A bond has a coupon, a price and a single quoted yield. That yield is an average. A cash flow due next year and one due in ten years are not priced alike, because lenders want more for tying money up longer. The yield curve shows the required return at each maturity. It prices bonds, gives forward rates and shows the cost of new debt.
Add the borrower’s credit risk, and you have the yield a company must actually pay.
This topic covers the link between bond price and required return, the shape and theories of the yield curve, valuing a bond from spot rates, bootstrapping a yield curve, credit risk, spreads, credit ratings and expected loss. The previous topic covered duration and convexity. The next covers forward rate agreements and interest rate swaps. Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on yield curves.
In 30 seconds
A bond’s market value is the PV of its cash flows at the lender’s required return. The required return is the IRR (gross redemption yield, pre-tax cost of debt). Price and yield move inversely. A yield curve plots risk-free yields against maturity; normally it slopes up, and may be steep or inverse. Theories: expectations, liquidity preference and market segmentation.
To value a bond, discount each flow at the spot rate for its own year. Bootstrapping builds spot rates from coupon bonds, year by year. The forward rate between years follows from spot rates: (1 + r2)² = (1 + r1)(1 + rf). Corporate yield = risk-free yield + credit spread (100 basis points = 1%). Ratings run from AAA (lowest spread) to D; below BBB- is junk. Expected loss = loss given default × probability of default.
Key terms in plain English
| Term | What it means |
|---|---|
| Yield curve | A plot of required yields against time to maturity |
| Spot rate | The average annual yield for a single cash flow due at a given date |
| Bootstrapping | Deriving spot rates one maturity at a time from bond prices |
| Credit spread | The extra yield over the risk-free rate for default risk |
| Credit rating | An agency’s grade of the likelihood of default |
| Loss given default (LGD) | The amount a lender would lose if the borrower defaults |
| Basis point | One hundredth of 1% |
Syllabus check
This topic is in syllabus area E (treasury and advanced risk management). It covers:
- explaining the shape of yield curves;
- valuing a bond using a yield curve;
- explaining credit risk and credit ratings;
- estimating the cost of debt using the term structure of interest rates.
CAMA 2020 and debt pricing
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 should be raised on terms that protect the company and its members |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should understand how credit risk affects the cost of 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 |
Price and required return
A bond’s market value is the present value of its cash flows at the lender’s required rate of return. ICAN’s example: a 6% bond redeemable in four years (cash flows 6, 6, 6, 106) is worth 103.54 at 5%, 100.00 at 6% and 96.62 at 7%. Price and required return move in opposite directions; the cash flows are fixed, so if lenders want more, they offer less.
If the market value is 103.54, the required return is the IRR of the flows, 5%. This is also called the pre-tax cost of debt, gross redemption yield and yield to maturity. When the yield on government debt falls, the state can borrow more cheaply.
The 5% is an average. Investors may require different rates for each year’s flow. ICAN’s two four-year bonds: A (5% coupon) yields 7.87%, price ₦904.72 per ₦1,000; B (10% coupon) yields 7.77%, price ₦1,074.40. A has the higher yield even though both are fairly priced, because 85.3% of A’s value arrives in year 4, where the required return (8%) is highest, against 75.25% for B. Discounting at spot rates of 5%, 6%, 7% and 8% for years 1 to 4 reproduces both prices.
Shape of the yield curve
The yield curve plots interest yields on similar debt against remaining term. Two points: yields are gross (pre-tax) and the curve is built from risk-free securities such as government bonds of the highest rating. A normal curve slopes upward, because lenders want more for tying up funds longer (equivalently, a one-year bond is worth more than a ten-year one of the same coupon).
It may be steeply positive if rates are expected to rise, or inverse (negative) if they are expected to fall.
- Expectations theory: the curve reflects expected future short rates. If rates are expected to rise, the curve steepens.
- Liquidity preference theory: longer-term yields and prices are more volatile, so investors want extra compensation; this explains the usual upward slope.
