A company wants to borrow for two years. It can take a two-year loan, or a one-year loan now and another next year. The bank will not offer a deal in which one route is plainly cheaper than the other. That equality gives the forward rate: the rate for next year that is already hidden in today’s yield curve.
The same rate prices FRAs and swaps. And a currency swap lets a company borrow in the cheapest market and still end up with liabilities in the currency it wants.
This topic covers forward interest rates derived from a spot yield curve, the pricing of an interest rate swap from forward rates, currency swaps and how they differ from interest rate swaps and FX swaps, and reasons for using them. The previous topic covered FRAs and interest rate swaps. The next covers futures and hedging with futures. Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on these.
In 30 seconds
A forward rate is the rate for a later period implied by two spot rates: (1 + rn)n = (1 + rn−1)n−1 × (1 + f), so f = (1 + rn)n ÷ (1 + rn−1)n−1 − 1. These are the rates for FRAs (for example a 12v24).
A swap is priced so that the PV of the expected payments in each direction is zero at the start: discount the expected floating payments (forward rates, with any margin) at the spot rates, then find the fixed rate x with x × (sum of discount factors) = that PV.
A currency swap exchanges interest payments in two currencies on equivalent principal amounts, with an actual exchange of principal at the end (at a rate fixed at the start) and sometimes at the start. It can swap fixed for fixed, and it hedges long-term currency risk. An FX swap exchanges principal at the start and re-exchanges it at the end, with no interest exchanged.
Key terms in plain English
| Term | What it means |
|---|---|
| Spot rate | The average annual yield for a single flow due at a given future date |
| Forward rate | The rate for a future period implied by the spot rates |
| Discount factor | 1 ÷ (1 + spot rate)n, the PV of 1 due in year n |
| Currency swap | An exchange of interest (and principal) in two currencies |
| FX swap | A spot exchange of principal re-exchanged later, with no interest payments |
Syllabus check
This topic is in syllabus area E (treasury and advanced risk management). It covers:
- estimating the cost of debt using the term structure of interest rates;
- explaining how swaps, including currency swaps, can hedge interest rate and foreign exchange risk.
CAMA 2020 and treasury contracts
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 | Treasury contracts should be entered into for the benefit of the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should approve and monitor the use of swaps and other hedging contracts |
| 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 |
Forward rates from the yield curve
A company wanting a two-year loan could instead borrow for one year and roll the loan over, locking the second-year rate with a 12v24 FRA. A bank would not offer a two-year loan that is better or worse than that route, so: (1 + r2)² = (1 + r1)(1 + rf).
ICAN’s example. Spot yields: one year 3.0%, two years 3.6%, three years 4.3%, four years 5.1%, five years 5.8%.
- 12v24: 1.036² ÷ 1.03 = 1.0420, so 4.2%.
- 24v36: 1.043³ ÷ 1.036² = 1.0571, so 5.7%.
- 36v48: 1.051⁴ ÷ 1.043³ = 1.0754, so 7.5%
48v60: 1.058⁵ ÷ 1.051⁴ = 1.0865, so 8.7%.
| Maturity | Spot yield | Forward rate for that year |
|---|---|---|
| 1 year | 3.0% | 3.0% (= spot) |
| 2 years | 3.6% | 4.2% |
| 3 years | 4.3% | 5.7% |
| 4 years | 5.1% | 7.5% |
| 5 years | 5.8% | 8.7% |
Pricing an interest rate swap
Any swap is based on the yield curve for the company’s debt. It is priced so that at the start the PV of the expected inflows and outflows is zero. Steps: (1) take the forward rates; (2) adjust them for any margin agreed; (3) discount the expected floating payments at the spot rates; (4) set the PV of the fixed payments equal to that figure and solve.
ICAN’s example. A company has $100m of variable-rate debt for five years and wants a variable-to-fixed swap. The bank will pay the company the variable rate less 40 basis points. The forward rates are as in the table above.
- Expected payments from the bank = forward rate − 0.4%: 2.6%, 3.8%, 5.3%, 7.1%, 8.3% on $100m, that is $2.6m, $3.8m, $5.3m, $7.1m and $8.3m.
- Discount at the spot-rate factors (1 ÷ 1.03 = 0.971
1 ÷ 1.036² = 0.932
1 ÷ 1.043³ = 0.881
1 ÷ 1.051⁴ = 0.820
1 ÷ 1.058⁵ = 0.754): PV = 2.52 + 3.54 + 4.67 + 5.82 + 6.26 = $22.81m. - The sum of the factors = 4.358. Fixed payment x: 4.358x = 22.81, so x = $5.23m a year, a fixed rate of 5.23%.
Currency swaps
An interest rate swap is in one currency, with interest on a notional principal.
A currency swap differs in three ways:
- (a) it is between two currencies, with interest on equivalent principal amounts
- (b) the interest need not be fixed against floating, and may be fixed in one currency against fixed in the other
- (c) there is an actual exchange of principal at the end of the swap at a rate fixed at the start (sometimes at the start too)
ICAN’s example. A UK company issues a seven-year, US$180m bond but wants sterling liabilities. It arranges a seven-year currency swap at £1 = $1.80.
- During the swap it receives dollar interest on $180m (used to pay the bond interest) and pays sterling interest on £100m (= 180 ÷ 1.80).
- At the end it receives $180m (used to redeem the bond) and pays £100m.
- Net: the company has sterling interest and sterling principal obligations only.
Uses: hedging long-term currency risk; borrowing at a favourable rate in the international market and swapping into the preferred currency, which can be cheaper than borrowing directly; and, if fixed in one currency is swapped for floating in the other, hedging interest rate risk. Currency swaps are far less common than interest rate swaps.
