Topic 2 Continued: Advanced Financial Mathematics and Annuity Valuation

Quick Recap of Financial Mathematics Principles

• Time Value of Money (TVM): The fundamental concept that money has a time value. This is managed by re-expressing expected future cash flows at the same point in time.

• Compounding Forward: Future value can be determined using the formula:

$FV_n = PV_0 \times (1 + r)^n$

*   **Project A Example:** $FV_4 = 376451 \times 1.085^1 = 408449$

• Discounting Backward: Present value is determined by the formula:

$PV_0 = \frac{FV_n}{(1 + r)^n}$

*   **Project A Example:** $PV_0 = \frac{376451}{1.085^3} = 294726$
*   **Project B Example:** $PV_0 = \frac{393768}{1.085^4} = 284132$

Manipulating the Present Value Formula to Solve for Unknown Parameters

When the Present Value ($PV_0$) and the Future Value ($FV_n$) are known, we can solve for either the interest rate ($r$) or the number of periods ($n$).

Solving for the Unknown Interest Rate ($r$)

To find the interest rate required for an investment to grow to a specific target, use the formula:

$r = \left(\frac{FV_n}{PV_0}\right)^{\frac{1}{n}} - 1$

• Example Case: You invest $\$10000$ for a five-year period. What interest rate is required for the funds to double?

  • Setting up the equation: $\$10000 = \frac{\$20000}{(1 + r)^5}$

  • Rearranging: $(1 + r)^5 = \frac{\$20000}{\$10000}$

  • $(1 + r)^5 = 2$

  • $1 + r = 2^{\frac{1}{5}}$

  • $r = 2^{0.2} - 1 = 0.1487$

  • Result: $14.87\% \text{ p.a.}$

Solving for the Unknown Time Period ($n$)

To find how long it takes for a sum to grow to a specific value at a given interest rate, use the logarithm-based formula:

$n = \frac{\log\left(\frac{FV_n}{PV_0}\right)}{\log(1 + r)}$

• Example Case: You invest $\$10000$ at an interest rate of $10\% \text{ p.a.}$. How long will it take for these funds to triple in value?

  • Setting up the equation: $\$10000 = \frac{\$30000}{1.10^n}$

  • Rearranging: $1.10^n = \frac{\text{\$30000}}{\text{\$10000}} = 3$

  • Using logarithms: $n \times \log(1.10) = \log(3)$

  • $n = \frac{\log(3)}{\log(1.10)} = 11.53 \text{ years}$

Ordinary Annuities

Definition of an Ordinary Annuity

An ordinary annuity is a series of equal, periodic cash flows ($CF$) occurring at the end of each period for a total of $n$ periods.

Present Value of an Ordinary Annuity

From first principles, the Present Value ($PV_0$) is the sum of discounted individual cash flows:

$PV_0 = \frac{CF}{(1 + r)^1} + \frac{CF}{(1 + r)^2} + \cdots + \frac{CF}{(1 + r)^n}$

This is more commonly expressed using the closed-form formula:

$PV_0 = \frac{CF}{r} \times \left(1 - \frac{1}{(1 + r)^n}\right)$

• Standard Example: How much must you invest today at a locked-in rate of $10\% \text{ p.a.}$ to generate an annuity of $\$1000 \text{ p.a.}$ for 10 years?

  • $PV_0 = \frac{\$1000}{0.10} \times \left(1 - \frac{1}{1.10^{10}}\right) = \$6144.57$

• Long-term Example: Investing at $10\% \text{ p.a.}$ for a 50-year annuity of $\$1000 \text{ p.a.}$.

  • $PV_0 = \frac{\$1000}{0.10} \times \left(1 - \frac{1}{1.10^{50}}\right) = \$9914.81$

Future Value of an Ordinary Annuity

The Future Value ($FV_n$) is calculated by compounding the Present Value of the annuity forward to time $n$:

$FV_n = \left[\frac{CF}{r} \times \left(1 - \frac{1}{(1 + r)^n}\right)\right] \times (1 + r)^n$

• 10-Year Example: Invest $\$1000$ at the end of each year for 10 years at $10\% \text{ p.a.}$.

  • $FV_{10} = \$6144.57 \times 2.5937 = \$15937.43$

• 50-Year Example: Invest $\$1000$ at the end of each year for 50 years at $10\% \text{ p.a.}$.

