Corporate Finance - Bonds Valuation

Corporate Finance: Big Picture

• Objective: Maximize the value of the business (firm).

Investment Decision

• Invest in assets that earn a return greater than the minimum acceptable hurdle rate.
• The hurdle rate should reflect the riskiness of the investment and the mix of debt and equity used to fund it.
• The return should reflect the magnitude and the timing of the cashflows as well as all side effects.

Financing Decision

• Find the right kind of debt for your firm and the right mix of debt and equity to fund your operations.
• The optimal mix of debt and equity maximizes firm value.
• The right kind of debt matches the tenor of your assets.

Dividend Decision

• If you cannot find investments that make your minimum acceptable rate, return the cash to owners of your business.
• How much cash you can return depends upon current & potential investment opportunities.
• How you choose to return cash to the owners will depend on whether they prefer dividends or buybacks.

Basic Principles

• Objective of corporate decisions is to maximize the value of the business
• If you can borrow money at 8%, do not borrow at 9%
• If your cashflows are in euro, take euro denominated debt
• If your assets are long-term, take long-term debt
• If you borrow at 8%, do not invest in projects that earn you less than 8% (This is the focus of this course)
• If you can’t find such project, please return the money
• Repay debt, pay dividends or buyback stocks
• Safe cashflows are more valuable than risky cashflows.
• A rupee earned today is more valuable than a rupee earned in future.
• Present value (PV) is the current worth of a future cashflow or a stream of future cash flows.
• The value of an asset is the present value of the future expected cash flows.

Bonds

• Bonds named after food:
  • Dim sum: Renminbi bond issued in Hong Kong by a Chinese entity
  • Kimchi: Non-Korean won bond issued in the Korean market by a foreign entity
  • Baklava: Turkish lira bond issued in the Turkish market by a domestic or foreign entity
  • Maple: Canadian dollar bond issued in Canada by a foreign entity
  • Kiwi: New Zealand dollar bond available only to New Zealand residents
  • Masala: INR denominated bond issued outside India
• Bonds named after animals:
  • Bulldog: British pound bond issued in the UK market by a foreign entity
  • Panda: RMB-denominated bond issued in the Chinese market by a foreign entity
  • Kangaroo: Australian dollar bond issued in the Australian market by a foreign entity
• Bonds named after people:
  • Yankee: US dollar denominated bond issued by a foreign entity in the US market
  • Samurai: Japanese yen denominated bond issued by a foreign entity in the Japanese market

What is a Bond?

• A bond is a type of investment that represents a loan made by an investor to a borrower, typically a corporation, government, or municipality.
• When you buy a bond, you are essentially lending money to the issuer in exchange for periodic interest payments (called the coupon) and the return of the principal amount (the face value) when the bond matures.

Key Features of a Bond

• Issuer – The entity borrowing the money (e.g., a government, corporation).
• Face Value (Par Value) – The amount the issuer agrees to repay at maturity.
• Coupon Rate – The interest rate paid to the bondholder, usually annually or semiannually.
• Maturity Date – The date when the issuer must repay the principal.
• Market Price – The price a bond is traded at, which may be different from its face value

Simple Coupon Bond

• A simple coupon bond is a type of bond that pays fixed periodic interest (coupon payments) to the bondholder until maturity, at which point the face value (principal) is repaid.
• Face Value (FV) – Principal amount or the maturity payment from a bond
• Coupon – Periodic cash flow from the bond. It equals face value * periodic coupon rate (think of it as periodic interest rate)
  • Coupons are always quoted as annualized percentage rates (APR).
  • Periodic interest rate = APR/number of periods in the year
• 2-year bond with 5% semiannual coupon vs. 2 year bond with 5% annual coupon
• FV is not same as Price.
• Price – price the bond is trading at in the market
  • For a bond with FV of 100, what does a price of 106.8 mean?
  • If price = 100, Trading at Par (Price = FV)
  • If price = 105, Trading at Premium (Price > FV)
  • If price = 94, Trading at discount (Price < FV)

U.S. Treasury Bond Example

• A five-year, $1000 bond with a 5$25
• Period 10: 25 + $1000

Valuing a Bond

• The value of the bond is the present value of future cash flows
• Value = \sum_{t=1}^{N} \frac{C}{(1+r)^t} + \frac{M}{(1+r)^N}
  • Where C is coupon (25), M is the maturity amount or face value (1000), r is the discount rate (Annual rate/2), N is the number of periods (10 half-year periods)
  • r is the discount rate, or market rate, or required rate/yield, yield to maturity (YTM).

