Interest Rates, Security Valuation, and the Federal Reserve System Notes

Nominal Interest Rates and Fundamentals

• Nominal interest rates are the interest rates observed in financial markets.

• Nominal rates directly affect the value of financial securities.

• Fluctuations in interest rates significantly influence performance, decision-making, and financial planning across individual investors, corporate entities, and governmental institutions.

• Historical trends in key U.S. interest rates (1972–2022) demonstrate cyclical co-movements across various debt instruments, including the Federal Funds Rate, 3-Month T-Bill, Moody's AAA Bond Yield, and the 30-Year Fixed Rate Mortgage Average during economic expansions and recessions.

Loanable Funds Theory and Market Dynamics

• Loanable Funds Theory posits that market equilibrium interest rates are determined by the aggregate supply and demand for loanable funds across financial market participants.

• Market participants are categorized into four principal sectors:

  • Households (consumers)

  • Businesses

  • Governments

  • Foreign participants

• Interest rate changes drive inverse and direct responses in supply and demand quantities:

  • An increase in interest rates ($\uparrow$) results in an increase in quantity supplied ($\uparrow$) and a decrease in quantity demanded ($\downarrow$).

  • A decrease in interest rates ($\downarrow$) results in a decrease in quantity supplied ($\downarrow$) and an increase in quantity demanded ($\uparrow$).

• Value of Loanable Funds by Sector (2019 Data, in trillions of USD):

  • Households: Funds Supplied = $84.66$, Funds Demanded = $16.05$, Net Funds Supplied = $68.61$ (Net Supplier).

  • Businesses: Funds Supplied = $126.53$, Funds Demanded = $178.33$, Net Funds Supplied = $-51.80$ (Net Demander).

  • Governments: Funds Supplied = $5.66$, Funds Demanded = $28.86$, Net Funds Supplied = $-23.20$ (Net Demander).

  • Foreign Investors: Funds Supplied = $27.20$, Funds Demanded = $20.81$, Net Funds Supplied = $6.39$ (Net Supplier).

• Specific economic factors influence supply and demand curves, shifting equilibrium interest rates:

  • Wealth of Suppliers: An increase shifts the supply curve right ($+$, inverse impact on equilibrium rate).

  • Risk of Security: An increase shifts supply left ($-$) and demand left ($-$), creating direct upward pressure on equilibrium rates.

  • Near-Term Spending Needs: An increase reduces supply ($-$) by shifting the supply curve left (direct impact on equilibrium rate).

  • Monetary Expansion / Policy: Expansionary monetary policy increases supply ($+$) by shifting the supply curve right (inverse impact on equilibrium rate).

  • Economic Growth / Conditions: Economic expansion increases both the supply ($+$) and demand ($+$) for loanable funds. Supply shift exerts inverse pressure, while demand shift exerts direct upward pressure on rates.

  • Utility Derived from Asset: Increased utility decreases supply ($-$) and increases demand ($+$), exerting direct upward pressure on equilibrium rates.

  • Restrictive Nonprice Conditions: Higher restrictiveness increases supply ($+$) and decreases demand ($-$), shifting the demand curve to exert inverse pressure on rates.

  • Expected Inflation: Higher expected inflation decreases supply ($-$) and increases demand ($+$).

Determinants of Interest Rates for Individual Securities

• Inflation: The continual rise in the overall price level of a standardized basket of goods and services. Measured in the U.S. using the Consumer Price Index (CPI) and Producer Price Index (PPI):

\text{Inflation (IP)} = \n\frac{\text{CPI}_{t+1} - \text{CPI}_t}{\text{CPI}_t} \times 100

• Real Risk-Free Rate (RFR): The theoretical interest rate on a risk-free security assuming zero expected inflation during the investment holding period. Reflects society's time preference for consumption (a stronger preference to consume today elevates the real risk-free rate). U.S. Treasury Bills serve as the standard benchmark.

• Fisher Effect: Defines the relationship between nominal interest rate ($i$), real risk-free rate ($RFR$), and expected inflation ($E(IP)$):

$i = RFR + E(IP)$

• Default Risk Premium (DRP): Compensation for the risk that a security issuer may fail to make scheduled interest or principal payments. $DRP_j$ represents the spread between the quoted interest rate on security $j$ and a U.S. Treasury security of identical maturity, liquidity, tax, and provision profiles.