  • $FV_{50} = \$9914.81 \times 117.39 = \$1163908.50$

Historical Context: Interest Rates

While $10\% \text{ p.a.}$ might seem high today, historical data for Australian Government 10-Year Bond Yields (1970–2025) shows yields peaked well above $14\% \text{ p.a.}$ in the early 1980s before trending downward toward levels below $2\% \text{ p.a.}$ after 2015.

Specialized Annuities: Due, Deferred, and Growth

Annuities Due

An annuity due is an annuity where cash flows occur at the beginning of each period, effectively shifting all cash flows one period earlier than an ordinary annuity.

• Formula Options for PV:

  1. Compounding an ordinary annuity by one period: $PV_0 = \frac{CF}{r} \times \left(1 - \frac{1}{(1 + r)^n}\right) \times (1 + r)^1$

  2. Treating the first payment as a standalone cash flow: $PV_0 = CF + \frac{CF}{r} \times \left(1 - \frac{1}{(1 + r)^{n-1}}\right)$

• Example Case: $\$1000$ annually for 4 years at $10\% \text{ p.a.}$.

  • Ordinary Annuity PV: $\$3169.87$

  • Annuity Due PV: $\$3169.87 \times 1.10 = \$3486.85$

• Comparison of Balances (at $10\% \text{ p.a.}$):

  • 10-Year PV Due: $\$6759.027$; FV: $\$17531.18$

  • 50-Year PV Due: $\$10906.29$; FV: $\$1280298.80$

Deferred Annuities

A deferred annuity is an ordinary annuity with a delayed start. It is intuitively worth less than a standard ordinary annuity starting today.

• Definitions:

  • $t$: Number of periods until the first cash flow is paid.

  • $N$: Number of cash flows paid.

• PV Formula:

$PV_0 = \frac{CF}{r} \times \left(1 - \frac{1}{(1 + r)^N}\right) \times \frac{1}{(1 + r)^{t-1}}$

• Example Case: Invest $\$1000$ annually for 5 years at $10\% \text{ p.a.}$, but the first investment is deferred for 4 years (meaning payments at end of years 4, 5, 6, 7, and 8).

  • $PV_0 = \frac{\$1000}{0.10} \times \left(1 - \frac{1}{1.10^5}\right) \times \frac{1}{1.10^3} = \$2848.07$

  • Balance on final deposit day ($FV_8$): $\$2848.07 \times 1.10^8 = \$6105.10$

Constant Growth Annuities

Used when assuming constant cash flows is unrealistic (e.g., valuing a firm). Cash flows grow at a constant rate $g$ per period.

• PV Formula:

$PV = \frac{CF}{r - g} \times \left(1 - \left(\frac{1 + g}{1 + r}\right)^n\right)$

• Example Case: A lottery guarantees $\$1000$ p.a. for three years, starting in one year, growing at $5\%$ p.a. Assume $r = 10\% \text{ p.a.}$.

  • $PV = \frac{\$1000}{0.10 - 0.05} \times \left(1 - \frac{1.05^3}{1.10^3}\right) = \$2605.18$

Perpetuities

Standard Perpetuities

A perpetuity is an annuity that continues infinitely ($n \rightarrow \infty$).

• PV Formula:

$PV_0 = \frac{CF}{r}$

• Historical Case (British Consols): Non-redeemable bonds issued by the UK government.

  • 1803 Issue: Bonds promised $3\% \text{ p.a.}$ interest on a face value of $\pounds 100$ at a yield of $5\% \text{ p.a.}$.

  • $PV_0 = \frac{\pounds 3}{0.05} = \pounds 60$

  • 2015 Redemption: In 2015, the UK decided to redeem all consols. At that time, the recorded yield was $2.11\% \text{ p.a.}$.

  • Redemption Price: $\frac{\pounds 3}{0.0211} = \pounds 142.18$

Specialized Perpetuities

• Deferred Perpetuity: $PV_0 = \frac{CF}{r} \times \frac{1}{(1 + r)^{t-1}}$

  • Example: A bond paying $6\% \text{ p.a.}$ on $\$100$ face value starting in Year 4 with a $5\%$ yield.