Two Parts of a Coupon Bond

• Present Value of coupons (you can use annuity formula)
• Present value of face value/maturity value
• Practice Problem
  • Value a 12% semi-annual 5-year bond, with a Face Value of 100, if market interest rate is 8% per annum, compounded semiannually
  • 6 * PVIFA (4%, 10) + 100 * PVIF (4%,10)

PV of Annuity Formula

• When the same cash flow C is paid out for n periods
• PV = C * \frac{1 - (1 + r)^{-n}}{r}
  • C is the periodic cash-flow
  • r is the discount rate
  • n is the number of periods for which the cash flow will last
  • PV is the present value of this cash flow stream

Practice Problem

• ABC Corporation issues ten-year bonds (face value 1,000) with a coupon rate of 8.6% that pay interest semiannually. Similar ten-year bonds with semi-annual coupons are paying 8.0% interest (annually). What is the value of the bond
  • Coupon per period = 4.3% of $1000 =$43
  • Discount rate = 4% per half-year period
  • Value of the bond = 584.37 + $456.40 = $1,040.77

Types of Bonds

• Fixed Rate Bonds
• Fixed rate zero coupon (deep discount)
• Floating rate
• Bonds with options
  • Puttable/Callable/Convertible

Fixed Rate Bonds

• Pay a fixed coupon rate over the life of the bond.
• Investors receive predictable income regardless of market interest rate changes.
• Example: A 5-year bond with a 6% coupon rate pays 60 per year (if FV = $1,000).
• Risk: If market interest rates rise, fixed-rate bond prices fall (and vice versa)

Fixed Rate Zero Coupon Bonds (Deep Discount Bonds)

• Issued at a deep discount and matures at face value.
• No periodic interest payments (zero coupon).
• Investors earn return from the price difference between purchase price and face value.
• Example: Buy a zero-coupon bond for ₹700, get ₹1,000 at maturity.
• Risk: Interest rate risk (bond price fluctuates with market rates).

Floating Rate Bonds (FRBs)

• Coupon rate changes periodically based on a benchmark interest rate.
• Protects investors from interest rate risk.
• Example: FRB with 6-month MIBOR + 2% margin → If MIBOR is 3%, the bond pays 5% for that period.
• Risk: Variable income (cash flows are unpredictable).

Bonds with Embedded Options

• Bonds that give rights to the issuer or investor to alter terms.

Callable Bonds (Issuer’s Right)

• Issuer can redeem the bond before maturity.
• Used when interest rates fall (company refinances debt at lower rates).
• Investors demand higher yield as compensation.
• Example: A 10-year 6% callable bond may be redeemed after 5 years if rates drop.
• Risk: Investors may not get future interest if called early.

Puttable Bonds (Investor’s Right)

• Investor can sell back the bond before maturity.
• Used when interest rates rise (investors reinvest at higher rates).
• Investors accept lower yields in exchange for this protection.
• Example: A 7% 10-year puttable bond allows selling back at year 5 if rates increase.
• Risk: Lower returns compared to regular bonds.

Risks of Investing in Bonds

• Default risk
• Reinvestment risk
• Interest rate risk
• Inflation
• Liquidity risk

Default Risk (Credit Risk)

• The risk that the bond issuer fails to make interest or principal payments.
• Who is affected? Investors in corporate bonds, lower-rated bonds, and junk bonds.
• Higher risk in: Bonds with low credit ratings (BB or below) or issuers with weak financials.
• Example:
  • A company issues a 10-year bond at 8% interest.
  • After 5 years, the company goes bankrupt → Investors lose their money.

Reinvestment Risk

• The risk that interest or principal payments from a bond will be reinvested at a lower rate than the original bond’s yield.
• Who is affected? Investors in bonds with high coupon payments or callable bonds.
• Example:
  • You buy a 10-year bond at 6%.
  • After 5 years, interest rates fall to 3%.
  • When you reinvest your coupon payments, you earn only 3% instead of 6%.

