / ($1 * 91/360)

Simpl,

=0.03 / ($1 * 0.2528)

ℓ ≈ 0.7 or 11.87%

Therefore, the interest rate ℓ in the Actual/360 convention is approximately 11.87%.

Problem 2: Continuous-Compounding Interest Rate Convention

To find the continuous-compounding rate r with a three-month horizon treated as 1/4 years, we can use the formula:

r = ln(Future Value / Present Value) / Time

Given that the Present Value (PV) is $1, the Future Value (FV) is $1.03, and the Time is 1/4 years, we can substitute these values into the formula:

r = ln($1.03 / $1) / (1/4)

Simplifying,

r = ln(1.03) / 0.25

Using a calculator,

r ≈ 0.1197 or 11.97%

Therefore, the continuous-compounding rate r is approximately 11.97%.

Problem 3: Actual/360 Interest Rate Convention

To find the value to which the investment grows over a one-month horizon using the Actual/360 convention, we can use the formula:

Future Value = Present Value * (1 + ℓ * Number of Days / 360)

Given that the Present Value (PV) is $1, the interest rate ℓ is 4%, and the Number of Days is 31, we can substitute these values into the formula:

Future Value = $1 * (1 + 0.04 * 31/360)

Simplifying,

Future Value = $1 * (1 + 0.0114)

Future Value ≈ $1.0114

Therefore, the investment grows to approximately $1.0114.

Problem 4: Continuous-Compounding Interest Rate Convention

To find the continuously-compounded rate r to achieve the same outcome as in Problem 3, we can use the formula:

r = ln(Future Value / Present Value) / Time

Given that the Present Value (PV) is $1, the Future Value is $1.0114, and the Time is 1/12 years, we can substitute these values into the formula:

r = ln($1.0114 / $1) / (1/12)

Simplifying,

r = ln(1.0114) / (1/12)

Using a calculator,

r ≈ 0.0095 or 0.95%

Therefore, the continuously-compounded rate r is approximately 0.95%.

Problem 5: Actual/360 Interest Rate Convention

To find the interest rate ℓ in the Actual/360 convention with a one-month horizon of 31 days, we can use the formula:

ℓ = (Future Value – Present Value) / (Present Value * Number of Days / 360)

Given that the Present Value (PV) is $1, the Future Value is $1.0114 (from Problem 3), and the Number of Days is 31, we can substitute these values into the formula:

ℓ = ($1.0114 – $1) / ($1 * 31/360)

Simplifying,

ℓ = $0.0114 / ($1 * 0.0861)

ℓ ≈ 0.1316 or 13.16%

Therefore, the interest rate ℓ in the Actual/360 convention is approximately 13.16%.

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