• Liquidity Risk Premium (LRP): Compensation required for holding securities that cannot be converted into cash on short notice at a predictable price with low transaction costs.

• Special Provisions or Covenants ($SCP_j$):

  • Taxability: Interest on municipal securities is exempt from federal income taxes, enabling lower nominal yields relative to taxable bonds.

  • Convertibility: Allows holders to exchange bonds for corporate stock at a preset price, benefiting the holder and lowering the required yield.

  • Callability: Gives issuers the right to redeem bonds prior to maturity, creating reinvestment risk for holders and requiring higher yields.

  • General Rule: Features benefiting the security holder decrease required interest rates; features benefiting the issuer increase required interest rates.

• Term to Maturity / Maturity Premium ($MP_j$): Reflects yield differences attributable strictly to time remaining until maturity. The maturity premium can be positive, negative, or zero.

• Functional Formula for Fair Interest Rate ($i_j^*$):

$i_j^* = f(IP, RFR, DRP_j, LRP_j, SCP_j, MP_j)$

Term Structure of Interest Rates and Yield Curves

• The term structure of interest rates compares market yields across securities that differ solely in their remaining term to maturity.

• Yield curves plot yields against maturity and take distinct structural shapes over time:

• Three theoretical frameworks explain yield curve structure:

  • Unbiased Expectations Theory: Assumes investors are risk-neutral, meaning holding an $N$-period bond is identical to holding a sequence of $N$ one-year bonds. The yield curve reflects market expectations of future short-term interest rates.

  • Liquidity Premium Theory: Extends unbiased expectations by accounting for risk aversion. Investors demand an increasing liquidity premium ($L_N$) to lock up funds for longer durations due to price uncertainty:

$L_2 < L_3 < L_4 < \dots < L_N$

• Market Segmentation Theory: Rejects perfect substitutability across maturities. Assumes investors and institutions have rigid maturity preferences matching their liability profiles. Shifts in yield curve slopes are determined by supply and demand equilibrium within isolated maturity segments.

Principles of Time Value of Money and Security Valuation

• Fundamentals of Time Value of Money (TVM):

  • A dollar received today is worth more than a dollar received at a future date due to earning capacity.

  • Security valuation utilizes two primary TVM cash flow structures:

  • Lump Sum: A single cash payment occurring at a specific point in time.

  • Annuity: A uniform series of periodic cash flows occurring over a set horizon.

• Present Value of a Lump Sum:

$PV = \frac{FV_t}{(1 + r)^t}$

• Present Value of an Annuity:

$PV = \frac{PMT}{r} \left[ 1 - \frac{1}{(1 + r)^t} \right]$

Measures of Interest Rates and Return Definitions

• Coupon Rate: The nominal annual interest rate promised by a bond issuer, expressed as a percentage of face value.

• Required Rate of Return ($r$): The discount rate that determines the fair present value ($PV$) of a security, incorporating all associated risk factors ex-ante:

$PV = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}$

• Expected Rate of Return ($E(r)$): The interest rate expected by market participants purchasing a security at current market price ($\bar{P}$) and holding it across their investment horizon:

$\bar{P} = \sum_{t=1}^{n} \frac{CF_t}{[1 + E(r)]^t}$

• Realized Rate of Return ($\bar{r}$): The ex-post rate of return earned on an investment over a completed holding period:

$\bar{P} = \sum_{t=1}^{n} \frac{RCF_t}{[1 + \bar{r}]^t}$

• If $\bar{r} > r$, the investor earned a return greater than required for the ex-ante risk.

• If $\bar{r} < r$, the investor earned less than necessary to compensate for risk.

  • Market Efficiency: Describes the speed with which security prices adjust to new information to align current market prices with fair present values. In efficient markets, current market price ($\bar{P}$) equals fair present value ($PV$).

Bond Valuation Mechanics and Price Sensitivity

• Bond Cash Flows: Derived from periodic interest payments ($INT$) and lump sum face value payment ($M$, standard at $1,000 in U.S. markets) paid at maturity ($T$).

• New bond issues typically set coupon rates equal to the prevailing required rate of return.