Interest Rate Risk

• The risk that bond prices will change due to fluctuations in market interest rates.
• Who is affected? Investors in fixed-rate bonds.
• Key Rule: When interest rates rise, bond prices fall (because new bonds offer higher returns). When interest rates fall, bond prices rise (because older bonds offer better returns).
• Example:
  • You buy a 5-year bond with a 6% coupon.
  • After 1 year, interest rates rise to 8%.
  • Your bond becomes less attractive, and its market price drops.

Inflation Risk (Purchasing Power Risk)

• The risk that inflation reduces the real value of bond returns.
• Who is affected? Investors in fixed-rate bonds.
• Example:
  • You invest in a 10-year bond at 5%.
  • If inflation rises to 6%, your real return is negative (-1%).

Liquidity Risk

• The risk that you cannot sell the bond easily without a price discount.
• Who is affected? Investors in corporate or low-rated bonds.
• Example:
  • You buy a corporate bond and want to sell it before maturity.
  • There are no buyers or you have to sell at a discount to attract interest.

Yield to Maturity (YTM)

• Yield to maturity is that single discount rate which when used to discount all future cash flows of a bond, makes the price equal to the present value of cash flows
• This is the rate that market wants the bond to pay.
• Suppose this bond is trading at 990 (at issuance)
• YTM is the discount rate at which the present value of cashflows are equal to the market price of the bond
• \frac{25}{(1+YTM)} + \frac{25}{(1+YTM)^2} + … + \frac{1025}{(1+YTM)^{10}}
• Easiest way of calculating YTM is to use IRR function with cashflows -990, +25, +25, … , +1025
• Note that you will get a semi-annual YTM!

Yield to Maturity (YTM): Shortcut Formula

• YTM \approx \frac{C + \frac{FV - P}{N}}{\frac{FV + P}{2}}
  • Where:
    • C = Annual coupon payment
    • FV = Face value of the bond
    • P = Current market price of the bond
    • N = Number of years to maturity

Calculation of YTM

• Q: A five year, 8% annual coupon government bond is priced at 950 per 1000 of par value. Calculate the yield-to-maturity ?

Calculation of YTM

• Q: A five year, 4.50% semiannual coupon government bond is priced at 98 per 100 of par value. Calculate the yield-to-maturity on
  • Semi-annual basis (Quoted as APR)
  • annual basis
  • quarterly basis
  • 2-year compounding basis

Yield to Maturity

• What happens to price of a bond when YTM rises?
  • A bond with 8% coupon rate and ₹1,000 face value trades at ₹1,000 when YTM is 8%.
  • If YTM rises to 10%, new bonds offer better returns, so the price of the old bond must decrease to make its effective yield match 10%.
• What happens to price of a bond when YTM falls?
  • The same bond with an 8% coupon rate and ₹1,000 face value trades at ₹1,000 when YTM is 8%.
  • If YTM drops to 6%, new bonds only offer 6% return, so investors bid up the price of the 8% bond above ₹1,000.

Price Changes with Interest Rate

• Value a Face value =100, 12% annual coupon, 5 year bond if
  • YTM is 14% - Trading at discount
  • YTM is 10% - Trading at premium
  • YTM is 12% - Trading at par
• YTM can also be referred to as current cost of debt, or prevailing interest rate for the issuer

Current Yield

• Current yield equals annual coupons received divided by bond price
• Its importance and use:
  • It is a quick and simple way to estimate the return on a bond without considering time to maturity or capital gains/losses.
  • Investors can quickly compare bonds with different prices and coupon rates.
• Current yield > YTM when bond is selling at premium
• Current yield < YTM when bond is selling at discount

Current Yield: Calculation

• A 5-year bond has:
  • Face Value = ₹1,000
  • Coupon Rate = 8% → Annual Coupon = ₹80
  • Current Market Price = ₹950
• Notably, there exists an inverse relationship between price and current yield

Current Yield, Coupon Rate, and YTM

• If bond is selling at discount → Price < FV → Coupon Rate < Current Yield < YTM
• If bond is selling at a premium → Price > FV → Coupon Rate > Current Yield > YTM

Questions

• 10 yr bond with a FV of 1000, Coupon of 60. If market interest rate increases
  • What happens to coupon rate?
  • What happens to bond price?
  • What happens to YTM?

