Interest Rate vs Present Value
Interest Rate and Present Value are two Money, Banking & Finance concepts in AP Economics that students often mix up. An interest rate is the cost of borrowing money or the reward for saving it, expressed as a percentage of the principal per year. Present value is what a future sum of money is worth today, after discounting for the interest that could be earned in the meantime. Here is how they compare side by side.
Interest rates are set in money and loanable-funds markets and steered by the central bank. Lower rates encourage borrowing, investment, and spending; higher rates encourage saving and slow the economy. The real interest rate (nominal minus inflation) reflects the true cost of borrowing.
Because money available now can earn interest, a dollar today is worth more than a dollar in the future. Present value is used to compare investments and value bonds. A higher interest rate lowers present value.
Interest Rate vs Present Value: The Input You Choose and the Number It Produces
| Interest Rate | Present Value | |
|---|---|---|
| Role in the calculation | The input you discount with | The output the calculation produces |
| Units | Percent per year | A sum of money, stated for today |
| Which way they move | Rises | Falls, for any positive amount promised in the future |
| What it measures | The reward for waiting, or the cost of not waiting | What a future payment is worth once that cost is subtracted |
| Effect of a longer horizon | One quoted rate can cover any horizon | Sensitivity to the rate compounds, so distant payments swing hardest |
| What a mistake in it costs | Using the wrong rate is a judgment error | A wrong value can reverse an accept-or-reject decision |
| Where the exam uses it | The vertical axis of the money market and loanable funds | The logic behind a downward sloping investment demand curve |
Doubling the rate barely dents a payment due in two years and destroys most of one due in twenty
Present value falls when the rate rises, but the size of the fall depends almost entirely on how far away the payment is. Price a promise of 100. Due in two years, discounted at 5 percent, it is worth 90.70 today. Discount the same promise at 10 percent and it is worth 82.64, a decline of roughly 9 percent. Now push the payment out to twenty years. At 5 percent, compounding for two decades multiplies the money by about 2.65, so the present value is 37.69. At 10 percent the multiplier is about 6.73 and the present value collapses to 14.86, a decline of roughly 61 percent from the 5 percent figure. Same promise, same issuer, same doubling of the rate, and one calculation shrugged while the other lost most of its value. The reason is compounding: the discount is applied once per year, so an extra percentage point gets multiplied through every one of those years. Anything whose payoff is decades out, a pension liability, a power plant, a forestry investment, is dominated by the rate rather than merely adjusted by it. Work through the arithmetic step by step at /calculate/present-value.
Choosing a different rate can reverse the decision without changing anything about the project
A firm can accept and reject the same project depending only on the rate applied to it. Take a project that costs 90 today and pays a certain 100 in two years. Discount at 5 percent and the payoff is worth 90.70 now, so the project clears its cost by 0.70 and gets built. Discount at 10 percent and the payoff is worth 82.64, falling short of the 90 outlay by 7.36, so the same project is rejected. The break-even sits at roughly 5.4 percent, the rate at which the payoff exactly equals the cost. Nothing about the machinery, the customers or the payoff changed. What changed was the return available elsewhere, and that is what a discount rate represents: the opportunity cost of tying money up here rather than lending it out. Rank every project in an economy by its break-even rate and the downward sloping investment demand curve appears immediately, since a rising rate knocks out the projects with the thinnest margins first. This is the mechanism behind the loanable funds diagram at /glossary/loanable-funds-market, and it is why /glossary/opportunity-cost is doing more work in finance than most students expect.
Frequently asked questions
What happens to present value when the interest rate rises?
Present value falls, because a larger slice gets discounted away before the future payment is counted. How far it falls depends on the horizon. A payment due next year loses a little, while a payment due in twenty years can lose most of its value, since the discount compounds through every intervening year. A promise of 100 due in two decades is worth 37.69 at 5 percent and 14.86 at 10 percent, which is a collapse of roughly 61 percent from doubling the rate.
Which interest rate should be used for discounting?
The rate should match the return available on an alternative of similar risk and similar timing. Discounting a risky payoff at a risk-free rate overstates its present value, because the risk-free alternative was never a genuine substitute. Discounting a twenty-year payoff at an overnight rate makes the same mistake in the time dimension. Picking the rate is the judgment call in the whole exercise, and the arithmetic that follows is mechanical by comparison.
Why does present value matter for investment decisions?
Firms compare a cost paid now against revenue arriving later, and those amounts are not directly comparable until the future stream is converted to present value. A project costing 90 that returns 100 in two years is worth doing at 5 percent, where the payoff is worth 90.70, and not worth doing at 10 percent, where it is worth 82.64. Every project has such a cutoff rate, which is exactly why higher rates reduce total investment across the economy.
Live Loanable Funds graph. Drag the curves, or open the full version.
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