• General Valuation Formula for Semiannual Coupon Bonds:

$V_b = \frac{INT}{2} \left[ \frac{1 - \frac{1}{(1 + r_b/2)^{2T}}}{r_b/2} \right] + \frac{M}{\left(1 + \frac{r_b}{2}\right)^{2T}}$

• Zero-Coupon Bond Valuation ($INT = 0$):

$V_b = \frac{M}{\left(1 + \frac{r_b}{2}\right)^{2T}}$

• Bond Price Classifications:

  • Premium Bond ("Above Par"): Price > Face Value $\iff$ Coupon Rate > Yield to Maturity (YTM).

  • Par Bond ("At Par"): Price = Face Value $\iff$ Coupon Rate = YTM.

  • Discount Bond ("Below Par"): Price < Face Value $\iff$ Coupon Rate < YTM.

• Worked Example 1 (Zero-Coupon Bond):

  • Given: 5-year maturity ($T = 5$), $1,000 face value ($M = 1000$), purchase price $800 ($V_b = 800$).

  • Calculation:

$800 = \frac{1000}{\left(1 + \frac{r_b}{2}\right)^{10}}$

$r_b = \left[ \left(\frac{1000}{800}\right)^{\frac{1}{10}} - 1 \right] \times 2 = 0.0451 \text{ or } 4.51\%$

• Worked Example 2 (Coupon Bond):

  • Given: 10-year maturity ($T = 10$, $2T = 20$), 5% coupon rate ($INT/2 = \$25$), $1,000 face value ($M = 1000$), 4% annual market yield ($r_b/2 = 0.02$).

  • Calculation:

$V_b = 25 \left[ \frac{1 - \frac{1}{(1.02)^{20}}}{0.02} \right] + \frac{1000}{(1.02)^{20}} = \$1,081.76$

• Bond Price Sensitivity Rules:

  • Interest Rate Changes: Bond prices move inversely to changes in required yields. Price increases when yield decreases; price decreases when yield increases.

• Maturity: Longer-maturity bonds experience greater price sensitivity to interest rate changes than shorter-maturity bonds. However, sensitivity increases at a decreasing rate. As maturity approaches, bond price converges toward face value.

* *Example*: 30-year zero-coupon bond at 5% YTM ($100 face value):

$P_{30} = \frac{100}{(1.025)^{60}} = \$22.62$

  * Price 5 years later (25 years to maturity):

$P_{25} = \frac{100}{(1.025)^{50}} = \$29.54$

• Coupon Rate: Higher coupon rates yield higher present values and smaller percentage price sensitivity for a given change in interest rates.

  • Example: 30-year bonds evaluated at 5% YTM ($100 face value):

    • 10% Coupon Bond Price = $177.27

    • 5% Coupon Bond Price = $100.00

    • 3% Coupon Bond Price = $69.09

Equity Valuation Models

• Common stock valuation equals the present value of an infinite stream of future dividends discounted at required return $r_s$:

$P_0 = \sum_{t=1}^{\infty} \frac{Div_t}{(1 + r_s)^t}$

• Zero Growth Model: Dividends remain constant forever ($Div_t = Div$):

$P_0 = \frac{Div}{r_s}$

• Example: Longs Drug Stores pays an annual dividend of $0.56 per share; expected return is 5%.

$P_0 = \frac{0.56}{0.05} = \$11.20$

• Constant Growth Model (Gordon Growth Model): Dividends grow at a constant annual rate $g$ forever:

$P_0 = \frac{Div_0 (1 + g)}{r_s - g} = \frac{Div_1}{r_s - g}$

• Expected rate of return formulation:

$r_s = \frac{Div_1}{P_0} + g$

• Example: Longs Drug Stores expects 3% dividend growth, $Div_1 = \$0.56$, return = 5%.

$P_0 = \frac{0.56}{0.05 - 0.03} = \$28.00$

• Nonconstant / Supernormal Growth Model: Three-step valuation process:

  1. Calculate the present value of dividends during the supernormal growth period.

  2. Calculate stock price at the end of the supernormal growth period using the constant growth model, then discount this terminal value to present value.

  3. Sum the present value of nonconstant dividends and the discounted terminal stock price.

Duration and Convexity Analysis

• Duration ($D$): The weighted-average time to maturity on a fixed-income security using relative present values of cash flows as weights. Measures price sensitivity for small interest rate changes.

