Loan schedules, capital and interest components, APR

Q: What is the general equation of value for a loan repaid by level annual payments in arrear?

The equation of value is $L = X a_{\overline{n}|}$, where $L$ is the loan amount, $X$ is the level annual payment, and $a_{\overline{n}|} = \frac{1 - v^n}{i}$.

Q: How is the equation of value formulated when loan repayments are made $p$ times per year?

The equation of value is $L = X a_{\overline{n}|}^{(p)}$, where $X$ is the total annual payment amount made in level instalments of $\frac{X}{p}$ each period.

Q: What is the definition and formula for the prospective loan balance at time $t$?

The prospective loan balance $L_t$ is the present value at time $t$ of all remaining future repayments: $L_t = X a_{\overline{n - t}|}$ for level annual payments.

Q: What is the definition and formula for the retrospective loan balance at time $t$?

The retrospective loan balance is the accumulated value of the original loan minus the accumulated value of all repayments made up to time $t$: $L_t = L(1 + i)^t - X s_{\overline{t}|}$.

Q: How are the interest and capital components of a loan instalment at time $t$ calculated?

The interest component is $I_t = i \cdot L_{t-1}$, and the capital repayment component is $C_t = X_t - I_t$.

Q: What are the closed-form algebraic expressions for $I_t$ and $C_t$ under a level annual repayment schedule?

$C_t = X v^{n - t + 1}$ and $I_t = X(1 - v^{n - t + 1})$.

Q: At what rate do the capital repayment components grow over time in a level repayment loan schedule?

Capital repayment components grow geometrically at the effective interest rate $i$ per period, such that $C_{t+1} = C_t(1 + i)$.

Q: What is the Annual Percentage Rate (APR) in the context of loan evaluation?

APR is the effective annual interest rate that equates the net loan amount received by the borrower to the present value of all repayments and compulsory expenses.

Q: Why is the APR substantially higher than a quoted flat rate of interest on an instalment loan?

Because a flat rate charges interest on the original principal across the entire term, ignoring the fact that regular instalments reduce the outstanding balance over time.

Q: How are revised loan terms determined when the contractual interest rate changes midway through the term?

Calculate the outstanding balance $L_k$ immediately after the change date using the old rate, then set $L_k$ equal to the present value of future repayments evaluated at the new interest rate.

CM1 – Actuarial Mathematics for Modelling · IFoA (Institute and Faculty of Actuaries) · 10 min read · Updated: Oct 22, 2026

Introduction to Loan Schedules, Capital & Interest, and APR

Welcome to this core chapter of CM1! Whether taking out a mortgage to buy a home, financing a car, or funding corporate projects, loans form the backbone of commercial finance. In this chapter, we explore how loans are structured, how each regular repayment is split between paying off interest and repaying capital, and how to measure the true annual cost of borrowing via the Annual Percentage Rate (APR).

Don't worry if this seems tricky at first. At its heart, the entire chapter relies on one straightforward idea you have already met: the equation of value, equating the loan amount borrowed to the present value of repayments.

Note on syllabus placement: This chapter focuses strictly on Section 2.2 of the syllabus ("loans repaid by regular instalments of interest and capital — repayments, interest and capital components, the effective interest rate (APR) and a schedule of repayments"). Other applications in this section, such as bond pricing, project appraisal, and share valuation, are covered in neighbouring chapters.

1. Fundamental Loan Principles and Repayment Structures

The Equation of Value for a Loan

When an amount $L$ is borrowed at an effective interest rate $i$ per period and repaid by a stream of payments $X_1, X_2, \dots, X_n$ at the end of each period, the equation of value at time $t = 0$ is:

$L = \sum_{t=1}^{n} X_t v^t$

where $v = \frac{1}{1 + i}$ is the discount factor per period.

If the loan is repaid by level annual repayments of amount $X$ in arrear over $n$ years at annual effective rate $i$, this simplifies to the familiar actuarial formula:

$L = X a_{\overline{n}|}$

where $a_{\overline{n}|} = \frac{1 - v^n}{i}$. From this, the annual level instalment is readily found as:

$X = \frac{L}{a_{\overline{n}|}}$

Payments Payable $p$-thly

In practice, loans like mortgages are often repaid monthly or quarterly. If level repayments of $\frac{X}{p}$ are made $p$ times a year for $n$ years (totalling $X$ per year), the equation of value at annual effective rate $i$ is:

$L = X a_{\overline{n}|}^{(p)}$

Recall the standard identity relating $a_{\overline{n}|}^{(p)}$ and $i^{(p)}$:

$a_{\overline{n}|}^{(p)} = \frac{1 - v^n}{i^{(p)}}$

where $(1 + i) = \left(1 + \frac{i^{(p)}}{p}\right)^p$.

