Financial Mathematics - Calculation of YTM | PAPER III – ACCOUNTING & FINANCIAL MANAGEMENT FOR BANKERS | MODULE C: FINANCIAL MANAGEMENT

Financial Mathematics: Yield to Maturity (YTM) and Bond Concepts

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Meaning of Debt and Introduction to Bonds

Debt represents borrowed funds that must be repaid over time, usually with interest. A bond is a fixed-income instrument representing a loan made by an investor to a borrower. Bonds are used by companies, municipalities, states, and sovereign governments to finance projects and operations.

Terms Associated with Bonds

• Face Value: The nominal value of the bond, usually ₹1,000.
• Coupon Rate: Annual interest rate paid on the bond's face value.
• Maturity: The time at which the bond principal is repaid.
• Yield: The rate of return on the bond investment.

Types of Bonds

• Government Bonds
• Corporate Bonds
• Zero-Coupon Bonds
• Convertible Bonds

Optionality in Bonds

Bonds may include options such as call or put features:

• Callable Bonds: Issuer can redeem before maturity.
• Putable Bonds: Holder can sell back before maturity.

Valuation of Bonds

The value of a bond is the present value of future cash flows (interest and principal). The formula is:

Bond Price = ∑ [C / (1+r)^t] + [F / (1+r)^n]

Where:
C = annual coupon payment
F = face value
r = discount rate
n = number of years

Bond Value with Semi-Annual Interest

When interest is paid semi-annually, adjustments are made:

Bond Price = ∑ [C/2 / (1 + r/2)^(2t)] + [F / (1 + r/2)^(2n)]

Current Yield on Bond

It is the ratio of annual coupon payment to current market price.

Current Yield = (Annual Interest / Current Price) × 100

Yield to Maturity (YTM)

YTM is the internal rate of return (IRR) earned by an investor if the bond is held until maturity.

It equates the present value of future payments to the bond’s current price. The equation is solved iteratively or using financial calculators.

Example 1:

Face Value = ₹1,000, Coupon = 10%, Years to Maturity = 5, Current Price = ₹950

Approximate YTM = [C + (F - P)/n] / [(F + P)/2]

YTM = [100 + (1000 - 950)/5] / [(1000 + 950)/2] = 110 / 975 = 11.28%

Example 2:

Bond with ₹1,200 price, ₹100 coupon, 4 years to maturity, face value ₹1,000.

YTM = [100 + (1000 - 1200)/4] / [(1000 + 1200)/2] = 50 / 1100 = 4.55%

Example 3:

Zero-coupon bond, Price = ₹800, Face = ₹1,000, Years = 5

YTM = [(1000 / 800)^(1/5)] - 1 = (1.25)^(0.2) - 1 ≈ 4.56%

Example 4:

Coupon = ₹60, Price = ₹920, Face = ₹1,000, Years = 3

YTM ≈ [60 + (1000 - 920)/3] / [(1000 + 920)/2] = 86.67 / 960 = 9.03%

Example 5:

Semi-annual bond: Face = ₹1,000, Coupon = ₹80/year, Price = ₹950, Years = 4

YTM approximation with semi-annual interest:

Coupon = ₹40 every 6 months; Total Periods = 8

Use trial-and-error or IRR function in Excel/Calculator for accurate result

Theorems for Bond Value

• Bond prices and interest rates are inversely related.
• Longer maturity = higher sensitivity to interest rate changes.
• Low-coupon bonds are more volatile than high-coupon ones.

Duration of Bond

Duration is a measure of interest rate sensitivity. It reflects the weighted average time to receive cash flows.

Properties of Duration

• Higher coupon = lower duration
• Longer maturity = higher duration
• Zero-coupon bond duration = maturity

Bond Price Volatility

Price volatility is affected by duration, coupon rate, and time to maturity. Modified duration helps measure this impact precisely.

MCQs on Financial Mathematics – YTM and Bonds

1. What does YTM represent?
A. Market value
B. Return until maturity
C. Dividend yield
D. Book value
Answer: B

2. What is the formula for current yield?
A. Coupon / Face Value
B. Coupon / Market Price
C. Face Value / Coupon
D. Market Price / Face Value
Answer: B

3. In a semi-annual bond, how often is interest paid?
A. Annually
B. Monthly
C. Quarterly
D. Twice a year
Answer: D

4. Duration measures:
A. Time to maturity
B. Interest income
C. Sensitivity to rate changes
D. Price appreciation
Answer: C

5. If a bond’s price increases when interest rates fall, this demonstrates:
A. Positive correlation
B. Inverse relationship
C. Flat yield curve
D. Premium pricing
Answer: B

6. Which bond has the highest interest rate sensitivity?
A. Short-term, high-coupon
B. Long-term, zero-coupon
C. Short-term, zero-coupon
D. Long-term, high-coupon
Answer: B

7. What happens to duration as interest rates rise?
A. Increases
B. Decreases
C. Stays same
D. Becomes negative
Answer: B

8. Callable bonds benefit the issuer when:
A. Rates rise
B. Rates fall
C. Stock market falls
D. Inflation increases
Answer: B

9. What does a zero-coupon bond not offer?
A. Price
B. Maturity
C. Interest payments
D. Face value
Answer: C

10. Modified duration is:
A. Duration × yield
B. Macaulay Duration / (1 + yield per period)
C. Duration + coupon
D. Present value of bond
Answer: B