FIM TVM Masterclass

Future and Present Values of a Single Cash Flow

These are the most fundamental concepts in TVM. All other calculations, including those for annuities, are built upon these two formulas. It's crucial to understand that present value (PV) is the value of a future cash flow today, while future value (FV) is the value of a present cash flow at a future date.

Future Value (FV):

$$FV = PV(1+r)^{t}$$

Present Value (PV):

$$PV = \frac{FV}{(1+r)^{t}}$$

Where PV is Present Value, FV is Future Value, r is the interest rate per period, and t is the number of periods.

Annuities, Perpetuities, and their Variations

The exam heavily tests your ability to distinguish between ordinary annuities, annuity dues, and perpetuities. An ordinary annuity is a series of constant payments at the end of each period, such as a loan payment. An annuity due has payments at the beginning of each period, like a lease. A perpetuity is a stream of payments that continues forever.

Present Value of an Ordinary Annuity:

$$PV = C \times \left[ \frac{1 - 1/(1+r)^{t}}{r} \right]$$

Future Value of an Ordinary Annuity:

$$FV = C \times \left[ \frac{(1+r)^{t}-1}{r} \right]$$

Annuity Due Conversion:

To find the value of an annuity due, simply multiply the corresponding ordinary annuity value by $ (1+r) $.

$$PV_{due} = PV_{ordinary} \times (1+r)$$ $$FV_{due} = FV_{ordinary} \times (1+r)$$

Present Value of a Perpetuity:

$$PV = C/r$$

The exam also includes questions on growing annuities and growing perpetuities, where cash flows increase at a constant rate ($g$). This is a more complex application of the perpetuity concept and is often used for stock valuation.

Growing Perpetuity (Gordon Growth Model):

$$PV = \frac{C_1}{r-g}$$

Growing Annuity Present Value:

$$PV = C_1 \left[ \frac{1 - \left(\frac{1+g}{1+r}\right)^t}{r-g} \right]$$

Note: $C_1$ is the cash flow at the end of the first period.

Compounding, EAR vs. APR, and Loan Amortization

A key lesson from the exam problems is that the stated interest rate (APR) is not the true rate. The Effective Annual Rate (EAR) is what matters. Always convert to EAR when comparing loans or investments with different compounding frequencies. The concept of loan amortization is also heavily tested, requiring you to calculate a fixed payment that pays off both principal and interest over time. A core insight is that the interest portion of the payment decreases with each period while the principal portion increases.

Effective Annual Rate (EAR):

$$EAR = \left[ 1 + \frac{\text{Quoted rate}}{m} \right]^m - 1$$

Loan Amortization Payment:

$$ C = \frac{PV}{\frac{1 - [1/(1+r)^{t}]}{r}} $$

Multi-stage Problems and Strategic Advice

Some problems require combining multiple concepts. For example, a retirement savings problem involves calculating the FV of an annuity for a savings period, and then letting that lump sum compound for a later period. The key takeaway from these problems is the power of early investment. You may also be asked to provide financial advice, requiring you to calculate and compare the present or future values of different investment options.