Quick Review: Every instalment paid by a borrower does two distinct jobs:

1. It pays the lender interest on the balance outstanding over the period just ended.

2. Any remaining portion goes towards reducing the loan capital (the principal).

2. Outstanding Loan Balances: Prospective vs Retrospective

At any time $t$ ($0 < t < n$), we frequently need to determine the outstanding loan balance, denoted $L_t$. There are two mathematically equivalent ways to calculate this: the prospective method (looking forward) and the retrospective method (looking backward).

The Prospective Loan Balance

The prospective method values all remaining future repayments:

$L_t = \text{Present value at time } t \text{ of all future repayments yet to be made}$

For a level annual payment $X$ with $n - t$ payments remaining:

$L_t = X a_{\overline{n - t}|}$

Why students love this method: It is almost always quicker and simpler to compute in exam questions whenever future payments remain level.

The Retrospective Loan Balance

The retrospective method values past cashflows brought forward to time $t$:

$L_t = \text{Accumulated value at time } t \text{ of the original loan} - \text{Accumulated value at time } t \text{ of all repayments already made}$

For a level annual payment $X$ made for $t$ periods:

$L_t = L(1 + i)^t - X s_{\overline{t}|}$

where $s_{\overline{t}|} = \frac{(1 + i)^t - 1}{i}$.

Equivalence of Both Methods

As long as the rate of interest $i$ remains constant throughout, both methods yield identical results:

$L(1 + i)^t - X s_{\overline{t}|} = (X a_{\overline{n}|})(1 + i)^t - X \frac{(1 + i)^t - 1}{i} = X \left[ \frac{1 - v^n}{i}(1 + i)^t - \frac{(1 + i)^t - 1}{i} \right] = X \left[ \frac{1 - v^{n-t}}{i} \right] = X a_{\overline{n - t}|}$

Key Takeaway: Use the prospective method when future payments are fixed and simple. Use the retrospective method if past payments were simple but future payments alter or are subject to unknown changes.

3. Capital and Interest Components of an Instalment

Consider the payment $X_t$ made at time $t$. It consists of two parts:

$X_t = I_t + C_t$

where $I_t$ is the interest component and $C_t$ is the capital repayment component.

Step-by-Step Breakdown

Step 1: Interest component ($I_t$)
Interest is charged on the outstanding balance at the beginning of the period (which is time $t - 1$):

$I_t = i \cdot L_{t-1}$

Step 2: Capital component ($C_t$)
The capital repaid is whatever remains of the payment after interest is deducted:

$C_t = X_t - I_t = X_t - i \cdot L_{t-1}$

Step 3: Updating the balance
The balance remaining at the end of the period is the old balance minus the capital repaid:

$L_t = L_{t-1} - C_t$

Algebraic Expressions for a Level Annuity Loan

When the loan is repaid by level annual instalments $X$, we can substitute the prospective balance $L_{t-1} = X a_{\overline{n - t + 1}|}$ into the formula for $I_t$:

$I_t = i \cdot X a_{\overline{n - t + 1}|} = i \cdot X \left( \frac{1 - v^{n - t + 1}}{i} \right) = X (1 - v^{n - t + 1})$

Then, the capital component is:

$C_t = X - I_t = X - X (1 - v^{n - t + 1}) = X v^{n - t + 1}$

Notice the elegant geometric progression! For level annual instalments:

$C_t = X v^{n - t + 1}$

$C_{t+1} = X v^{n - (t+1) + 1} = X v^{n - t} = C_t(1 + i)$

Did you know? Under a level repayment schedule, the capital component grows geometrically at rate $i$ each year: $C_{t+1} = C_t (1 + i)$, while the interest component $I_t$ declines over time as the principal outstanding falls.

4. The Loan Schedule

A loan schedule (or amortization table) tracks the evolution of the loan over each time step. Candidates sitting CM1B in Microsoft Excel will frequently build these schedules, while CM1A candidates in Microsoft Word may be asked to construct individual rows or explain the relationships between entries.

