Amortization Formula Derivation

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Questions and Answers

What fundamental principle underlies the derivation of the amortization formula?

  • The total interest paid equals the sum of all repayments minus the principal.
  • The sum of all repayments equals the principal plus total interest paid.
  • The principal equals the sum of the present values of all repayments. (correct)
  • The future value of all repayments equals the principal.

In the context of deriving the amortization formula, what does 't' represent?

  • The total number of repayments. (correct)
  • The principal amount of the loan.
  • The total amount of interest paid over the loan term.
  • The annual interest rate.

Why is the sum of a geometric series formula necessary when deriving the amortization formula?

  • To adjust the interest rate for compounding periods.
  • To calculate the total interest paid on the loan.
  • To determine the principal amount of the loan.
  • To consolidate the present values of all future repayments into a single expression. (correct)

What is the first term ('a') in the geometric series when deriving the amortization formula?

<p>A / (1 + i)^t (C)</p> Signup and view all the answers

What is the common ratio ('r') of the geometric series used in deriving the amortization formula?

<p>1 + i (B)</p> Signup and view all the answers

When simplifying the equation to isolate 'A' in the amortization formula derivation, what is the result of (1 - (1 + i))?

<p>-i (D)</p> Signup and view all the answers

Which formula is essential for calculating the present value of individual repayments in the amortization formula derivation?

<p>PV = FV / (1 + i)^t (A)</p> Signup and view all the answers

What algebraic manipulation is most helpful in simplifying the derivation of the amortization formula?

<p>Reversing the order of terms in the geometric series (D)</p> Signup and view all the answers

If the numerator and denominator are multiplied by -1, which part of the amortization formula is changed?

<p>The sign of the terms involving 'i' and '(1 + i)^t' (B)</p> Signup and view all the answers

Following the correct derivation of the amortization formula, which variable is isolated on one side of the equation?

<p>A (monthly repayment) (A)</p> Signup and view all the answers

The standard amortization formula is: $A = P * (i * (1 + i)^t) / ((1 + i)^t - 1)$. Which component represents the effect of compounding interest over the loan term?

<p>$(1 + i)^t$ (B)</p> Signup and view all the answers

Suppose you are about to derive the amortization formula. What is the FIRST step?

<p>Start with the premise that the principal equals the sum of the present value of all repayments. (B)</p> Signup and view all the answers

How does increasing the interest rate 'i' affect the monthly repayments 'A', assuming the principal 'P' and the number of months 't' remain constant?

<p>'A' increases. (B)</p> Signup and view all the answers

Consider two loans with the same principal 'P' and interest rate 'i', but Loan 1 has a term of 't' months and Loan 2 has a term of '2t' months. How does the monthly repayment 'A' differ between the two loans?

<p>Loan 1 has a higher 'A'. (D)</p> Signup and view all the answers

When deriving the amortization formula, why is it important to discount future repayments to their present value?

<p>To reflect opportunities to earn interest on the money. (B)</p> Signup and view all the answers

How does the amortization formula reflect the trade-off between the interest rate 'i' and the loan term 't'?

<p>For a fixed principal, increasing 'i' will increase 'A', but increasing 't' can decrease 'A'. (D)</p> Signup and view all the answers

What key insight does the amortization formula provide regarding the allocation of each payment?

<p>Each payment covers a varying amount of principal and interest, with the interest portion decreasing over time. (D)</p> Signup and view all the answers

What adjustment is needed when the effective interest rate changes during the term of the loan?

<p>The remaining payments are recalculated with a new amortization schedule. (C)</p> Signup and view all the answers

During the derivation, you have: $P = (A / (1 + i)^t) * (1 - (1 + i)^t) / (1 - (1 + i))$. What is the next step to isolate A?

<p>Multiply both sides by $ (1 + i)^t * (1 - (1 + i)) / (1 - (1 + i)^t)$ (B)</p> Signup and view all the answers

In the amortization formula $A = P * (i * (1 + i)^t) / ((1 + i)^t - 1)$, which of the following demonstrates the direct effect of the amount of principal?

<p>If P increases, A tends to increase proportionally, unless other factors change. (B)</p> Signup and view all the answers

Flashcards

Amortization Formula

A = P * (i * (1 + i)^t) / ((1 + i)^t - 1), where A = monthly repayments, P = principal, i = interest rate, t = number of months.

Present Value Formula

PV = FV / (1 + i)^t, calculates today's worth of a future sum.

Geometric Series Sum

Sn = a * (1 - r^n) / (1 - r), finds the sum of 'n' terms in a series.

Amortization Formula Derivation Key

The loan amount should equal the sum of the present values of each repayment.

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Present Value of Repayments

Each repayment's value today, considering interest and time.

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Repayments as a Geometric Series

A sequence of repayments forms a geometric series, where each term is multiplied by (1 + i).

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First Term in Reversed Series

First term is A / (1 + i)^t.

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Common Ratio in Series

Each term is multiplied by 1 + i to get the next term

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Number of Terms

The total number of repayments made.

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Substitution into Geometric Series Formula

Substitute a, r, and n into Sn = a * (1 - r^n) / (1 - r) to find the total present value of all repayments.

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Isolating 'A'

Isolating A in the equation allows you to derive the amortization formula.

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Simplification

(1 - (1 + i)) simplifies to -i.

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Study Notes

Amortization Formula Derivation

  • The amortization formula derivation is a common exam question.
  • It is located on page 31 of the log tables.
  • The formula is: A = P * (i * (1 + i)^t) / ((1 + i)^t - 1)
    • A = monthly repayments
    • P = principal (total mortgage amount)
    • i = interest rate
    • t = number of months
  • The principal equals the sum of the present value of all repayments.

Present Value and Geometric Series

  • Present value formula (page 30 of log tables): PV = FV / (1 + i)^t
  • Applying this formula requires using the sum of a geometric series.
  • Sum of a geometric series formula (page 22 of log tables): Sn
  • 'a' represents individual repayments
  • Need to calculate each repayment's present value
  • The Sn formula sums all present values.

Derivation Steps

  • The first mortgage repayment occurs one month after the loan begins.
  • Present value of the first repayment: A / (1 + i)^1
  • Present value of the second repayment: A / (1 + i)^2
  • The sum of all present values equals P.
  • Reversing the terms clarifies the algebra.
  • First term (a), common ratio (r), and number of terms (n) are required.
  • First term: a / (1 + i)^t
  • Common ratio: 1 + i
  • Number of terms: t
  • Use the formula Sn = a * (1 - r^n) / (1 - r) for the sum of the geometric series.

Substitution and Simplification

  • Substitute the values into the geometric series formula to find P.
  • P = (A / (1 + i)^t) * (1 - (1 + i)^t) / (1 - (1 + i))
  • Isolate A to match the amortization formula.
  • To isolate A, rearrange the equation: P * (1 + i)^t * (1 - (1 + i)) / (1 - (1 + i)^t) = A
  • Simplify the bracketed term (1 - (1 + i)) to (1 - 1 - i) = -i.
  • Multiply the numerator and denominator by -1 to adjust the signs.
  • The amortization formula: A = P * (i * (1 + i)^t) / ((1 + i)^t - 1)

Conclusion

  • The principal equals the sum of the present values of all future repayments.
  • Present value and sum of a geometric series formulas are required.

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