Nominal versus Real Interest Rates and Financial Asset Pricing

Distinction Between Nominal and Real Interest Rates

Nominal Interest Rates Defined:
- Nominal interest rates are interest rates that are not adjusted for changes in purchasing power caused by changes in the price level.
- They represent the stated monetary amount of interest paid or received per dollar invested or borrowed, without accounting for inflation's erosive effect.
The Impact of Inflation on Investment Returns:
- Inflation reduces the purchasing power of returns on any investment across the financial system.
- When an investor buys a fixed-income asset, inflation diminishes the real value of both the regular interest payments and the return of the principal.
- Example of Purchasing Power Erosion:
- Suppose an investor purchases a $1,000$ bond that pays $50$ in interest each year for $20$ years.
- If the purchasing power of the dollar declines over time due to inflation, the investor effectively loses part of the interest income to higher price levels.
- In addition to eroding interest payments, inflation erodes the purchasing power of the principal.
- If the inflation rate is $5\%$ per year, the purchasing power of the $1,000$ principal falls by $50$ each year.
Real Interest Rates Defined:
- Real interest rates are interest rates that have been adjusted for changes in purchasing power.
- The real interest rate represents the true economic return to lending and the true economic cost of borrowing.
- Because lenders and borrowers make decisions that extend into the future, and because future price levels are uncertain, market participants must base their lending, borrowing, saving, and investing decisions on expected real interest rates rather than known actual real interest rates.

The Fisher Equation and Expected Real Interest Rates

Estimating Expected Real Rates:
- To determine the expected real interest rate, savers and borrowers must formulate an expectation of what the inflation rate will be over the duration of the loan or investment.
- The expected real interest rate equals the nominal interest rate minus the expected rate of inflation:

$r = i - \pi^e$

Where:
- $r$ is the expected real interest rate.
- $i$ is the nominal interest rate.
- $\pi^e$ is the expected rate of inflation.
- Rearranged Formulation for the Nominal Interest Rate:
Rearranging the relationship demonstrates that the nominal interest rate equals the expected real interest rate plus the expected inflation rate:

$i = r + \pi^e$

Hypothetical Car Loan Application:
- Suppose a borrower applies for a car loan from a local commercial bank.
- The borrower is willing to pay, and the bank is willing to accept, a real interest rate of $3\%$ .
- Both the borrower and the bank forecast and expect that the inflation rate will be $2\%$ .
- Consequently, the agreed nominal interest rate set on the loan contract is:

$i = 3\% + 2\% = 5\%$

Outcome of Unexpected Higher Inflation:
- Suppose actual inflation turns out to be $4\%$ , which is higher than the expected $2\%$ .
- The actual real interest rate paid by the borrower and received by the bank is:

$5\% - 4\% = 1\%$

Because $1\% < 3\%$ , the actual real interest rate is less than the expected real interest rate.
As a direct result, the borrower gains by paying a lower real interest rate, and the lending bank loses by receiving a lower real interest rate.

Redistributional Effects of Unexpected Inflation

Systematic Redistribution Between Borrowers and Lenders:
- Deviations between realized inflation and anticipated inflation shift wealth between borrowers and lenders because loan agreements fix nominal interest payments in advance.
Condition 1: Realized Inflation Exceeds Expected Inflation ( $\pi > \pi^e$ ):
- When the actual inflation rate is greater than the expected inflation rate, the actual real interest rate is less than the expected real interest rate.
- Borrowers gain: The debt is repaid with dollars that have less purchasing power than anticipated.
- Lenders lose: The return on capital yields less purchasing power than anticipated.
Condition 2: Realized Inflation Falls Short of Expected Inflation ( $\pi < \pi^e$ ):
- When the actual inflation rate is less than the expected inflation rate, the actual real interest rate is greater than the expected real interest rate.
- Borrowers lose: Borrowers must pay back their debts using dollars that retain more purchasing power than originally expected, raising their real cost of borrowing.
- Lenders gain: Lenders receive loan repayments with higher-than-expected purchasing power, increasing their true return on lending.

