Difficulty: Introductory | Prerequisites: Basic algebra, familiarity with percentages.
Big Picture
Interest rates sit at the heart of macroeconomics. They determine the cost of borrowing, the return on saving, and the price of bonds in financial markets. This section covers two things you need cold for any macro exam: how to convert a nominal interest rate into a real one (adjusting for inflation), and why bond prices move in the opposite direction to interest rates. If you understand these mechanics, the Federal Reserve's policy decisions and their ripple effects will make far more sense.
TL;DR
The real interest rate strips out inflation from the nominal rate so you can see what a saver or borrower gains or loses in purchasing power. Bond prices and interest rates have an inverse relationship: when rates fall, existing bonds become more valuable, and vice versa.
Nominal interest rate
The stated rate of interest on a loan, bond, or deposit before any adjustment for inflation. This is the number the bank advertises.
Think of it as the "sticker price" of borrowing or saving, before you account for what inflation does to your purchasing power.
Real interest rate
The rate of return on saving (or cost of borrowing) after removing the effect of inflation. It tells you how much your purchasing power changes over time.
In simple terms, this is what you actually earn or pay once rising prices are factored in.
Expected inflation rate
The rate at which prices are anticipated to rise over a given period. Used in the Fisher equation to convert nominal rates to real rates.
Fisher equation (exact form)
The precise formula linking real, nominal, and inflation rates:
Real interest rate = [(1 + nominal rate) / (1 + inflation rate)] - 1
The approximate shortcut (real ≈ nominal - inflation) works for small values but the exact form is what the exam expects.
Bond
A debt instrument where the issuer (e.g. a government) promises to pay the holder a fixed stream of interest payments plus the face value at maturity. Bond prices move inversely to interest rates.
Inverse relationship (bond prices and interest rates)
When nominal interest rates fall, the price of existing bonds rises, because the fixed coupon payments on those bonds become relatively more attractive compared to newly issued bonds at lower rates. The reverse also holds: rising rates push bond prices down.
The nominal rate is what you see quoted. The real rate is what matters for purchasing power.
To find the real rate precisely, use the Fisher equation:
Real rate = [(1 + nominal) / (1 + inflation)] - 1
Worked example from the practice test:
A 5-year CD pays 3% nominal. Expected inflation is 1%.
(1.03 / 1.01) - 1 = 1.0198 - 1 = 0.0198 = 1.98%
The approximate shortcut (3% - 1% = 2%) is close, but the exact answer is 1.98%.
Bond prices and nominal interest rates move in opposite directions. This is one of the most-tested relationships in introductory macro.
When banks decrease nominal interest rates:
New bonds are issued at lower coupon rates.
Existing bonds, which still pay the older (higher) coupon, become more valuable.
Result: bond prices rise.
When banks increase nominal interest rates:
New bonds offer higher returns.
Existing bonds with lower coupons become less attractive.
Result: bond prices fall.
The mechanism is straightforward: a bond's coupon payment is fixed at issue. If prevailing rates drop, that fixed payment is worth more relative to what new buyers could get elsewhere, so the bond's market price rises.
Fisher Equation (exact)
Real interest rate = [(1 + nominal rate) / (1 + inflation rate)] - 1
Fisher Equation (approximation)
Real interest rate ≈ Nominal interest rate - Inflation rate
The approximation is convenient for mental maths but the exact formula is more accurate and is the version tested on this course's exams.
When central banks cut interest rates (as during the 2008 financial crisis or the COVID-19 pandemic), bond prices surge, which is why investors often pile into bonds as a "safe haven." Understanding the real interest rate is also how you judge whether a savings account is keeping up with inflation or quietly losing you money.
Students often think the real interest rate is simply the nominal rate minus inflation. The approximate version works for small numbers, but the exact Fisher equation divides rather than subtracts. On exams, using the wrong method can cost you marks.
Some students assume that when interest rates fall, bond prices also fall. The relationship is inverse: falling rates mean rising bond prices.
A common slip is confusing the interest rate on a bond with the bond's price. The coupon rate is fixed at issue. The price is what changes in the secondary market.
Students sometimes think inflation always reduces the real rate to near zero. If nominal rates are high enough relative to inflation, the real return can still be substantial.
⚠️ Expect a calculation question using the exact Fisher equation. Show each step: plug in, divide, subtract 1, convert to a percentage.
⚠️ The inverse relationship between bond prices and interest rates is a near-certain exam topic. Be able to explain the mechanism, not just state the direction.
⚠️ Know the distinction between nominal and real interest rates and be prepared to explain why the real rate is the one that matters for economic decisions.
True or false: The real interest rate is always lower than the nominal interest rate.
True (provided inflation is positive, which is the normal assumption in these problems).
Fill in the blank: When nominal interest rates decrease, bond prices ______.
Rise.
True or false: The approximate Fisher equation gives the same answer as the exact Fisher equation.
False. The approximation is close for small rates but not identical.
Fill in the blank: The exact Fisher equation is: Real rate = [(1 + ______) / (1 + ______)] - 1.
Nominal rate; inflation rate.
Q: A bank offers a 5-year CD at 3% nominal interest. Expected inflation is 1%. What is the exact real interest rate? Show your work.
A: (1 + 0.03) / (1 + 0.01) - 1 = 1.03 / 1.01 - 1 = 1.0198 - 1 = 0.0198 = 1.98%.
Q: Banks across the country are decreasing nominal interest rates. What happens to the price of government bonds, and why?
A: The price of government bonds will rise. Existing bonds pay fixed coupon rates that are now higher than the rates on newly issued bonds, making the existing bonds more valuable in the market.
Q: A savings account pays 5% nominal interest. Inflation is 3%. Using the exact Fisher equation, what is the real interest rate?
A: (1.05 / 1.03) - 1 = 1.0194 - 1 = 0.0194 = 1.94%. (The approximation would give 2%, which is close but not exact.)
Q: If the central bank raises interest rates sharply, would you expect an investor holding a portfolio of long-term government bonds to see gains or losses?
A: Losses. Rising interest rates push bond prices down because the fixed coupon on existing bonds becomes less attractive compared to newly issued higher-rate bonds.
This material links directly to monetary policy: the Federal Reserve's decisions to raise or lower the federal funds rate ripple through to bond markets and real returns on savings. It also connects to the loanable funds market, where real interest rates determine equilibrium borrowing and lending.
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