A bond pays a fixed coupon and a fixed sum at the end. Everything about its cash flows is known, so its value is the present value of those flows at the return investors require. A preference share is nearly the same, with a fixed dividend. If market yields rise, bond prices fall, and if they fall, prices rise.
The valuation rules are the same ones used for shares, only more exact.
This topic covers the valuation principle for debt, irredeemable and redeemable fixed-rate bonds (including semi-annual interest), convertible bonds and preference shares, and why tax is ignored in the market valuation. The previous topic covered dividend valuation models. The next covers the efficient market hypothesis. Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on bond valuation.
In 30 seconds
A bond’s value is the present value of its interest and redemption, discounted at the pre-tax yield investors require (tax is ignored because investors’ tax positions differ). Irredeemable: MV = interest ÷ yield. For half-yearly interest, use the half-yearly interest and about half the annual yield. Redeemable: PV of the interest annuity plus PV of the principal.
Convertible: the higher of (a) the straight-bond value and (b) the PV of interest to conversion plus the PV of the expected share value on conversion. Yields and prices move in opposite directions. Preference shares: irredeemable ones are valued like a perpetuity, dividend ÷ required return; redeemable ones by discounting dividends and the redemption value.
Key terms in plain English
| Term | What it means |
|---|---|
| Coupon (nominal rate) | The fixed interest as a percentage of nominal value |
| Yield (required return) | The return investors demand, used as the discount rate |
| Irredeemable bond | A bond that never repays its principal |
| Redeemable bond | A bond repaid at a set date, usually at par |
| Convertible bond | A bond that can be turned into shares |
| Ex-interest value | The value excluding interest that is about to be paid |
Syllabus check
This topic is in syllabus area B (business analysis). It covers:
- estimating the market value of debt and preference shares.
CAMA 2020 and debt securities
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Debt terms should be set in the interest of the company and its creditors |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should understand how the market values the company’s debt |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
The basic principle
Debt is valued on the same principle as shares: the present value of all future cash flows from it. For bonds, those cash flows are the interest and the repayment of principal, discounted at the cost of the debt, which here is the investors’ pre-tax required return. Tax is ignored because the personal tax positions of investors differ.
Because the cash flows on fixed-rate bonds are known, not estimated, the valuation is more exact than for shares. Bonds are usually valued per ₦100 or ₦1,000 nominal. This is the reverse of the cost of debt calculation, which finds the yield from a known price.
Irredeemable debt
The value is the PV of interest for ever: MV = interest ÷ required yield.
- ICAN’s 7% bond paying annually with a required yield of 8%: (100 × 7%) ÷ 8% = ₦87.50 per ₦100 nominal.
- A 10% bond of ₦1,000 paying half-yearly with a required annual yield of 8% (about 4% half-yearly): half-yearly interest = ₦50, so value = 50 ÷ 0.04 = ₦1,250.
If the company’s own view is wanted, use the post-tax cost of the debt, but the market value uses the lenders’ pre-tax return.
Redeemable debt
The value is the PV of all interest to maturity plus the PV of the principal, discounted at the yield. ICAN’s example: a 6% bond with three years to run, redeemed at par, with a required yield of 5%. Per ₦1,000: interest ₦60 a year. Year 1: 60 × 0.952 = 57.1; year 2: 60 × 0.907 = 54.4; year 3: 1,060 × 0.864 = 915.8. Value = ₦1,027.3. The bond is worth more than par because its coupon (6%) is above the required yield (5%).
Convertible bonds
The market value of a convertible bond is the higher of: (a) its value as a straight bond redeemed at maturity; and (b) the PV of interest up to conversion plus the PV of the expected share value on conversion. Value it both ways and take the higher.
ICAN’s example. 4% convertible bonds convert in four years at 25 shares per ₦1,000. The share price is ₦46 and dividends are expected to grow 5% a year, so the share price is expected to grow at the same rate. Bondholders require 6%.
- Expected share price in four years.
46 × 1.054 = ₦55.9.
Value of 25 shares = ₦1,397.5. - As a straight bond.