Bond Strips

• Bond Strips: A bond strip refers to separating (or "stripping") a bond's principal and coupon payments into individual zero-coupon securities. Each stripped component trades separately as a zero-coupon bond.
• Example of Bond Stripping
  • A 10-year bond has: Face Value = ₹1,000 and Coupon Rate = 8% (₹80 annually)
  • If this bond is stripped, it creates 11 separate zero-coupon bonds:
    • 10 bonds representing each ₹80 coupon payment (one for each year).
    • 1 bond representing the ₹1,000 principal repayment at the end of 10 years.
  • Each of these components trades separately in the market.

Macaulay Duration

• What is Macaulay Duration?
  • Macaulay Duration measures the weighted average time (in years) that an investor must wait to receive a bond’s cash flows (coupons + principal repayment).
• Formula:
  • Duration = \frac{\sum [t * PV(CFt)]}{\sum PV(CFt)} = \frac{\sum [t * \frac{CF_t}{(1+YTM)^t}]}{Price}

Macaulay Duration : Calculation

• For Example: Consider a 2-year bond with: Face Value = ₹1,000, Coupon Rate = 10% (₹100 per year), Yield to Maturity (YTM) = 8% and Market Price = ₹1,018.52

Interest Rate Sensitivity

• Interest Rate Sensitivity measures how much a bond's price changes when interest rates fluctuate.
• Key Concept:
  • Bond prices and interest rates move inversely—when rates rise, bond prices fall, and vice versa.
  • The degree of price change depends on duration
  • Modified Duration = \frac{Macaulay Duration}{1 + \frac{yield}{n}}
  • Measures % change in bond price when yield changes by 1 percentage point
  • If interest rate increases (decreases) by m%, then bond price decreases (increases) by (modified duration*m%)

Question

• A 3-year, 8% coupon bond with a face value of INR 100, has the following cash flows to be received at the end of next three years.
  • Year 1: 8
  • Year 2: 8
  • Year 3: 108
• The bond is currently trading at par, that is, the current price of the bond is INR 100. What is the approximate increase in the bond’s price, if the yield declines by 1% (today)?

Question

• Rank the following in order of increasing duration
  • 5-year, 8% coupon bond
  • 5-year, zero-coupon bond
• Rank the following in order of increasing duration
  • 5-year, 8% coupon bond
  • 10-year, 8% coupon bond

Forward Rates

• Forward rate is rate at which you can agree to borrow or lend money in future
  • e.g. a rate agreed upon to borrow INR 100 for 3 years at a time 5 years from now is called a three year forward rate starting five years later
  • A forward rate is the implied interest rate for a future period
• Spot rate is rate at which you can agree to borrow or lend money today
  • The cost of money today for a specific period.

Forward Rates and Spot Rates

(1 + 5%)^2 = (1 + 4%)^1 \times (1+ F_{1,1})

Forward Rates and Spot Rates

• (1 + 5%)^2 = (1 + 4%)^1 \times (1+ F_{1,1})
• (1.05)^2 = (1.04) \times (1 + F_{1,1})
• 1.1025 = 1.04 \times (1+ F_{1,1})
• F_{1,1} = \frac{1.1025}{1.04} - 1
• F_{1,1} = 5.96\%

Forward Rates and Spot Rates: Formula

• (1 + Sn)^n = (1 + Sm)^m \times (1+ F_{m,n-m})^{n-m}
  • Where:
    • S_m = Spot rate for maturity m
    • S_n = Spot rate for maturity n
    • F_{m,n-m}$$ = Forward rate from year m to n
    • m, n = Maturity years

Question

• You observe that a one-year bond with annual coupon of Rs. 6 and par value of Rs. 100 has a current price of Rs.102.91. A two-year bond with annual coupon of Rs.6.50 and par value of Rs.100 has a current market price of Rs.101.10.
• What are the one period and two period spot rates?