  • $PV_3 = \frac{\$6}{0.05} = \$120$

  • $PV_0 = \frac{\$120}{1.05^3} = \$103.66$

• Perpetuity Due: $PV_0 = CF + \frac{CF}{r}$

• Constant Growth Perpetuity: $PV_0 = \frac{CF_1}{r - g}$ (Note: r > g must hold).

  • Example: $\$12$ payment in one year, increasing at $6\% \text{ p.a.}$ with a discount rate of $10\% \text{ p.a.}$.

  • $PV_0 = \frac{\$12}{0.10 - 0.06} = \$300$

Effective Interest Rates

Formula and Definitions

To account for different compounding frequencies, express rates in terms of an effective rate per period ($r_e$).

$r_e = \left(1 + \frac{r}{m}\right)^m - 1$

• $r$: Stated interest rate (often the Annual Percentage Rate or APR).

• $m$: Number of sub-periods per period.

• $\frac{r}{m}$: The effective rate per sub-period.

Comparison Table ($8\% \text{ p.a.}$ Stated Rate)

The Need for Consistency

Discount rates must be consistent with the time period of the cash flows.

• Example: Project promising $\$100$ in 6 months and $\$100$ in 12 months. Quoted rate: $10\% \text{ p.a.}$ compounding semi-annually.

  • Effective Semi-annual Rate: $\frac{10\%}{2} = 5\%$

  • Effective annual rate (APY): $(1.05)^2 - 1 = 10.25\% \text{ p.a.}$

  • PV Calculation (Years): $PV_0 = \frac{100}{1.1025^{0.5}} + \frac{100}{1.1025^1} = \$185.94$

  • PV Calculation (Half-Years): $PV_0 = \frac{100}{1.05^1} + \frac{100}{1.05^2} = \$185.94$

Comprehensive Examples and Applications

Lawsuit Settlement Example

Quarterly payments of $\$10000$ for 10 years (40 payments). Interest rate: $6\% \text{ p.a.}$ compounding quarterly.

• Effective Quarterly Rate: $\frac{0.06}{4} = 0.015$

• $PV_0 = \frac{\$10000}{0.015} \times \left(1 - \frac{1}{1.015^{40}}\right) = \$299158.45$

• $FV_{40} = \$299158 \times 1.015^{40} = \$542678.94$

Boulderado Snowboard Project

• Development Phase: Annuity due. $4$ annual investments of $\$250000$, first payment today ($t=0$). Discount rate $10\% \text{ p.a.}$.

  • $PV_0 = -\$250000 - \frac{\$250000}{0.10} \times \left(1 - \frac{1}{1.10^3}\right) = -\$871713$

• Production Phase: Deferred 10-year annuity. Annual cash flows of $\$200000$ for 10 years, starting in year 4.

  • $PV_3 = \frac{\$200000}{0.10} \times \left(1 - \frac{1}{1.10^{10}}\right) = \$1228913.42$

  • $PV_0 = \frac{\$1228913.42}{1.10^3} = \$923300.84$

• Total Project Value:

  • $Total = -\$871713 + \$923300.84 = \$51587.84$

Savings Accumulation Example

Invest $\$5000$ at end of Year 1, growing at $6\% \text{ p.a.}$. Final flow at end of Year 3. Return on investment: $8\% \text{ p.a.}$.

• PV Method:

  • $PV = \frac{\$5000}{0.08 - 0.06} \times \left(1 - \frac{1.06^3}{1.08^3}\right) = \$13633.27$

  • $FV_3 = \$13633.27 \times 1.08^3 = \$17174$

• First Principles Method:

  • $FV_3 = (\$5000 \times 1.08^2) + (\$5000 \times 1.06 \times 1.08) + (\$5000 \times 1.06^2) = \$17174$

Problem-Solving Steps for Financial Mathematics

1. Timeline: Draw a timeline and transpose information from the question.

2. Packages: Break cash flows into smaller, manageable packages.

3. Rates: Ensure the use of the correct effective interest rate per period.

4. Formulae: Apply appropriate formulae (Annuity, Perpetuity, etc.).

5. Review: Check that the specific question asked has been fully answered.

Questions & Discussion

• Calculator Policy: Only the Casio FX82 (any suffix) is permitted for use.

• Reading Requirements:

  • Berk et al, Chapter 4 and Chapter 5 (section 5.1).

  • Lamba, A. S. (2026), Teaching Note 1: Sections 3, 4.1, 4.2, and 5.