Session 08-02 - Annuities & Loan Amortization

Section 08: Financial Mathematics

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review from Session 08-01

Test your understanding of Compound Interest

Find the 6th term of the geometric sequence: \(4, 12, 36, \ldots\)
Calculate the future value of 5,000 invested at 6% compounded quarterly for 5 years.
Find the effective annual rate for 8% compounded monthly.
How much should you invest today at 5% annual interest to have 15,000 in 8 years?

Learning Objectives

What You’ll Master Today

Understand ordinary annuities and annuities due
Calculate future value of regular payment streams
Calculate present value of annuity payments
Solve for payment amounts and number of payments
Build loan amortization schedules
Analyze interest vs. principal in loan payments

Part A: Introduction to Annuities

What is an Annuity?

An annuity is a series of equal payments made at regular intervals.

. . .

Examples:

Monthly rent payments
Salary deposits
Loan repayments
Retirement contributions
Insurance premiums

. . .

The key: Equal payments at equal intervals

Types of Annuities

Two Main Types

Ordinary Annuity (Nachschussig): Payments made at the end of each period

Annuity Due (Vorschussig): Payments made at the beginning of each period

. . .

Part B: Future Value of Annuities

Future Value of Ordinary Annuity

Question: If you save 500 per month at 6% annual interest, how much will you have in 10 years?

. . .

Future Value Formula (Ordinary Annuity)

\[FV = PMT \cdot \frac{(1 + r)^n - 1}{r}\]

where:

\(PMT\) = Payment per period
\(r\) = Interest rate per period
\(n\) = Total number of payments

FV Calculation Example

Given: 500/month, 6% annual (0.5% monthly), 10 years (120 months)

\[FV = 500 \cdot \frac{(1.005)^{120} - 1}{0.005}\]

. . .

\[FV = 500 \cdot \frac{1.8194 - 1}{0.005} = 500 \cdot \frac{0.8194}{0.005}\]

. . .

\[FV = 500 \cdot 163.88 = 81,939.67\]

. . .

Total deposits: \(500 \times 120 = 60,000\)

Interest earned: \(81,939.67 - 60,000 = 21,939.67\)

FV of Annuity Due

Payments at beginning of period earn one extra period of interest:

. . .

Future Value Formula (Annuity Due)

\[FV_{\text{due}} = PMT \cdot \frac{(1 + r)^n - 1}{r} \cdot (1 + r)\]

Or simply: \[FV_{\text{due}} = FV_{\text{ordinary}} \cdot (1 + r)\]

. . .

Same example as annuity due: \[FV = 81,939.67 \times 1.005 = 82,349.36\]

Part C: Present Value of Annuities

Present Value of Ordinary Annuity

Question: How much is a stream of future payments worth today?

. . .

Present Value Formula (Ordinary Annuity)

\[PV = PMT \cdot \frac{1 - (1 + r)^{-n}}{r}\]

. . .

Example: What is the present value of receiving 1,000/month for 5 years at 8% annual?

\[PV = 1000 \cdot \frac{1 - (1.00667)^{-60}}{0.00667} = 1000 \cdot 49.32 = 49,318.43\]

PV of Annuity Due

Present Value Formula (Annuity Due)

\[PV_{\text{due}} = PMT \cdot \frac{1 - (1 + r)^{-n}}{r} \cdot (1 + r)\]

Or simply: \[PV_{\text{due}} = PV_{\text{ordinary}} \cdot (1 + r)\]

. . .

Same example as annuity due: \[PV = 49,318.43 \times 1.00667 = 49,647.32\]

Break - 10 Minutes

Part D: Solving for Unknowns

Finding the Payment Amount

Question: How much must you save monthly to have 100,000 in 15 years at 5%?

. . .

Rearrange the FV formula:

\[PMT = \frac{FV \cdot r}{(1 + r)^n - 1}\]

. . .

\[PMT = \frac{100000 \times 0.004167}{(1.004167)^{180} - 1} = \frac{416.67}{1.1137} = 374.26\]

. . .

You need to save 374.26 per month!

Finding Number of Payments

Question: How many months to pay off a 20,000 loan at 6% with 450/month payments?

. . .

From the PV formula, solve for \(n\):

\[n = -\frac{\ln(1 - \frac{PV \cdot r}{PMT})}{\ln(1 + r)}\]

. . .

\[n = -\frac{\ln(1 - \frac{20000 \times 0.005}{450})}{\ln(1.005)} = -\frac{\ln(0.7778)}{0.00499}\]

. . .

\[n = -\frac{-0.2513}{0.00499} = 50.4 \text{ months} \approx 51 \text{ payments}\]

Part E: Loan Amortization

What is Amortization?

Amortization = Paying off a loan through regular payments

. . .