40 × 3.465 + 1,000 × 0.792 = 138.6 + 792.0 = ₦930.6. - If converted.
138.6 + 1,397.5 × 0.792 = 138.6 + 1,106.8 = ₦1,245.4.
The higher is conversion, so the price is about ₦1,245.
Preference shares
A preference share has a fixed dividend. If it is irredeemable, value = dividend ÷ required return, the same as an irredeemable bond. If redeemable, discount the dividends and the redemption value at the required return.
The rules in more depth
Why price and yield move opposite ways
The cash flows are fixed. If investors want a higher return, the same flows are worth less today.
Why a bond can trade above or below par
If the coupon exceeds the required yield, the bond is worth more than par. If it is lower, it is worth less. At maturity, the price returns to par.
Why a convertible is worth the higher of two values
The holder can choose. The option to convert has a value, so the price cannot fall below the straight-bond value.
Why tax is ignored
Investors pay different rates of tax, so a market price cannot depend on one investor’s tax position.
Effect on analysis
Set out the cash flows by year, state the discount rate, compute the value and compare it with par or the current price.
Worked examples
Example 1: irredeemable. A 9% irredeemable bond has a required yield of 12%.
- Interest per ₦100 = ₦9.
- Value = 9 ÷ 0.12.
- Value = ₦75.00.
Example 2: redeemable. An 8% bond is redeemed at par in four years. The required yield is 10% (annuity factor 3.170; discount factor 0.683).
- PV of interest.
8 × 3.170 = 25.36. - PV of principal.
100 × 0.683 = 68.30. - Value = ₦93.66 per ₦100.
Example 3: convertible. A 5% bond of ₦1,000 converts in three years into 20 shares. The share price is ₦60 and grows 6% a year. The required return is 8% (annuity factor 2.577; discount factor 0.794).
- Share price at conversion.
60 × 1.063 = ₦71.46.
Value of shares = 20 × 71.46 = ₦1,429. - Converted.
50 × 2.577 + 1,429 × 0.794 = 128.9 + 1,134.8 = ₦1,263.7. - Straight.
128.9 + 794 = ₦922.9.
The higher is ₦1,263.7.
Example 4: preference. An 8% irredeemable preference share of ₦1 has a required return of 10%.
- Dividend = ₦0.08.
- Value = 0.08 ÷ 0.10.
- Value = ₦0.80.
How to score full marks
- Use the pre-tax required return.
- Per ₦100 or ₦1,000 nominal, state which.
- Check the number of interest payments a year.
- For convertibles, compute both values and take the higher.
- Compare with par and explain why.
- State that yield and price move oppositely.
Common mistakes in the exam
- Using the post-tax cost of debt for market value.
- Using the coupon rate as the discount rate.
- Forgetting the redemption payment.
- Using the wrong period for half-yearly interest.
- Not comparing the straight and conversion values.
- Discounting share value at the wrong date.
Quick revision
- Value = PV of interest and principal at the required pre-tax yield.
- Irredeemable: interest ÷ yield.
- Redeemable: annuity plus PV of principal.
- Convertible: the higher of straight and conversion value.
- Preference: dividend ÷ required return if irredeemable.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. A bond is valued at the present value of its:
- Interest and redemption payments
- Dividends only
- Earnings
- Net assets
Show the answer
A. At the required yield.
Question 2. The discount rate for market value is:
- The pre-tax required return
- The post-tax cost of debt
- The coupon
- Zero
Show the answer
A. Tax positions differ between investors.
Question 3. A 7% irredeemable bond with a required yield of 8% is worth, per ₦100:
- ₦87.50
- ₦100
- ₦114.29
- ₦107
Show the answer
A. 7 ÷ 0.08.
Question 4. When the coupon exceeds the required yield, the bond trades:
- Above par
- Below par
- At par
- At zero
Show the answer
A. It pays more than the market demands.
Question 5. If market yields rise, bond prices:
- Fall
- Rise
- Stay the same
- Double
Show the answer
A. They move in opposite directions.
Question 6. A convertible bond is worth:
- The higher of its straight and conversion values
- The lower of the two
- Par only
- Its coupon
Show the answer
A. The holder has the option.
Question 7. Half-yearly interest of ₦50 at a half-yearly yield of 4% gives an irredeemable value of:
- ₦1,250
- ₦625
- ₦500
- ₦2,500
Show the answer
A. 50 ÷ 0.04.
Question 8. An irredeemable preference share is valued at:
- Dividend ÷ required return
- Dividend × required return
- Par
- Earnings
Show the answer
A. A perpetuity.
Question 9. Bond cash flows are more certain than share cash flows, so the valuation is:
- More exact
- Less exact
- Impossible
- Based on earnings
Show the answer
A. Interest and principal are fixed.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) State the principle of debt valuation and why tax is ignored. (5 marks) (b) Explain why a bond may trade above par. (5 marks)
Question 12 (20 marks). (a) A 10% irredeemable bond of ₦1,000 pays interest annually. The required yield is 8%. Calculate its value. (3 marks) (b) A 7% bond is redeemed at par in three years. The required yield is 9% (factors 0.917, 0.842, 0.772). Calculate its value per ₦100. (5 marks) (c) A 6% convertible bond of ₦1,000 converts in three years into 25 shares. The share price is ₦40 and grows 5% a year. The required return is 8% (annuity factor 2.577; discount factor 0.794).
Calculate its value and state whether holders would convert. (7 marks) (d) A 12% irredeemable preference share of ₦1 has a required return of 15%. Calculate its value. (3 marks) (e) State why the valuation of bonds is more exact than that of shares. (2 marks)
Question 13 (10 marks). (a) A bond pays ₦50 every six months and the required yield is 8% a year. Calculate its value if irredeemable. (3 marks) (b) Explain how a change in required yield affects a bond’s price. (3 marks) (c) Explain how a convertible bond’s value is found. (4 marks)
Model answers
Show the model answer to Question 11
(a) Principle (5 marks). The value is the present value of all future interest and the redemption payment, discounted at the investors’ required return. (3) Tax is ignored because investors’ personal tax positions differ. (2)
(b) Above par (5 marks). A bond trades above par when its coupon is higher than the yield investors now require. (2.5) The extra interest makes the stream worth more than par today, and the price returns to par at redemption. (2.5)
Show the model answer to Question 12
(a) Irredeemable (3 marks).
- Interest = ₦100.
- Value = 100 ÷ 0.08 = ₦1,250. (3)
(b) Redeemable (5 marks).
- 7 × 0.917 = 6.42
- 7 × 0.842 = 5.89
- 107 × 0.772 = 82.60. (3)
- Value = ₦94.91 per ₦100. (2)
(c) Convertible (7 marks).
- Share price at conversion.
40 × 1.053 = ₦46.31. - Value of 25 shares = ₦1,157.6. (2)
- Interest = 60 × 2.577 = 154.6.
- Converted value.
154.6 + 1,157.6 × 0.794 = 154.6 + 919.2 = ₦1,073.8. (3) - Straight value.
154.6 + 794 = ₦948.6, which is lower, so holders would convert. (2)
(d) Preference (3 marks).
- Dividend = ₦0.12.
- Value = 0.12 ÷ 0.15 = ₦0.80. (3)
(e) Exactness (2 marks). Interest and principal on fixed-rate bonds are known amounts, not estimates, while dividends must be forecast. (2)
Show the model answer to Question 13
(a) Value (3 marks).
- The half-yearly yield is about 4%.
- Value = 50 ÷ 0.04 = ₦1,250. (3)
(b) Yield and price (3 marks). The cash flows are fixed, so a higher required yield lowers their present value and the price falls. (2) A lower yield raises the price. (1)
(c) Convertible (4 marks). Value it as a straight bond and as a bond that converts, using the expected share price at conversion. (2) The value is the higher of the two. (2)
Pathfinder on Valuing irredeemable, redeemable and convertible bonds and preference shares
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on this topic. Bond values appear inside financing questions. Worked examples 1 to 4 show each type.
Revise first: Worked examples
Questions ask for the price of a convertible. See the section on convertible bonds.
Revise first: Convertible bonds
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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