Empirical Analysis of Nominal and Real Interest Rates (1990–2019)

Nominal, real, and expected real interest rates from 1990 through 2019

Measurement Framework for Aggregate Rates:
- Economists frequently gauge the benchmark nominal interest rate for the U.S. economy using the interest rate on U.S. Treasury bills with a maturity of three months.
- The actual real interest rate is calculated as the nominal interest rate on three-month U.S. Treasury bills minus the actual inflation rate, where inflation is measured by changes in the Consumer Price Index (CPI).
- The expected real interest rate is calculated as the nominal interest rate minus the expected inflation rate, where inflation expectations are drawn from the survey of professional forecasters conducted by the Federal Reserve Bank of Philadelphia.
- Historical data tracking these three rates extends from the first quarter of $1990$ through the fourth quarter of $2019$ (sourced from the Federal Reserve Bank of St. Louis and the Federal Reserve Bank of Philadelphia).
Key Historical Patterns and Co-Movement:
- Nominal and real interest rates broadly display positive co-movement, tending to rise and fall together across economic cycles.
Negative Real Interest Rates:
- During specific historical periods, particularly following the onset of the $2007$ financial crisis, the real interest rate turned negative.
- Economic Rationale for Holding Negative Real-Yield Assets:
- Rational investors purchased Treasury bills yielding negative expected real returns because alternative market investments carried elevated default risk.
- During and after the $2007$ financial crisis, the low yields offered on riskier financial assets were deemed insufficient to compensate for their risk of default.
- Institutional investors deliberately accepted a negative real interest rate on U.S. Treasury bills to preserve principal capital safely rather than face high probabilities of substantial capital loss in corporate bonds or other private securities.
Deflation Dynamics:
- A nominal interest rate can drop below the real interest rate when the inflation rate is negative.
- Deflation Defined: A sustained decline in the overall price level.
- When deflation occurs, the subtraction of a negative inflation rate from the nominal rate causes the real interest rate to surpass the nominal interest rate:

$r = i - (-\pi) = i + \pi$

Historical occurrences in the United States include the first $10$ months of $2009$ and the early months of $2015$ , during which real interest rates exceeded nominal rates.

Treasury Inflation-Protected Securities (TIPS)

Value of TIPS as a percentage of all Treasury securities from 1996 through 2019

Mechanics and Inception of TIPS:
- In January $1997$ , the U.S. Department of the Treasury introduced index-linked sovereign bonds known as TIPS (Treasury Inflation-Protected Securities) to mitigate investor exposure to inflation risk.
- The principal value of a TIPS bond is adjusted periodically in proportion to movements in the price level, as measured by the Consumer Price Index (CPI).
- While the coupon (stated interest) rate remains fixed at issuance, the effective dollar return increases with inflation because this fixed percentage is applied to an upwardly adjusted principal.
- If the general price level decreases due to deflation, the principal value of the TIPS bond is adjusted downward.
Mathematical Illustration of TIPS Returns:
- Consider a newly issued $10$ -year TIPS bond with an initial principal of $1,000$ and a stated interest rate of $3\%$ .
- If the annual inflation rate over the subsequent year is $2\%$ , the bond's principal is adjusted upward by $2\%$ :

\1,000 \times (1 + 0.02) = \ $1,020$

The fixed $3\%$ coupon rate applies directly to the new adjusted principal amount:

$0.03 \times 1{,}020 = \$30.60$

Relative to the initial $1,000$ investment, the actual realized interest rate received by the investor is:

$\frac{\$30.60}{\$1{,}000} = 3.06\%$

Evolution of the TIPS Market Share (1996–2019):
- Starting near $0\%$ at introduction in $1997$ , TIPS grew steadily as a proportion of total U.S. Treasury securities, reaching approximately $9\%$ by $2008$ .
- Between $2008$ and $2010$ , the market share of TIPS contracted significantly (dropping toward $6.5\%$ ), driven by falling inflation expectations during the acute phases of the Great Recession.
- Between $2010$ and $2015$ , investor demand rebounded as market participants sought insulation against future inflation risks, driving the proportion back toward $9\%$ .
- From $2015$ through $2019$ , TIPS stabilized at a steady equilibrium just below $9\%$ of total outstanding U.S. Treasury securities.

Asset Pricing and the Inverse Relationship with Interest Rates

Present Value Foundation of Security Pricing:
- The market price of any financial security is fundamentally equal to the present discounted value of all future cash flows and payments that the owner will receive.
- Future cash flows are discounted back to the present using prevailing market interest rates.
The Mechanism of Opposing Movements:
- When market interest rates rise, the discount factor applied to future payments increases, which directly causes the present discounted value of those future cash flows to fall; hence, security prices fall.
- When market interest rates fall, the discount factor decreases, which causes the present discounted value of future cash flows to rise; hence, security prices rise.
- Because security values are mathematically determined by discounted cash flow valuation, interest rates and the prices of financial securities necessarily move in opposite directions.