Nominal and Real Interest Rates
Emma lends her cousin $\$100$ for a year, and he promises to pay back $\$105$. At the end of the year, the same snacks she could have bought with $\$100$ now cost $\$106$. Did Emma actually come out ahead? By the end of this page you will be able to say exactly how much purchasing power she gained or lost.
Financial assets: return, risk, and liquidity
A financial asset is a claim on future payments. When you put money in a savings account, buy a bond, or buy shares of stock, you give up money today in exchange for a promise or a share of future income.
Three features separate one financial asset from another:
The three main types of assets you will see on the AP exam:
| Asset | What it is | Typical return | Typical risk | Liquidity |
|---|---|---|---|---|
| Savings account | Money deposited at a bank (FDIC insured) | Low | Very low | Very high |
| Bond | A loan to a corporation or the government | Medium | Medium | Medium |
| Stock (or index fund) | Partial ownership of firms | High on average | High | High if the stock is traded daily, but the price can fall |
Why do riskier assets pay more? Nobody would hold a risky asset unless it offered a higher expected return to make up for the chance of loss.
Liquidity matters too. Cash is the most liquid asset. A savings account is almost as liquid: you can withdraw the money at any time. A house is not liquid, since selling it takes weeks or months.
Rank a savings account, a corporate bond, and a stock index fund from lowest to highest in typical risk and in typical expected return. Explain in one sentence what it…
Rank a savings account, a corporate bond, and a stock index fund from lowest to highest in typical risk and in typical expected return. Explain in one sentence what it means for an asset to be liquid, and say which of the three is the most liquid.
Emma has $\$10{,}000…
Emma has $\$10{,}000$ that she will need in two years for a car. She is comparing a bank CD earning 3% (FDIC insured), a corporate bond earning 5%, and a stock index fund with an average long-run return of 8% that can lose value in any given year. Expected inflation is 2.5%. Compute the approximate expected real return on each option. Then argue which she should choose, using the ideas of risk, return, and liquidity.
Nominal vs. real interest rates
Back to Emma. She lent $\$100$ and got back $\$105$, so the interest rate is 5%. But prices rose 6%, so $\$105$ buys less than $\$100$ bought a year ago.
Why subtract? The lender earns $i$ more dollars, but each dollar buys $\pi$ percent less. The gain in purchasing power is roughly the difference.
Emma's case: $r \approx 5\% - 6\% = -1\%$. She lost about 1% of her purchasing power. The question from the start is answered: no, she did not come out ahead.
The real rate can be negative. It can even be computed in years of deflation (negative inflation): subtracting a negative number adds to the real rate. With $i = 3\%$ and $\pi = -1\%$, $r \approx 3\% - (-1\%) = 4\%$. Real rates can be higher than nominal rates when prices fall.
A bank offers a loan at a nominal interest rate of 6% when expected inflation is 2%. Compute the expected real interest rate. Write the formula you use.
A bank offers a loan at a nominal interest rate of 6% when expected inflation is 2%. Compute the expected real interest rate. Write the formula you use.
Define the nominal interest rate and the real interest rate. Which one tells a lender how much purchasing power she gains from lending, and why?
Define the nominal interest rate and the real interest rate. Which one tells a lender how much purchasing power she gains from lending, and why?
An economy has the following data. | Year | Nominal rate | Inflation rate | |---|---|---| | 1 | 5% | 3% | | 2 | 4% | 4.5% | | 3 | 3% | $-1\%$ | Compute the real interest…
An economy has the following data.
| Year | Nominal rate | Inflation rate |
|---|---|---|
| 1 | 5% | 3% |
| 2 | 4% | 4.5% |
| 3 | 3% | $-1\%$ |
Compute the real interest rate in each year. In which year do lenders lose purchasing power, and what does the result for year 3 show about deflation?
Expected vs. actual inflation
When a loan is made, nobody knows what inflation will be. Lenders and borrowers agree on a nominal rate based on expected inflation. Later, actual inflation turns out to be something else.
-
Expected real rate: $i - \pi^e$ (what both sides planned on).
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Actual real rate: $i - \pi$ (what really happened).
| If actual inflation is... | Actual real rate is... | Who gains? |
|---|---|---|
| Higher than expected ($\pi > \pi^e$) | Lower than planned | Borrower (repays with dollars worth less) |
| Lower than expected ($\pi < \pi^e$) | Higher than planned | Lender (gets dollars worth more) |
| Equal to expected | As planned | Nobody gains or loses |
Fully expected inflation redistributes nothing, because the nominal rate already includes it. The surprise is what moves wealth between borrowers and lenders.
A loan carries a nominal interest rate of 9%. When it was made, borrower and lender expected inflation of 3%. Actual inflation turned out to be 5%. Compute the expected…
A loan carries a nominal interest rate of 9%. When it was made, borrower and lender expected inflation of 3%. Actual inflation turned out to be 5%. Compute the expected real interest rate and the actual real interest rate. Who benefited from the difference?
An economy's nominal interest rate is 5% and expected inflation is 2%. Then households and firms come to expect inflation of 4%. (a) Compute the real rate before the…
An economy's nominal interest rate is 5% and expected inflation is 2%. Then households and firms come to expect inflation of 4%. (a) Compute the real rate before the change and the real rate after the change, if the nominal rate stays at 5%. (b) Describe the effect on borrowing, investment, and aggregate demand in that case. (c) Now suppose lenders fully adjust, and the nominal rate rises to 7%. Compute the real rate, and explain the Fisher effect that this illustrates. (d) Explain why unexpected inflation can redistribute income between borrowers and lenders but fully expected inflation generally does not.
Interest rates and bond prices
Why do news reports say "bond prices fall when interest rates rise"? Imagine you own a bond that pays a fixed amount each year. A new bond comes out paying more because rates went up. Who would pay full price for your old, lower-paying bond? Nobody. To sell it, you must cut the price.
For a bond that pays a fixed amount forever (a perpetuity):
$$\text{price} = \frac{\text{annual payment}}{\text{market interest rate}}$$Notice that a 1-point rise in the rate cut the price by 20%. Small changes in rates can cause big changes in bond prices.
Who bears the risk when rates rise? The person who already holds the bond: its market value falls. A person who is about to buy is helped, since she can buy at the lower price and get a higher yield. If you hold the bond until it matures, you still get the promised payments, but you lose if you must sell early.
A bond pays a fixed $\$50…
A bond pays a fixed $\$50$ per year forever (a perpetuity), and its price equals the annual payment divided by the market interest rate. (a) Compute its price when the market rate is 5%. (b) Compute its price when the market rate rises to 6.25%. (c) Compute the percent change in the price. (d) Explain the general relationship between bond prices and interest rates, and explain who bears the risk when interest rates rise, the investor who holds the bond or the one who is about to buy it.
An economy's nominal interest rate is 5% and expected inflation is 2%. Then households and firms come to expect inflation of 4%. (a) Compute the real rate before the…
An economy's nominal interest rate is 5% and expected inflation is 2%. Then households and firms come to expect inflation of 4%. (a) Compute the real rate before the change and the real rate after the change, if the nominal rate stays at 5%. (b) Describe the effect on borrowing, investment, and aggregate demand in that case. (c) Now suppose lenders fully adjust, and the nominal rate rises to 7%. Compute the real rate, and explain the Fisher effect that this illustrates. (d) Explain why unexpected inflation can redistribute income between borrowers and lenders but fully expected inflation generally does not.