Structure of a Level Loan Schedule

Year $t$	Outstanding Balance at Start $L_{t-1}$	Total Instalment $X_t$	Interest Paid $I_t = i \cdot L_{t-1}$	Capital Repaid $C_t = X_t - I_t$	Outstanding Balance at End $L_t = L_{t-1} - C_t$
$1$	$X a_{\overline{n}\|}$	$X$	$X (1 - v^n)$	$X v^n$	$X a_{\overline{n-1}\|}$
$2$	$X a_{\overline{n-1}\|}$	$X$	$X (1 - v^{n-1})$	$X v^{n-1}$	$X a_{\overline{n-2}\|}$
$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$t$	$X a_{\overline{n-t+1}\|}$	$X$	$X (1 - v^{n-t+1})$	$X v^{n-t+1}$	$X a_{\overline{n-t}\|}$
$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$n$	$X a_{\overline{1}\|}$	$X$	$X (1 - v)$	$X v$	$0$
Total	—	$n \cdot X$	$n X - L$	$L$	—

Useful Check: The sum of all capital repayments must equal the original loan amount:

$\sum_{t=1}^n C_t = \sum_{t=1}^n X v^{n - t + 1} = X (v^n + v^{n-1} + \dots + v) = X a_{\overline{n}|} = L$

And the total interest paid across the entire loan is simply:

$\sum_{t=1}^n I_t = \sum_{t=1}^n X_t - \sum_{t=1}^n C_t = n X - L$

5. Step-by-Step Worked Example: Calculating Components & Balances

A borrower takes out a loan of \$10,000 to be repaid by 5 equal annual instalments payable in arrear at an effective annual rate of interest of $6\%$.

Step 1: Calculate the level annual instalment $X$

$a_{\overline{5}|} = \frac{1 - (1.06)^{-5}}{0.06} = \frac{1 - 0.747258}{0.06} = 4.212364$

$X = \frac{10000}{4.212364} = 2373.96$

Step 2: Find the components in Year 1

$I_1 = 0.06 \times 10000 = 600.00$

$C_1 = X - I_1 = 2373.96 - 600.00 = 1773.96$

$L_1 = 10000 - 1773.96 = 8226.04$

Step 3: Find the components in Year 3 directly

Rather than working through Year 2, use our standard formulas directly:

$C_3 = X v^{5 - 3 + 1} = X v^3 = 2373.96 \times (1.06)^{-3} = 2373.96 \times 0.839619 = 1993.22$

$I_3 = X - C_3 = 2373.96 - 1993.22 = 380.74$

Check with prospective balance at time $t = 2$:

$L_2 = X a_{\overline{3}|} = 2373.96 \times \frac{1 - (1.06)^{-3}}{0.06} = 2373.96 \times 2.673012 = 6345.62$

$I_3 = 0.06 \times L_2 = 0.06 \times 6345.62 = 380.74$. (Matches exactly!)

6. The Effective Interest Rate: APR (Annual Percentage Rate)

What is APR?

Lenders frequently quote headline interest rates that do not reflect the true economic cost to the borrower due to mandatory upfront administration fees, valuation expenses, arrangement costs, or compounding frequencies. To protect consumers and provide a uniform basis for comparison, legislation requires lenders to quote the APR (Annual Percentage Rate).

In actuarial mathematics, the APR is defined as the effective annual rate of interest that equates the net amount received by the borrower to the present value of all repayments and compulsory expenses paid by the borrower.

The APR Equation of Value

If a borrower takes a nominal loan $L$, but pays upfront arrangement fees $E_0$, and subsequently pays instalments $X_t$ alongside periodic fees or charges $e_t$ at times $t$, the APR is the unique rate $i$ satisfying:

$L - E_0 = \sum_{t} (X_t + e_t)(1 + i)^{-t}$

Alternatively viewed:

$\text{Net proceeds received at } t = 0 = \text{PV of all borrower outgoes discounted at APR}$

Important Note: Because fees reduce the net loan proceeds received upfront, the APR is always greater than or equal to the nominal or contractual rate of interest quoted on the debt.

Common Pitfall: Nominal Flat Rates vs APR

A classic trick in commercial consumer credit is quoting a "flat interest rate".