• Formula for Annual Cash Flows:

$D = \frac{\sum_{t=1}^{N} \frac{CF_t \times t}{(1 + r)^t}}{\sum_{t=1}^{N} \frac{CF_t}{(1 + r)^t}}$

• Formula for Semiannual Cash Flows:

$D = \frac{\sum_{t=1/2}^{N} \frac{CF_t \times t}{(1 + r/2)^{2t}}}{\sum_{t=1/2}^{N} \frac{CF_t}{(1 + r/2)^{2t}}}$

• Worked Examples and Properties of Duration:

  • Base Bond: 4-year maturity, 10% semiannual coupon ($50/period), $1,000 face value, 8% rate of return ($r_b = 8\%$), price $1,067.34 $\rightarrow D = 3.42\text{ years}$.

  • Impact of Coupon: Lowering coupon to 6% ($30/period), price $932.68 $\rightarrow D = 3.60\text{ years}$. Lower coupon rates increase duration.

  • Impact of Yield: Increasing return to 10%, price $1,000.00 $\rightarrow D = 3.39\text{ years}$. Higher yields decrease duration.

  • Zero-Coupon Bond: Duration equals remaining term to maturity ($D = T_{ZC}$).

• Elasticity and Economic Meaning of Duration:

  • Annual Compounding:

$\frac{\Delta P}{P} = -D \left[ \frac{\Delta r_b}{1 + r_b} \right]$

• Semiannual Compounding:

$\frac{\Delta P}{P} = -D \left[ \frac{\Delta r_b}{1 + r_b / 2} \right]$

• Modified Duration ($MD$):

$\frac{\Delta P}{P} = -MD \times \Delta r_b$

• Convexity: The degree of curvature in the price-yield relationship.

  • Duration accurately measures price changes only for minor interest rate fluctuations (several basis points, where 1 basis point = 0.01%).

  • Convexity causes capital losses from interest rate increases to be smaller than capital gains from equal interest rate decreases.

  • Ignoring convexity introduces error into duration-matching portfolio immunization strategies when rate shifts are large.

Structure and Organization of the Federal Reserve System

• The Federal Reserve (the Fed) was created under the Federal Reserve Act of 1913 as the central bank of the United States.

• Four Core Duties of the Federal Reserve System:

  1. Conducting monetary policy

  2. Supervising and regulating depository institutions

  3. Maintaining financial system stability

  4. Providing payment and financial services to the U.S. government, public, financial institutions, and foreign official institutions

• Organizational Governance:

  • 12 Federal Reserve Banks (FRBs): Distributed geographically across defined districts. Each operates under a 9-member Board of Directors. Nationally chartered banks are required to hold membership.

  • Board of Governors: Located in Washington, D.C. Consists of 7 members appointed by the U.S. President and confirmed by the Senate to nonrenewable 14-year terms. Primary focus: monetary policy formulation and supervisory regulation.

  • Federal Open Market Committee (FOMC): The primary monetary policy-making body. Composed of the 7 Board Governors, the President of the Federal Reserve Bank of New York, and a rotating group of 4 other FRB Presidents. Required to meet at least 4 times per year in Washington, D.C. Publishes economic assessments in the Beige Book.

Functions and Balance Sheet of Federal Reserve Banks

• Eight Primary Operational Functions of FRBs:

  1. Assistance in conducting monetary policy (setting discount rates, operating discount windows).

  2. Supervision and regulation of state-chartered member banks and bank holding companies.

  3. Consumer protection and community affairs law enforcement.

  4. Government commercial banking services for the U.S. Treasury.

  5. New currency issue and replacement of damaged currency.

  6. Central check clearing system operations.

  7. Wire transfer services via the Federal Reserve Communications System (Fedwire).

  8. Economic research services.

• Balance Sheet Composition:

  • Liabilities: Monetary Base = Currency in Circulation + Total Reserves (Required Reserves + Excess Reserves).

  • Assets: U.S. Treasury securities, U.S. Government Agency securities (Fannie Mae, Freddie Mac), Treasury currency, Gold and foreign exchange, Interbank loans.

  • Federal Reserve Balance Sheet Trends (2007 vs. 2018, in Billions of USD):

  • Total Assets: $951.3B (2007) $\rightarrow$ $4,236.00B (2018) (Peak of $4,542.2B in 2016).

