3.2 Future Value and Present Value

Time value of money is the foundation of nearly every financial calculation you'll encounter. It explains why a dollar today is worth more than a dollar in the future: that dollar can be invested and earn a return in the meantime. Future value and present value are the two core tools for quantifying this idea, letting you figure out how money grows over time and what future cash flows are really worth right now.

These calculations show up everywhere in finance: comparing investments, planning for retirement, pricing loans, and evaluating business projects. The math isn't complicated, but you need to be precise about the inputs.

Future Value of a Single Sum

Calculating Future Value

Future value tells you how much a lump sum today will be worth at some point in the future, given a specific interest rate and compounding frequency. The formula is:

$FV = PV \times (1 + r)^n$

• $FV$ = future value
• $PV$ = present value (the amount you start with)
• $r$ = periodic interest rate
• $n$ = total number of compounding periods

Two details trip students up here: $r$ and $n$ must match the compounding frequency, not just the annual figures.

Finding the periodic rate ($r$): Divide the annual interest rate by the number of compounding periods per year.

• 6% annual with monthly compounding → $r = 6\% / 12 = 0.5\%$
• 8% annual with quarterly compounding → $r = 8\% / 4 = 2\%$

Finding the number of periods ($n$): Multiply the number of years by the compounding periods per year.

• 5 years, monthly compounding → $n = 5 \times 12 = 60$
• 10 years, semi-annual compounding → $n = 10 \times 2 = 20$

Interpreting Future Value

The future value is the total amount your investment grows to, including all accumulated interest. A couple of examples:

• $\$1{,}000$ invested at 5% annually, compounded monthly for 10 years, grows to approximately $\$1{,}647$
• $\$5{,}000$ at 3% annually, compounded quarterly for 7 years, reaches about $\$6{,}166$

As long as the interest rate is positive, the future value will always exceed the present value. The gap between them is the total interest earned over the investment period. More time and higher rates both widen that gap, because compound interest builds on itself.

Present Value of a Single Sum

Calculating Present Value

Present value works in the opposite direction: it answers "what is a future amount of money worth today?" You're essentially reversing the compounding process, which is called discounting. The formula is:

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

• $PV$ = present value
• $FV$ = future value (the amount you expect to receive later)
• $r$ = periodic discount rate
• $n$ = total number of discounting periods

The discount rate represents your opportunity cost of capital, the return you could earn if you had the money now instead of later.

Finding the periodic discount rate ($r$): Same logic as future value. Divide the annual rate by compounding periods per year.

• 10% annual, monthly discounting → $r = 10\% / 12 \approx 0.833\%$
• 6% annual, semi-annual discounting → $r = 6\% / 2 = 3\%$

Finding the number of periods ($n$): Same approach as before.

• 4 years, quarterly discounting → $n = 4 \times 4 = 16$
• 8 years, annual discounting → $n = 8 \times 1 = 8$

Interpreting Present Value

The present value tells you the current worth of money you won't receive until later. Because you lose the opportunity to invest that money in the meantime, future cash flows are always worth less than their face value today (assuming a positive discount rate).

• $\$10{,}000$ received in 5 years, discounted at 8% compounded monthly, has a present value of approximately $\$6{,713}$
• $\$50{,}000$ received in 12 years, discounted at 5% compounded semi-annually, is worth about $\$27{,}920$ today

The difference between the future value and the present value is the total discount, reflecting the cost of waiting.

Applying Future and Present Value

Time Value of Money Problems

Most TVM problems give you three of the four variables ($PV$, $FV$, $r$, $n$) and ask you to solve for the missing one. Here's a general approach:

1. Identify which variable is unknown.
2. Write down the known values, making sure $r$ and $n$ reflect the compounding frequency.
3. Plug into the appropriate formula and solve algebraically (or use a financial calculator).

For example, to find the interest rate needed to grow $\$10{,}000$ to $\$15{,}000$ in 6 years with annual compounding, you'd set up $15{,}000 = 10{,}000 \times (1 + r)^6$ and solve for $r$. To find how many years it takes $\$5{,}000$ to reach $\$20{,}000$ at 8% compounded quarterly, you'd solve for $n$ in the future value formula and then divide by 4 to convert periods back to years.

Financial Applications

Retirement planning uses both FV and PV. You might calculate the future value of monthly $\$500$ contributions over 30 years at 7% compounded monthly to see what your savings balance will be. Or you might calculate the present value of a desired $\$60{,}000$ annual retirement income for 20 years, discounted at 5%, to figure out how much you need saved by the time you retire.

Capital budgeting compares the present value of a project's expected cash flows to its upfront cost. The difference is the net present value (NPV).

• A project costing $\$100{,}000$ with annual cash inflows of $\$30{,}000$ for 5 years, discounted at 10%, has an NPV of approximately $\$13{,}724$. Positive NPV means the project creates value, so you'd accept it.
• A project costing $\$500{,}000$ generating $\$75{,}000$ per year for 10 years at a 12% discount rate has a negative NPV, meaning it destroys value. Reject it.

Loan amortization relies on present value to determine periodic payments. The loan amount is the present value, and the payment formula solves for the amount that, when discounted back, exactly equals the loan balance.

• A $\$200{,}000$ mortgage at 4.5% over 30 years requires monthly payments of approximately $\$1{,}013$
• A $\$50{,}000$ car loan at 6% over 5 years requires monthly payments of about $\$967$

Future Value vs Present Value

Inverse Relationship

Future value and present value are two sides of the same coin. Compounding moves a value forward in time; discounting moves it backward. You can always convert between them using the same rate and time period.

• $\$1{,}000$ at 5% compounded annually for 10 years → FV ≈ $\$1{,}629$
• $\$10{,}000$ to be received in 7 years, discounted at 8% compounded semi-annually → PV ≈ $\$5{,}820$

Two factors control how far apart PV and FV sit:

• Interest/discount rate: A higher rate increases FV (more growth) and decreases PV (heavier discounting).
• Time horizon: A longer period amplifies both effects, because compounding has more time to build and discounting has more time to erode value.

Financial Decision-Making

Comparing present values is the standard way to evaluate financial choices, since it puts everything in today's dollars.

• If a project's PV of expected cash flows exceeds its cost (positive NPV), the project adds value and should be accepted.
• When choosing among multiple opportunities with similar risk, pick the one with the highest positive NPV.

The choice between taking money now versus later depends on your required rate of return and your opportunity cost. If you have access to high-return investments, a dollar today is especially valuable because you can put it to work immediately. If you have a long time horizon and no pressing need for cash, deferring payment in exchange for a larger future amount can make sense, but only if the implied return meets or exceeds what you could earn elsewhere.