How Much is Future Money Worth Today?

1. Discounting and Present Value

When we talk about the Time Value of Money, we often ask: “How much will my money grow?” But in finance, we frequently need to ask the opposite question:

“If I need a specific amount of money in the future (say, \(C\) dollars in \(T\) years), how much do I need to invest today?”

This amount is called the Present Value (PV).

The Logic of Indifference

If the current annual interest rate is \(r\), you should technically be “indifferent” between receiving the Present Value (\(PV\)) today or the future amount (\(C\)) later. Why? Because if you have the \(PV\) today, you can lend or invest it at rate \(r\) to generate exactly \(C\) in the future.

The Formula

To find the Present Value, we reverse the compounding process. Instead of multiplying our current money by the interest rate, we divide the future amount by the interest factor.

• For 1 Year: $$PV = \frac{C}{(1+r)}$$
• For T Years (General Formula): $$PV = \frac{C}{(1+r)^T}$$

Key Insight: Moving Money Through Time

Think of the interest rate factor \((1+r)^T\) as a mechanism to move cash flows through time:

• Compounding: Multiplying by \((1+r)^T\) moves money into the Future.
• Discounting: Dividing by \((1+r)^T\) moves money into the Past (to the Present).

We often refer to this divisor as the Discount Factor (\(DF_T\)):

$$DF_T = \frac{1}{(1+r)^T}$$

Note: The Discount Factor is simply the reciprocal (inverse) of the Future Value Factor.

2. Example: The European Tour Problem

The Scenario:

• History: 3 years ago, you deposited $8,000 into an investment account earning $10% annual interest, compounded quarterly.
• Goal: You plan to take a dream tour to Europe 2 years from now.
• The Cost: You estimate the trip will cost $20,000 at the time of departure (2 years from now).

Step 0: Where do we stand today?

First, let’s figure out how much money is currently in the account (at Time \(0\)).

• Time elapsed: 3 years.
• Frequency: Quarterly (\(m=4\)).
• Total periods: \(3 \times 4 = 12\) periods.

$$FV_{today} = \$8,000 \times \left(1 + \frac{0.10}{4}\right)^{12} = \$8,000 \times (1.025)^{12} \approx \$10,759.11$$

You currently have $10,759.11 in the account.

Step 1: The Lump Sum Target

How much more money do you need to deposit today to reach the $20,000 goal in 2 years?

We need to find the Present Value (at Time 0) of the $20,000 needed at Time 2.

• Time remaining: 2 years (\(8\) quarters).

$$PV_{needed} = \frac{\$20,000}{\left(1 + \frac{0.10}{4}\right)^{8}} = \frac{\$20,000}{(1.025)^{8}} \approx \$16,414.97$$

• The Shortfall: $$\$16,414.97 \text{ (Needed)} – \$10,759.11 \text{ (Have)} = \mathbf{\$5,655.86}$$

Result: You need to deposit $5,655.86 today.

Step 2: The Installment Plan

Now, suppose the travel agency offers a payment plan. Instead of $20,000 upfront, you can pay:

1. $5,000 when the trip starts (t=2).
2. $8,000 one year after the trip (t=3).
3. $8,000 two years after the trip (t=4).

With this new offer, how much money do you need to add to your account today?

We calculate the Present Value (at Time \(0\)) of these three distinct cash flows.

$$PV_{plan} = \frac{\$5,000}{(1.025)^{8}} + \frac{\$8,000}{(1.025)^{12}} + \frac{\$8,000}{(1.025)^{16}}$$

• Discounting Payment 1 (\(t=2\), 8 periods): \(\approx \)$4,103.74
• Discounting Payment 2 (\(t=3\), 12 periods): \(\approx \)$5,948.74
• Discounting Payment 3 (\(t=4\), 16 periods): \(\approx \)$5,387.52

Total PV Required:

$$PV_{plan} = \$4,103.74 + \$5,948.74 + \$5,387.52 = \$15,440.00$$

• The New Shortfall: $$\$15,440.00 – \$10,759.11 = \mathbf{\$4,680.89}$$

Result: The installment plan is cheaper in present value terms. You only need to deposit $$4,680.89$ today.

3. Multiple Cash Flows

Real-life financial problems are rarely as simple as “one deposit today, one withdrawal later.” Most financial instruments—like mortgages, car loans, or business projects—involve a series of cash flows over time.

The “Divide and Conquer” Strategy

How do we find the Present Value of a stream of different future payments \(C_1, C_2, …, C_T\)?

We use a principle called Value Additivity.

1. Treat each payment separately: Imagine each future cash flow is its own independent “single payment” problem.
2. Discount each one back to today: Calculate the PV for the Year 1 payment, then the PV for the Year 2 payment, and so on.
3. Sum them up: The total value of the investment today is simply the sum of all these individual present values.

The Formula

$$PV = \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} + … + \frac{C_T}{(1+r)^T}$$

In formal mathematical notation, we write this as:

$$PV = \sum_{t=1}^{T} \frac{C_t}{(1+r)^t}$$

Simple Example: “The Freelancer’s Contract”

The Scenario:

You have just completed a large consulting project. The client agrees to pay you over the next three years rather than all at once. The payment schedule is:

• Year 1: $10,000
• Year 2: $20,000
• Year 3: $30,000

Your opportunity cost of capital (interest rate) is \(10\%\) annually.

Question: What is this contract actually worth to you today?

Step-by-Step Calculation:

1. Discount Year 1 Payment: $$PV_1 = \frac{\$10,000}{(1.10)^1} = \$9,090.91$$
2. Discount Year 2 Payment: $$PV_2 = \frac{\$20,000}{(1.10)^2} = \frac{\$20,000}{1.21} = \$16,528.93$$
3. Discount Year 3 Payment: $$PV_3 = \frac{\$30,000}{(1.10)^3} = \frac{\$30,000}{1.331} = \$22,539.44$$

Total Present Value:

$$PV_{total} = \$9,090.91 + \$16,528.93 + \$22,539.44 = \mathbf{\$48,159.28}$$

Conclusion:

Even though the nominal sum of the payments is \($60,000(10k+20k+30k)\), the economic value of the contract today is only $48,159.28$ because of the delay in receiving the money.