- Market segmentation theory: supply and demand within each maturity sector drive its rate; it can explain any shape.
Spot and forward rates
With a one-year spot yield of 3.0% and a two-year spot yield of 3.5%, ₦1,000 grows to 1,030 in one year or 1,000 × 1.035² = 1,071.23 in two. The extra return for the second year is 1,071.23 ÷ 1,030 − 1 = about 4.0%, the forward rate. In general (1 + r2)² = (1 + r1)(1 + rf).
Valuing a bond from the yield curve
Discount each cash flow at the spot rate for its own maturity. ICAN’s example: a four-year bond with a 6% coupon, redeemable at par, when spot yields are 3.0%, 3.5%, 4.2% and 5.0%:
- 6 ÷ 1.03 = 5.83
6 ÷ 1.035² = 5.60
6 ÷ 1.042³ = 5.30
106 ÷ 1.05⁴ = 87.21. - Price = 103.94 per ₦100 nominal, so the company can raise ₦103.94 per ₦100 bond.
- The gross redemption yield, by interpolation between 4% (NPV +3.32) and 6% (NPV −3.94), is 4% + 3.32 ÷ (3.32 + 3.94) × 2% = 4.91%, the average cost of this debt.
Bootstrapping the curve
Spot rates can be derived from the prices of coupon bonds in one risk class, working from the shortest maturity. ICAN’s data: A, 1 year, coupon 6%, price 102; B, 2 years, 5%, price 101; C, 3 years, 4%, price 97 (all redeemed at par).
- One year.
106 ÷ 102 = 1.0392, so 3.92%. - Two years.
101 = 5 ÷ 1.0392 + 105 ÷ (1 + r)², so 105 ÷ (1 + r)² = 96.19 and (1 + r)² = 1.0916, giving 4.48%. - Three years.
97 = 4 ÷ 1.0392 + 4 ÷ 1.0448² + 104 ÷ (1 + r)³, so 104 ÷ (1 + r)³ = 89.49 and (1 + r)³ = 1.1622, giving 5.1%.
Credit risk, spreads and ratings
A company’s debt costs more than a risk-free bond of the same maturity because the company might default. Required yield = risk-free yield + credit spread. For a five-year bond, a risk-free rate of 5.4% and a spread of 80 basis points gives 5.40% + 0.80% = 6.20%. Spreads are lowest for the best ratings. Agencies such as Moody’s, Standard & Poor’s and Fitch rate debt:
| Grade | S&P | Moody’s |
|---|---|---|
| Prime | AAA | Aaa |
| High grade | AA+, AA, AA- | Aa1, Aa2, Aa3 |
| Upper medium | A+, A, A- | A1, A2, A3 |
| Lower medium (last investment grade) | BBB+, BBB, BBB- | Baa1, Baa2, Baa3 |
| Speculative (junk) | BB+ to BB- | Ba1 to Ba3 |
| Highly speculative | B+ to B- | B1 to B3 |
| Substantial risk to default | CCC+ to CCC-, CC, C, D | Caa1 to Caa3, Ca, C |
Models score variables such as firm size, gearing, industry risk, country risk, standing against peers, management strength, asset backing and cash generation. ICAN’s spread table for US construction bonds gives, for AA-/Aa3, spreads of 15, 24, 30, 40, 52 and 64 basis points for 1, 2, 3, 5, 7 and 10 years. A seven-year AA- bond with a risk-free rate of 6.6% yields 6.6% + 0.52% = 7.12%.
Expected loss
Expected loss = loss given default × probability of default. ICAN’s example: a ₦10m loan with ₦8m of collateral and a 2% probability of default has an expected loss of (10m − 8m) × 2% = ₦40,000. If the recovery rate is 80%, LGD is 20% and the answer is the same.
The rules in more depth
Why price and yield move inversely
The coupons are fixed. A higher required return lowers the present value.
