Interest Rates and Security Valuation

Chapter 3: Interest Rates and Security Valuation

1. Valuing a Financial Asset

To value any financial asset, two key ingredients are essential:

• Estimates of Future Cash Flows: This includes both the precise timing and the size of each cash flow the asset is expected to generate.

• An Appropriate Discount Rate: This rate must accurately reflect the inherent risk associated with the asset.

2. Bond Pricing

2.1 What is a Bond?

• A bond is fundamentally a form of a loan, specifically a debt security. It obligates the borrower (issuer) to pay the lender (bondholder) a principal amount (par value) and interest payments over a specified period.

• Bondholder Rights:

  • Receives coupon payments periodically before the bond's maturity. These payments are determined by the bond's coupon rate.

  • Receives the principal amount (par value) upon the bond's maturity.

• Cash Flow Stream: A bondholder essentially owns the right to a stream of cash flows that typically consists of:

  • An ordinary annuity of interest (coupon) payments.

  • A future lump sum payment of the par value at maturity.

• Bond Value (Price): The intrinsic value or price of a bond is calculated as the present value (PV) of all these future cash flows promised, discounted at the prevailing market rate of interest.

2.2 Bond Pricing Formulas

General Formula (for periodic compounding)

The price of a bond ($P<em>B$) is the sum of the present values of all future coupon payments and the present value of the par value at maturity. $P</em>B = \sum<em>{t=1}^{n \times m} \frac{C/m}{(1+i/m)^t} + \frac{F</em>n}{(1+i/m)^{n \times m}}$
Where:

• $P_B$ = Price of bond or present value of promised payments

• $C$ = Annual coupon payment ($C = \text{Coupon Rate} \times F_n$)

• $m$ = Number of times coupon payment is made per year (e.g., 1 for annual, 2 for semiannual, 4 for quarterly)

• $C/m$ = Coupon payment in each period $t$

• $F_n$ = Par value (principal amount) due at maturity

• $i$ = Market interest rate (discount rate or market yield, typically expressed as an annual rate)

• $i/m$ = Market interest rate per period

• $n$ = Number of years to maturity

• $n \times m$ = Total number of periods to maturity

Example: Bond Pricing Calculation

Consider a 3-year bond with a coupon rate of 8% and a face (par) value of $1000$. The market interest rate is 5%.

• Annually Compounding ($m=1$):

  • Annual Coupon Payment ($C$) = $0.08 \times 1000 = 80$

  • Market Rate per period ($i/m$) = $0.05/1 = 0.05$

  • Periods to Maturity ($n \times m$) = $3 \times 1 = 3$

  • Bond Price ($P<em>B$): $P</em>B = \frac{80}{(1+0.05)^1} + \frac{80}{(1+0.05)^2} + \frac{80}{(1+0.05)^3} + \frac{1000}{(1+0.05)^3}$

• Semiannually Compounding ($m=2$):

  • Semiannual Coupon Payment ($C/m$) = $80/2 = 40$

  • Market Rate per period ($i/m$) = $0.05/2 = 0.025$

  • Periods to Maturity ($n \times m$) = $3 \times 2 = 6$

  • Bond Price ($P<em>B$): $P</em>B = \sum_{t=1}^{6} \frac{40}{(1+0.025)^t} + \frac{1000}{(1+0.025)^6}$

2.3 Bond Pricing Principles

• Timing of Cash Flows: Cash flows are assumed to occur at the end of each period and are reinvested at the market interest rate ($i$).

• Par Bond: If the market interest rate equals the bond's coupon rate, the bond will trade at par (its price equals its face value).

  • $\text{Coupon Rate} = \text{Market Rate} \implies P<em>B = F</em>n$

• Premium Bond: If the coupon rate exceeds the market interest rate, the bond will trade above par (at a premium).

  • $\text{Coupon Rate} > \text{Market Rate} \implies P<em>B > F</em>n$

• Discount Bond: If the market interest rate exceeds the coupon rate, the bond will trade below par (at a discount).

  • $\text{Coupon Rate} < \text{Market Rate} \implies P<em>B < F</em>n$

2.4 Zero-Coupon Bonds

Zero-coupon bonds are simplified