Practice Problems of Bond Valuation and Analysis¶

1. Current Yield¶

a. Example 1¶

What is the current yield of a 10-year 12% coupon bond with a face value of Rs. 1000 and currently selling at Rs. 950?

Solution:
- Calculation of Current Yield:
  - \(CY = \frac{In}{Po} \times 100\)
  - \(In = \text{Annual Interest} = 1000 \times 12\% = 120\)
  - \(Po = 950\)
  - \(CY = \frac{120}{950} \times 100 = 12.63\%\)
- Interpretation: Since the bond is currently selling at a discount (\(950 < 1000\)), hence the current yield of 12.63% is greater than the coupon rate of 12%.

b. Example 2¶

The face value of a bond is Rs. 1000 and it is currently selling at Rs. 1200 with a coupon rate of 12% and 10 years to maturity. Calculate the current yield from the bond.

Solution:
- Calculation of Current Yield:
  - \(CY = \frac{In}{Po} \times 100\)
  - \(In = \text{Annual Interest} = 1000 \times 12\% = 120\)
  - \(Po = 1200\)
  - \(CY = \frac{120}{1200} \times 100 = 10\%\)
- Interpretation: Since the bond is currently selling at a premium (\(1200 > 1000\)), hence the current yield of 10% is lesser than the coupon rate of 12%.

2. Spot Interest Rate¶

a. Example¶

The face value of a zero coupon bond is Rs. 1000. It is currently selling at Rs. 797.19 and the investor will receive the principal amount of Rs. 1000 after two years of maturity of the bond. Calculate the spot interest rate on this zero coupon bond.

Solution:
- Calculation of spot interest rate on zero coupon bond:
  - \((1 + k)^2 = \frac{FV}{PV}\)
  - \((1 + k)^2 = \frac{1000}{797.19} = 1.2544\)
  - \((1 + k) = \sqrt{1.2544}\)
  - \(k = 1.12 - 1 = 0.12 \text{ or } 12\%\)

b. Example¶

The face value of a bond is Rs. 1000 and it is currently selling at Rs. 950 with a coupon rate of 10% and 5 years to maturity. Calculate the spot interest rate from the bond.

Solution:
- Calculation of spot interest rate:
  - \((1 + k)^5 = \frac{FV}{PV}\)
  - \((1 + k)^5 = \frac{1000}{950} = 1.0526\)
  - \((1 + k) = \sqrt[5]{1.0526}\)
  - \(k = 1.01 - 1 = 0.01 \text{ or } 1\%\)

3. Yield to Maturity¶

a. Example¶

Mr. Arun recently purchased a bond with Rs. 1000 face value at 10% coupon rate and 4 years to maturity. The bond makes annual interest payments. Mr. Arun paid Rs. 1032.40 for the bond. What is the bond yield to maturity?

Solution:
- Calculation of Bond yield to maturity:
  - \(YTM = \frac{I + \frac{(F - P)}{n}}{0.4F + 0.6P}\)
  - \(I = 1000 \times 10\% = 100\)
  - \(F = 1000\)
  - \(P = 1032.40\)
  - \(N = 4 \text{ Years}\)
  - \(T = 1032.40\)
  - \(YTM = \frac{100 + \frac{(1000 - 1032.40)}{4}}{0.4 \times 1000 + 0.6 \times 1032.40}\)
  - \(= \frac{100 + (-32.40)/4}{400 + 619.44}\)
  - \(= \frac{91.9}{1019.44}\)
  - \(YTM = 0.09 \text{ or } 9\%\)

b. Example¶

A bond with a face value of Rs. 1000 and a coupon rate of 12% is issued three years ago is redeemable after five years from now at a premium of 5%. The interest rate prevailing in the market currently is 14%. Estimate the bond yield to maturity.

Solution:
- Calculation of Bond yield to maturity:
  - \(YTM = \frac{I + \frac{(F - P)}{n}}{\frac{F + P}{2}}\)
  - \(I = 1000 \times 12\% = 120\)
  - \(F = 1000\)
  - \(P = 1050\)
  - \(N = 5 \text{ Years}\)
  - \(YTM = \frac{120 + \frac{(1000 - 1050)}{5}}{\frac{1000 + 1050}{2}}\)
  - \(= \frac{120 + (-10)/5}{1025}\)
  - \(= \frac{118}{1025}\)
  - \(YTM = 0.115 \text{ or } 11.5\%\)

4. Yield to Call¶

a. Example¶

Let's say you buy a bond with a face value of Rs. 1,000 and a coupon rate of 5%. The bond matures in 10 years, but the issuer can call the bond for face value (Rs. 1,000) in two years if they choose. You buy the bond for Rs. 960, a discount to face value. Calculate the yield to call.

Solution:
- Calculation of Yield to call:
  - Formula: \(YTC = \frac{Annual\ Interest + \frac{Redemption\ Value - Market\ Price}{N}}{\frac{Redemption\ Value + Market\ Price}{2}}\)
  - \(YTC = \frac{50 + \frac{(1000 - 960)}{2}}{\frac{(1000 + 960)}{2}}\)
  - \(= \frac{50 + 20}{980}\)
  - \(YTC = 0.071 \text{ or } 7.1\%\)

5. Bond Duration¶

a. Example 1¶

A bond with a face value of Rs.100, has a coupon rate of 12% issued three years ago is redeemable after five years from now at a premium of 5%. The interest rate prevailing in the market currently is 14%. Estimate the bond duration.

Solution:

Year	Cash Flow (Rs.)	PV Factor at 14%	Present Value	PV multiplied by year
1	12	0.8772	10.5264	10.5264
2	12	0.7695	9.2340	18.4680
3	12	0.6750	8.1000	24.3000
4	12	0.5921	7.1052	28.4208
5	12	0.5194	6.2328	31.1640
5	105	0.5194	54.5370	272.6850
Total			95.7354	386.0402

Bond Duration = \(\frac{\text{Sum of PV multiplied by year}}{\text{Sum of present value}} = \frac{386.0402}{95.7354} = 4.03 \text{ years}\)

b. Example 2¶

A bond with a face value of Rs. 1000, a coupon rate of 8%, and 5 years to maturity is currently selling at Rs. 950. The prevailing interest rate in the market is 10%. Calculate the bond duration.

Solution:

Year	Cash Flow (Rs.)	PV Factor at 10%	Present Value	PV multiplied by year
1	80	0.9091	72.7272	72.7272
2	80	0.8264	66.1120	132.2240
3	80	0.7513	60.1040	180.3120
4	80	0.6830	54.6400	218.5600
5	80	0.6209	49.6720	248.3600
5	1000	0.6209	620.9000	3104.5000
Total			924.7552	3956.6832

Bond Duration = \(\frac{\text{Sum of PV multiplied by year}}{\text{Sum of present value}} = \frac{3956.6832}{924.7552} = 4.27 \text{ years}\)

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