  • U.S. Treasury Securities: $740.6B (2007) $\rightarrow$ $2,338.00B (2018; 55.2% of total assets).

  • U.S. Government Agency Securities: $0.0B (2007) $\rightarrow$ $1,644.60B (2018; 38.8% of total assets).

  • Depository Institution Reserves: $20.8B (2007) $\rightarrow$ $1,556.00B (2018; 36.7% of liabilities).

  • Currency Outside Banks: $773.9B (2007) $\rightarrow$ $1,637.20B (2018; 38.6% of liabilities).

Monetary Policy Tools and Implementation Mechanics

• Three Principal Tools of Monetary Policy:

  1. Open Market Operations (OMOs): Trading Desk at the Federal Reserve Bank of New York (FRBNY) buys or sells U.S. Treasury securities based on FOMC policy directives. Primary mechanism to adjust excess reserves, money supply, and short-term interest rates.

  • Large-Scale Asset Purchases (Quantitative Easing, QE1–QE4) expanded the balance sheet via open market purchases of Treasuries and Mortgage-Backed Securities (MBS).

2. Discount Rate: Interest rate charged by FRBs on short-term loans to depository institutions via three window facilities:

   • Primary Credit: Available overnight to sound institutions at a rate above fed funds target.

   • Secondary Credit: Backup liquidity for troubled institutions.

   • Seasonal Credit: For institutions with recurring seasonal liquidity needs.

3. Reserve Requirements: Mandated percentage of transaction deposits that depository institutions must maintain as vault cash or reserve deposits at local FRBs.

   • Deposit Multiplier Effect:

\text{Change in Bank Deposits} = \n\frac{1}{\text{New Reserve Requirement}} \times \text{Change in Reserves}

• Federal Funds Rate: The interest rate charged on overnight interbank trades of excess reserves held at local FRBs. Financial Services Regulatory Relief Act of 2006 granted authority to pay Interest on Reserve Balances (IOER and IORR).

• Summary of Monetary Policy Activities:

  • Expansionary Policy (OMO Purchases, Discount Rate $\downarrow$, Reserve Ratio $\downarrow$): Reserves ($\uparrow$), Money Supply ($\uparrow$), Credit Availability ($\uparrow$), Interest Rates ($\downarrow$), Security Prices ($\uparrow$).

  • Contractionary Policy (OMO Sales, Discount Rate $\uparrow$, Reserve Ratio $\uparrow$): Reserves ($\downarrow$), Money Supply ($\downarrow$), Credit Availability ($\downarrow$), Interest Rates ($\uparrow$), Security Prices ($\downarrow$).

• Money Supply vs. Interest Rate Targeting:

  • The Fed cannot target money supply and interest rates simultaneously. Targeting money supply forces interest rates to fluctuate; targeting interest rates forces money supply to fluctuate.

  • In 1993, the Fed formally adopted interest rate targeting (Federal Funds Rate) via the Taylor Rule, balancing actual inflation against targeted levels and economic output gaps.

Global Financial Crises and International Central Banking

• Central Bank Independence: Insulates monetary policy decisions from political pressure. Highly independent central banks include the Federal Reserve, European Central Bank (ECB), and Bank of England (BoE). Central banks with less independence include the People's Bank of China, Reserve Bank of India, and Central Bank of Brazil.

• Financial Crisis Systemwide Rescue Programs:

  1. Expansion of retail deposit insurance.

  2. Direct capital injections by central governments.

  3. Debt guarantees for bank medium-term funding access.

  4. Asset purchases/guarantees to remove distressed assets from balance sheets.

• Post-Global Financial Crisis Challenges:

  • Greek Sovereign Debt Crisis: Severe bond debt crisis spread to Portugal, Spain, and Italy. Greece exited its third bailout program in August 2018, owing $330 billion to the EU and IMF with public debt reaching 180% of GDP.

  • Brexit (June 2016):

  • British Pound (GBP) fell over 11% to its lowest level since 1985.

  • Dow Jones Industrial Average fell 610.32 points (3.4%).

  • Stoxx Europe 600 dropped 7%.

  • Nikkei Stock Average declined 7.9%.

  • Bank of England cut benchmark rates in August 2016 to a record low of 0.25%.