Each payment consists of:

Interest portion: Pays interest on remaining balance
Principal portion: Reduces the loan balance

. . .

Key insight: Early payments are mostly interest, later payments are mostly principal!

Loan Payment Formula

Monthly Payment Formula

\[PMT = PV \cdot \frac{r}{1 - (1 + r)^{-n}}\]

where:

\(PV\) = Loan amount (principal)
\(r\) = Monthly interest rate
\(n\) = Total number of payments

Payment Calculation Example

Loan: 200,000 at 4.5% annual for 30 years (mortgage)

\[r = 0.045/12 = 0.00375, \quad n = 360\]

. . .

\[PMT = 200000 \times \frac{0.00375}{1 - (1.00375)^{-360}}\]

. . .

\[PMT = 200000 \times \frac{0.00375}{1 - 0.2598} = 200000 \times \frac{0.00375}{0.7402}\]

. . .

\[PMT = 200000 \times 0.005067 = 1,013.37\]

Building an Amortization Schedule I

First few payments of the 200,000 loan:

Payment	Payment	Interest	Principal	Balance
0	-	-	-	200,000.00
1	1,013.37	750.00	263.37	199,736.63
2	1,013.37	749.01	264.36	199,472.27
3	1,013.37	748.02	265.35	199,206.92
…	…	…	…	…

Building an Amortization Schedule II

How it works:

Interest: Balance \(\times\) rate = \(200,000 \times 0.00375 = 750\)
Principal: Payment - Interest = \(1,013.37 - 750 = 263.37\)
New Balance: Old Balance - Principal = \(200,000 - 263.37\)

Visualizing the Amortization

Outstanding Balance After k Payments

Outstanding Balance Formula

\[B_k = PV \cdot \frac{(1 + r)^n - (1 + r)^k}{(1 + r)^n - 1}\]

Or equivalently: \[B_k = PMT \cdot \frac{1 - (1 + r)^{-(n-k)}}{r}\]

. . .

Example: Balance after 10 years (120 payments) on the 200,000 mortgage

\[B_{120} = 1013.37 \times \frac{1 - (1.00375)^{-240}}{0.00375} = 1013.37 \times 158.59 = 160,706.24\]

Part F: Applications

Retirement Planning

Scenario: You want 500,000 at retirement in 30 years. Investment earns 7% annually.

How much must you invest monthly?

. . .

\[PMT = \frac{FV \cdot r}{(1 + r)^n - 1} = \frac{500000 \times 0.005833}{(1.005833)^{360} - 1}\]

. . .

\[PMT = \frac{2916.67}{7.878} = 370.16\]

. . .

Total invested: \(370.16 \times 360 = 133,257.60\)

Interest earned: \(500,000 - 133,257.60 = 366,742.40\) (more than 2.5x your contributions!)

Lease vs. Buy Decision

Should a company lease or buy equipment?

Buy: 50,000 now

Lease: 1,200/month for 4 years, then return

. . .

At 6% annual rate, PV of lease payments:

\[PV = 1200 \times \frac{1 - (1.005)^{-48}}{0.005} = 1200 \times 42.58 = 51,096.08\]

. . .

Leasing costs 51,096 in present value terms - buying at 50,000 is cheaper (before considering resale value of owned equipment).

Guided Practice - 15 Minutes

Practice Problems

Work in pairs

Problem 1: Calculate the future value of saving 300/month for 20 years at 5% annual interest.

Problem 2: Find the monthly payment for a 25,000 car loan at 4.8% for 5 years.

Problem 3: Create the first 3 rows of an amortization schedule for a 10,000 loan at 6% annual for 2 years (monthly payments).

Problem 4: How much is a monthly pension of 2,000 for 25 years worth today at 4% annual interest?

Wrap-Up & Key Takeaways

Today’s Essential Formulas

Annuity Formulas

Future Value: \(FV = PMT \cdot \frac{(1 + r)^n - 1}{r}\)

Present Value: \(PV = PMT \cdot \frac{1 - (1 + r)^{-n}}{r}\)

Payment: \(PMT = PV \cdot \frac{r}{1 - (1 + r)^{-n}}\)

For annuity due: Multiply by \((1 + r)\)

Key Takeaways

Annuities are equal payments at regular intervals
FV of annuity tells you what regular savings will become
PV of annuity tells you what a payment stream is worth today
Amortization shows how loans are paid off
Early payments are mostly interest, later payments mostly principal

. . .

Coming Next

Session 08-03: Cost Analysis & Pricing Decisions - minimum pricing for profitability!

Homework Assignment

Tasks 08-02

Calculate future and present values of annuities
Solve for payment amounts and number of payments
Build complete amortization schedules
Apply concepts to retirement and loan decisions

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These calculations require careful attention to the interest rate per period!