Example: A lender offers a loan of \$1,000 for 1 year, stating "just $10\%$ flat interest rate, repaid in 12 equal monthly instalments".

The total interest charged is $10\% \times 1000 = \$100$. Total to repay = \$1,100, which gives monthly instalments of:

$\frac{1100}{12} = 91.67$

Is the effective rate (APR) $10\%$? No! The borrower does not hold the full \$1,000 for the whole year; the principal is being paid off every month. The equation of value to find the monthly effective rate $j$ is:

$1000 = 91.67 a_{\overline{12}|j}$

Solving this reveals $j \approx 1.498\%$ per month. The effective annual rate (APR) is:

$\text{APR} = (1 + j)^{12} - 1 \approx (1.01498)^{12} - 1 \approx 19.5\%$

The APR is almost double the flat rate! Always remember: whenever repayments reduce the outstanding balance over time, the true APR will be significantly higher than a flat rate.

7. Handling Changes in Loan Terms (Interest Rates and Extra Payments)

In actuarial exams, questions often test your ability to adapt when loan conditions change midway through the term.

Change in Interest Rate

Suppose an interest rate changes from $i$ to $i'$ at time $k$:

1. Calculate the outstanding balance $L_k$ just after the payment at time $k$ using the original rate $i$:

$L_k = X a_{\overline{n-k}|i}$

2. Determine the new payment $X'$ over the remaining $n - k$ periods at the new rate $i'$:

$L_k = X' a_{\overline{n-k}|i'} \implies X' = \frac{L_k}{a_{\overline{n-k}|i'}}$

3. If instead the payment is kept unchanged at $X$, calculate the new term $m$ to repay the remaining balance:

$L_k = X a_{\overline{m}|i'}$

Early Capital Repayment

If the borrower makes an extra lump-sum repayment $C^*$ at time $k$:

The new outstanding balance becomes $L_k' = L_k - C^*$. The borrower can either:

• Reduce future instalments: keep the original remaining term $n - k$ and solve $L_k' = X_{\text{new}} a_{\overline{n-k}|}$.

• Reduce the term: keep the instalment at $X$ and solve for the reduced number of remaining periods $m$ from $L_k' = X a_{\overline{m}|}$.

8. Summary of Essential Formulas

Concept	Formula
Equation of Value (Annual)	$L = X a_{\overline{n}\|i}$
Equation of Value ($p$-thly)	$L = X a_{\overline{n}\|i}^{(p)}$
Prospective Balance	$L_t = X a_{\overline{n-t}\|i}$
Retrospective Balance	$L_t = L(1 + i)^t - X s_{\overline{t}\|i}$
Interest Component at time $t$	$I_t = i \cdot L_{t-1} = X(1 - v^{n-t+1})$
Capital Component at time $t$	$C_t = X_t - I_t = X v^{n-t+1}$
Capital Growth Relation	$C_{t+1} = C_t(1 + i)$
Equation of Value for APR	$L - \text{Initial Fees} = \sum (\text{Instalments} + \text{Expenses}) \cdot (1 + \text{APR})^{-t}$

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the general equation of value for a loan repaid by level annual payments in arrear?

The equation of value is $L = X a_{\overline{n}|}$, where $L$ is the loan amount, $X$ is the level annual payment, and $a_{\overline{n}|} = \frac{1 - v^n}{i}$.

How is the equation of value formulated when loan repayments are made $p$ times per year?

The equation of value is $L = X a_{\overline{n}|}^{(p)}$, where $X$ is the total annual payment amount made in level instalments of $\frac{X}{p}$ each period.

What is the definition and formula for the prospective loan balance at time $t$?

The prospective loan balance $L_t$ is the present value at time $t$ of all remaining future repayments: $L_t = X a_{\overline{n - t}|}$ for level annual payments.

What is the definition and formula for the retrospective loan balance at time $t$?

The retrospective loan balance is the accumulated value of the original loan minus the accumulated value of all repayments made up to time $t$: $L_t = L(1 + i)^t - X s_{\overline{t}|}$.

How are the interest and capital components of a loan instalment at time $t$ calculated?

The interest component is $I_t = i \cdot L_{t-1}$, and the capital repayment component is $C_t = X_t - I_t$.

What are the closed-form algebraic expressions for $I_t$ and $C_t$ under a level annual repayment schedule?