Why the curve is built from risk-free bonds
It isolates the time value of money. Credit risk is added separately as a spread.
Why a spot rate is not a single-year rate
A two-year spot rate is the average annual rate for a flow in two years, not the rate in the second year.
Why bootstrapping works
Each bond price contains earlier flows at known rates. What is left identifies the new rate.
Effect on analysis
Discount each flow at its own spot rate, then derive the yield. Add the spread to get the cost of corporate debt.
Worked examples
Example 1: pricing from spot rates. A two-year bond pays a 10% coupon on ₦100. Spot yields are 5% (year 1) and 6% (year 2).
- 10 ÷ 1.05 = 9.52.
- 110 ÷ 1.06² = 97.90.
- Price = ₦107.42.
Example 2: the first bootstrapping step. A one-year bond has an 8% coupon and a price of ₦104.
- 108 ÷ 104 = 1.03846.
- The one-year spot rate = 3.85%.
- This rate is then used to strip out year 1 from longer bonds.
Example 3: a credit spread. The risk-free rate is 12% and a BBB company’s spread is 150 basis points.
- 150 basis points = 1.5%.
- Yield = 12% + 1.5%.
- Yield = 13.5%.
Example 4: expected loss. A ₦50m loan has an expected recovery of 60%. The probability of default is 3%.
- LGD = 40% × ₦50m = ₦20m.
- Expected loss = 20m × 3%.
- Expected loss = ₦0.6m.
How to score full marks
- Discount each flow at its own spot rate.
- Show the PVs and the total price.
- Calculate the yield by IRR when asked.
- In bootstrapping, work in order and show each rearrangement.
- Add the spread in basis points correctly.
- Name a yield curve theory when explaining shape.
Common mistakes in the exam
- Discounting every flow at the yield to maturity.
- Treating a two-year spot rate as the rate for year 2 alone.
- Confusing 80 basis points with 80%.
- Using the wrong year’s rate in the bootstrapping step.
- Forgetting the collateral in LGD.
- Saying the curve always slopes upward.
Quick revision
- Price = PV at required return; inverse link with yield.
- Yield curve from risk-free bonds; normal, steep or inverse.
- Theories: expectations, liquidity preference, market segmentation.
- Value at spot rates; bootstrap upward.
- Forward.
(1 + r2)² = (1 + r1)(1 + rf). - Corporate yield = risk-free + spread; expected loss = LGD × probability.
- 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. If the required return on a bond rises, its price:
- Falls
- Rises
- Is unchanged
- Doubles
Show the answer
A. The coupons are fixed.
Question 2. A yield curve is normally built from:
- Risk-free bonds
- Junk bonds
- Shares
- Loans
Show the answer
A. Credit risk is added as a spread.
Question 3. A normal yield curve slopes:
- Upward
- Downward
- Flat always
- Randomly
Show the answer
A. Lenders want more for longer.
Question 4. The theory that curve shape reflects supply and demand in each maturity sector is:
- Market segmentation
- Expectations
- Liquidity preference
- PPP
Show the answer
A. It can explain any shape.
Question 5. To value a bond from the yield curve, discount each flow at:
- The spot rate for its own year
- The coupon rate
- The yield to maturity
- The risk-free rate for year 1
Show the answer
A. Spot rates differ by maturity.
Question 6. One-year spot 3.0% and two-year spot 3.5% imply a forward rate of about:
- 4.0%
- 3.5%
- 3.0%
- 7.0%
Show the answer
A. 1.035² ÷ 1.03 − 1.
Question 7. 80 basis points equals:
- 0.80%
- 8%
- 80%
- 0.08%
Show the answer
A. 100 basis points = 1%.
Question 8. Debt rated below BBB- is called:
- Sub-investment grade or junk
- Prime
- High grade
- Risk-free
Show the answer
A. Yields are much higher.
Question 9. Expected loss equals:
- LGD × probability of default
- LGD + probability
- Collateral × rate
- Loan ÷ rate
Show the answer
A. A simple expected value.
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) Explain why bond prices and required returns move inversely. (3 marks) (b) Explain three theories of the shape of the yield curve. (6 marks) (c) State one meaning of an inverse yield curve. (1 mark)
Question 12 (20 marks). Three bonds in one risk class pay interest annually and are redeemed at par: A, 1 year, 6% coupon, price 102; B, 2 years, 5%, price 101; C, 3 years, 4%, price 97. (a) Derive the one-, two- and three-year spot rates. (10 marks) (b) A four-year 6% bond is redeemable at par; spot rates for years 1 to 4 are 3.0%, 3.5%, 4.2% and 5.0%. Calculate its price. (4 marks) (c) Calculate the forward rate for year 2 given spot rates of 3.0% and 3.5%. (3 marks) (d) Explain why the yield to maturity can mislead. (3 marks)
Question 13 (10 marks). (a) A five-year risk-free yield is 5.4% and the spread is 80 basis points. Calculate the corporate yield. (2 marks) (b) A ₦10m loan has ₦8m of collateral and a 2% probability of default. Calculate the expected loss. (3 marks) (c) Explain what drives a credit rating. (5 marks)
Model answers
Show the model answer to Question 11
(a) Inverse link (3 marks). The cash flows from a bond are fixed. (1) If lenders require a higher return, they discount the flows at a higher rate. (1) The present value, which is the price, falls. (1)
(b) Theories (6 marks). Expectations: the curve reflects expected future short-term rates, so it steepens if rates are expected to rise. (2) Liquidity preference: longer-dated yields are more volatile, so investors want extra return, giving an upward slope. (2) Market segmentation: supply and demand in each maturity sector set its rate, so any shape is possible. (2)
(c) Inverse (1 mark). Rates are expected to fall. (1)
Show the model answer to Question 12
(a) Spot rates (10 marks).
- Year 1: 106 ÷ 102 = 1.0392, so 3.92%. (2)
- Year 2: 5 ÷ 1.0392 = 4.81
- 101 − 4.81 = 96.19
- 105 ÷ 96.19 = 1.0916
- square root = 1.0448, so 4.48%. (4)
- Year 3: 4 ÷ 1.0392 = 3.85
- 4 ÷ 1.0448² = 3.66
- 97 − 7.51 = 89.49
- 104 ÷ 89.49 = 1.1622
- cube root = 1.051, so 5.1%. (4)
(b) Price (4 marks).
- 6 ÷ 1.03 = 5.83
- 6 ÷ 1.035² = 5.60
- 6 ÷ 1.042³ = 5.30
- 106 ÷ 1.05⁴ = 87.21. (3)
- Price = 103.94. (1)
(c) Forward rate (3 marks).
- (1 + rf) = 1.035² ÷ 1.03 = 1.0400. (2)
- Forward rate = about 4.0%. (1)
(d) Yield to maturity (3 marks). It assumes every flow is discounted at the same yield. (1.5) Lenders require different returns for different maturities, so it is only an average. (1.5)
Show the model answer to Question 13
(a) Yield (2 marks). 5.40% + 0.80% = 6.20%. (2)
(b) Expected loss (3 marks).
- LGD = 10 − 8 = ₦2m. (1.5)
- Expected loss.
2m × 2% = ₦40,000. (1.5)
(c) Drivers (5 marks). Models score variables, weight them and total them into a score expressed as a rating. (1) Variables include firm size, gearing, industry risk, country risk, standing against peers, management strength, asset backing and cash-generating ability. Any four. (4)
Pathfinder on Spot rates, bootstrapping, forward rates, credit spreads, ratings and expected loss
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 yield curves. Practise the valuation and bootstrapping steps in the worked sections.
Revise first: Bootstrapping the curve
Questions ask for the cost of debt from a risk-free rate and a spread. See the section on credit risk, spreads and ratings.
Revise first: Credit risk, spreads and ratings
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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