Discount Rate Explained: The Fed's Rate vs the Present Value Rate

Discount rate has two meanings in economics. In monetary policy, it is the interest rate the Federal Reserve charges commercial banks that borrow from it directly, usually overnight, through the discount window. In finance, it is the interest rate used to convert a future payment into its present value, so a higher discount rate makes money received later worth less today.

How to tell the two meanings apart

The two rates answer different questions, and context tells you which one is meant.

The Federal Reserve discount rate

What the discount window is

The discount window is the Fed's lending facility for banks. A bank that runs short of reserves can pledge collateral, such as government bonds, and borrow from the Fed. The interest rate on that loan is the discount rate.

Normally the window is a backstop. In a panic, when banks stop lending to each other, it is how the Fed acts as lender of last resort, keeping a sound bank from failing over a short cash gap.

Discount rate vs federal funds rate

• The discount rate is what a bank pays when it borrows from the Fed. The Fed announces it, so it is an administered rate.
• The federal funds rate is what a bank pays when it borrows reserves from another bank overnight. It is a market rate that the Fed steers toward a target.

The Fed normally sets the discount rate above the federal funds rate, which keeps the window a last resort. The discount rate vs federal funds rate comparison lays the two side by side.

The discount rate in an ample reserves system

AP Macro now teaches two frameworks. In today's ample reserves framework, banks hold so many reserves that small changes in their quantity no longer move the federal funds rate. The Fed steers it with administered rates, chiefly interest on reserve balances (IORB), plus the discount rate:

• IORB is the rate the Fed pays banks on the reserves they hold. A bank will not lend reserves to another bank for much less than it earns by leaving them at the Fed, so IORB holds the federal funds rate up near its level.
• The discount rate is the rate the Fed charges banks to borrow. It sits above IORB. A bank will not pay another bank much more than it would pay the Fed, so the discount rate limits how high short-term rates can go.

The ample reserves guide draws the reserve market graph for both frameworks, and discount rate vs IORB explains why one sits above the other. Move IORB yourself in the reserve market sandbox.

See it move

Make it move

The reserve market under limited reserves, where the supply of reserves sets the policy rate. Tap it to drag the curves, open the full version, or put it on your own site free, or turn it into a five-minute class activity.

Worked example: why banks borrow from each other first

Use hypothetical numbers. The federal funds rate is trading near 3.9% and the Fed's discount rate is 4.0%. A bank needs to borrow $20 million for one night. Use a 365-day year for simplicity.

Cost of borrowing from another bank:

$20,000,000 x 0.039 / 365 = $2,136.99

Cost of borrowing at the discount window:

$20,000,000 x 0.040 / 365 = $2,191.78

Difference: $2,191.78 - $2,136.99 = $54.79 more at the window.

The bank borrows from another bank. Now suppose lenders start asking 4.3% overnight. That night would cost $20,000,000 x 0.043 / 365 = $2,356.16, more than the window's $2,191.78, so the bank goes to the Fed. Because every borrower can switch, lenders cannot push the overnight rate far above 4.0%. That is the ceiling at work.

The discount rate in the limited reserves model

In the older limited reserves framework, reserves were scarce and the discount rate was taught as one of three tools, alongside open market operations and the reserve requirement.

• Lower it (expansionary). Borrowing from the Fed gets cheaper, banks may borrow and lend more, the money supply grows, rates fall, and aggregate demand shifts right.
• Raise it (contractionary). Borrowing from the Fed costs more, banks lend less, and aggregate demand shifts left.

A discount rate cut does not add reserves by itself; it only lowers the price of borrowing them. New money appears only if banks actually borrow and lend the reserves out. The monetary policy guide compares every tool, and AP Macro topic 4.6 lists them for both frameworks.

The discount rate in present value

Present value formula

A dollar today is worth more than a dollar next year, because today's dollar can earn interest in the meantime. Giving up that interest is an opportunity cost. To compare money at different dates, you discount future amounts back to today:

PV = FV / (1 + r)^n

where PV is present value, FV is the future amount, r is the discount rate as a decimal, and n is the number of years. Discounting is compounding run backward. Check your work with the present value calculator, or run the math forward with the future value calculator.

Worked example: a single future payment

A hypothetical relative promises you $5,000 in 4 years. Your discount rate is 6%, the rate you could earn by saving the money instead.

1. Convert the rate: r = 0.06.

2. Find the discount factor: (1.06)^4 = 1.06 x 1.06 x 1.06 x 1.06 = 1.26247696.

3. Divide: PV = $5,000 / 1.26247696 = $3,960.47.

Check it forward: $3,960.47 x 1.26247696 = $5,000.00.

Now raise the discount rate to 8%. The factor is (1.08)^4 = 1.36048896, so PV = $5,000 / 1.36048896 = $3,675.15. The promise did not change, yet it is worth $285.32 less today. A higher discount rate always lowers present value.

Worked example: is a project worth doing?

A hypothetical bakery can buy an oven for $9,000 today. The oven adds $3,500 of profit at the end of each of the next three years.

• At 7%: $9,185.11 - $9,000 = +$185.11. The profit is worth more than the oven, so the bakery buys it.
• At 9%: $8,859.53 - $9,000 = -$140.47. The same oven is now a losing project.

This is why investment demand slopes downward: higher interest rates mean heavier discounting, fewer projects pass, and borrowing falls. That is the demand side of the loanable funds market.

Why bond prices fall when rates rise

A bond's price is the present value of its fixed payments. Take a hypothetical 2-year bond with a $1,000 face value and a $40 coupon each year.

• At a 4% discount rate: $40 / 1.04 + $1,040 / 1.04^2 = $38.46 + $961.54 = $1,000.00
• At a 5% discount rate: $40 / 1.05 + $1,040 / 1.05^2 = $38.10 + $943.31 = $981.41

The payments are fixed, so a higher discount rate cuts the price by $18.59. This is the inverse relationship tested in topic 4.1; see the bonds guide.

Choosing a discount rate

The present value discount rate usually reflects the return on the next best use of the money, extra pay for risk, and expected inflation, so a riskier promise gets a higher rate. Nominal cash flows take a nominal rate; cash flows in today's purchasing power take a real interest rate, as covered in the Fisher equation guide.

Where the two meanings meet

When the Fed raises its administered rates, the federal funds rate rises, other market interest rates follow, and households and firms use a higher rate to discount the future. Projects like the bakery's oven stop passing the test, investment falls, and aggregate demand shifts left. The transmission mechanism guide traces each link.

How it shows up on the AP exam

Expect the Fed's discount rate in topic 4.6 questions that ask you to:

• Name the tool and direction. Lower the discount rate to expand, raise it to contract.
• Place it in a framework. Under ample reserves it is an administered rate alongside IORB. Under limited reserves it is one of three classic tools.

Common mistakes:

• Confusing the discount rate with the federal funds rate. One is charged by the Fed, the other by banks.
• Applying the money multiplier to a discount rate change. A rate cut by itself adds no reserves. Only the dollar amount banks actually borrow from the Fed enters the system, and that amount, not the rate change, is what the money multiplier applies to.
• Subtracting instead of dividing in present value. $5,000 discounted at 6% for one year is $5,000 / 1.06, not $5,000 minus 6%.
• Forgetting the exponent. Four years at 6% means dividing by 1.06^4, not by 1.24.