$C_t = X v^{n - t + 1}$ and $I_t = X(1 - v^{n - t + 1})$.

At what rate do the capital repayment components grow over time in a level repayment loan schedule?

Capital repayment components grow geometrically at the effective interest rate $i$ per period, such that $C_{t+1} = C_t(1 + i)$.

What is the Annual Percentage Rate (APR) in the context of loan evaluation?

APR is the effective annual interest rate that equates the net loan amount received by the borrower to the present value of all repayments and compulsory expenses.

Why is the APR substantially higher than a quoted flat rate of interest on an instalment loan?

Because a flat rate charges interest on the original principal across the entire term, ignoring the fact that regular instalments reduce the outstanding balance over time.

How are revised loan terms determined when the contractual interest rate changes midway through the term?

Calculate the outstanding balance $L_k$ immediately after the change date using the old rate, then set $L_k$ equal to the present value of future repayments evaluated at the new interest rate.

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Year \(t\)	Outstanding Balance at Start \(L_{t-1}\)	Total Instalment \(X_t\)	Interest Paid \(I_t = i \cdot L_{t-1}\)	Capital Repaid \(C_t = X_t - I_t\)	Outstanding Balance at End \(L_t = L_{t-1} - C_t\)
\(1\)	\(X a_{\overline{n}\|}\)	\(X\)	\(X (1 - v^n)\)	\(X v^n\)	\(X a_{\overline{n-1}\|}\)
\(2\)	\(X a_{\overline{n-1}\|}\)	\(X\)	\(X (1 - v^{n-1})\)	\(X v^{n-1}\)	\(X a_{\overline{n-2}\|}\)
\(\dots\)	\(\dots\)	\(\dots\)	\(\dots\)	\(\dots\)	\(\dots\)
\(t\)	\(X a_{\overline{n-t+1}\|}\)	\(X\)	\(X (1 - v^{n-t+1})\)	\(X v^{n-t+1}\)	\(X a_{\overline{n-t}\|}\)
\(\dots\)	\(\dots\)	\(\dots\)	\(\dots\)	\(\dots\)	\(\dots\)
\(n\)	\(X a_{\overline{1}\|}\)	\(X\)	\(X (1 - v)\)	\(X v\)	\(0\)
Total	—	\(n \cdot X\)	\(n X - L\)	\(L\)	—

Concept	Formula
Equation of Value (Annual)	\(L = X a_{\overline{n}\|i}\)
Equation of Value (\(p\)-thly)	\(L = X a_{\overline{n}\|i}^{(p)}\)
Prospective Balance	\(L_t = X a_{\overline{n-t}\|i}\)
Retrospective Balance	\(L_t = L(1 + i)^t - X s_{\overline{t}\|i}\)
Interest Component at time \(t\)	\(I_t = i \cdot L_{t-1} = X(1 - v^{n-t+1})\)
Capital Component at time \(t\)	\(C_t = X_t - I_t = X v^{n-t+1}\)
Capital Growth Relation	\(C_{t+1} = C_t(1 + i)\)
Equation of Value for APR	\(L - \text{Initial Fees} = \sum (\text{Instalments} + \text{Expenses}) \cdot (1 + \text{APR})^{-t}\)

Loan schedules, capital and interest components, APR

Introduction to Loan Schedules, Capital & Interest, and APR

1. Fundamental Loan Principles and Repayment Structures

The Equation of Value for a Loan

Payments Payable \(p\)-thly

2. Outstanding Loan Balances: Prospective vs Retrospective

The Prospective Loan Balance

The Retrospective Loan Balance

Equivalence of Both Methods

3. Capital and Interest Components of an Instalment

Step-by-Step Breakdown

Algebraic Expressions for a Level Annuity Loan

4. The Loan Schedule

Structure of a Level Loan Schedule

5. Step-by-Step Worked Example: Calculating Components & Balances

Step 1: Calculate the level annual instalment \(X\)

Step 2: Find the components in Year 1

Step 3: Find the components in Year 3 directly

6. The Effective Interest Rate: APR (Annual Percentage Rate)

What is APR?

The APR Equation of Value

Common Pitfall: Nominal Flat Rates vs APR

7. Handling Changes in Loan Terms (Interest Rates and Extra Payments)

Change in Interest Rate

Early Capital Repayment

8. Summary of Essential Formulas

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