FX swaps
An FX swap has an exchange of principal at the start and a re-exchange at the end, but no interest payments in between. It lets a company obtain a liability in a currency whose bond market is illiquid: issue bonds in dollars, swap the dollars into the other currency now, and repay the currency at the end.
The rules in more depth
Why forward rates are higher than spot on a rising curve
The two-year rate is an average of year 1 and year 2. If the average rises, the later year must be higher.
Why swaps start at zero value
Neither party should pay the other to enter the deal, so the PVs of both legs are equal.
Why principal is exchanged in a currency swap
The two principal amounts are in different currencies, so their values are not offsetting; exchange is needed to settle the bond.
Why FX swaps have no interest
The difference in interest rates is built into the rate at which the principal is re-exchanged.
Effect on analysis
Derive forward rates before pricing; state the discount factors; and show what each party pays and receives for a currency swap.
Worked examples
Example 1: a two-year forward. Spot yields are 2% (one year) and 3% (two years).
- Forward = 1.03² ÷ 1.02 − 1.
- 1.0609 ÷ 1.02 − 1.
- Forward = 4.01%.
Example 2: a three-year forward. The two-year spot rate is 3% and the three-year spot rate is 4%.
- Forward = 1.04³ ÷ 1.03² − 1.
- 1.124864 ÷ 1.0609 − 1.
- Forward = 6.03% for year 3.
Example 3: a par swap rate. Spot rates are 3.0%, 3.5% and 4.0% for one to three years. There is no margin.
- Forward rates: 3.00%, 4.00% and 5.01%.
- Discount factors: 0.9709, 0.9335, 0.8890 (sum 2.7934).
PV of floating = 0.0300×0.9709 + 0.0400×0.9335 + 0.0501×0.8890 = 0.1110. - Fixed rate.
0.1110 ÷ 2.7934 = 3.97%.
Example 4: currency swap principal. A company borrows $90m and swaps at £1 = $1.50.
- Sterling principal = 90 ÷ 1.50.
- = £60m.
- At the end it receives $90m and pays £60m.
How to score full marks
- Write the forward rate equation before substituting.
- Use year-by-year spot rates for discount factors.
- Adjust for any basis-point margin in the right direction.
- Equate the PVs to price a swap.
- For a currency swap, show interest and principal in both currencies.
- State the use: long-term currency hedge or cheaper borrowing.
Common mistakes in the exam
- Using the spot rate for a later year as the forward rate.
- Not squaring or cubing the spot factor.
- Adding the margin instead of deducting it.
- Forgetting that principal is exchanged in a currency swap.
- Confusing a currency swap with an FX swap.
- Dividing the PV by the wrong sum of factors.
Quick revision
- Forward: (1 + rn)n ÷ (1 + rn−1)n−1 − 1.
- Swap priced at zero PV: x × ΣDF = PV of floating payments.
- Currency swap: interest in two currencies, principal exchanged at the end.
- FX swap: principal exchanged and re-exchanged, no interest.
- Currency swaps hedge long-term currency risk.
- 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. The forward rate for year 2 equals:
- (1 + r₂)² ÷ (1 + r₁) − 1
- r₂ − r₁
- (1 + r₂) ÷ (1 + r₁)
- r₂ × r₁
Show the answer
A. From the two spot rates.
Question 2. Spot rates are 3.0% and 3.6% for years 1 and 2. The year 2 forward is about:
- 4.2%
- 3.6%
- 3.3%
- 6.6%
Show the answer
A. 1.036² ÷ 1.03 − 1.
Question 3. An interest rate swap is priced so that at the start the PV of the two sides is:
- Equal
- Zero for one side only
- Doubled
- Irrelevant
Show the answer
A. Neither party pays to enter.
Question 4. The discount factor for year 3 is:
- 1 ÷ (1 + r₃)³
- 1 ÷ (1 + r₃)
- r₃³
- (1 + r₃)³
Show the answer
A. Using the three-year spot rate.
Question 5. A currency swap differs from an interest rate swap because it:
- Exchanges principal at the end
- Never has interest
- Is in one currency
- Is for under a year
Show the answer
A. At a rate fixed at the start.
Question 6. A UK company with a $180m bond swapping at £1 = $1.80 has sterling principal of:
- £100m
- £180m
- £324m
- £1.8m
Show the answer
A. 180 ÷ 1.80.
Question 7. An FX swap has:
- No interest payments
- Interest each year
- No principal
- Only floating rates
Show the answer
A. Principal goes out and comes back.
Question 8. Currency swaps mainly hedge:
- Long-term currency risk
- Share price risk
- Tax risk
- Commodity risk
Show the answer
A. Futures and forwards cover the short term.
Question 9. A bank pays variable rate less 40 basis points; the expected payment for a 3.0% forward is:
- 2.6%
- 3.4%
- 0.4%
- 3.0%
Show the answer
A. 3.0% − 0.4%.
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 how a forward rate is derived from spot rates. (3 marks) (b) Calculate the year 2 and year 3 forward rates if spot rates are 3.0%, 3.6% and 4.3%. (4 marks) (c) Explain why swaps are priced to have zero value at the start. (3 marks)
Question 12 (20 marks). Spot yields for one to five years are 3.0%, 3.6%, 4.3%, 5.1% and 5.8%. A company has $100m of floating-rate debt for five years. A bank offers to pay variable rate less 40 basis points in exchange for fixed payments. (a) Derive the forward rates for years 2 to 5. (6 marks) (b) Calculate the fixed rate the bank will charge.
(8 marks) (c) Explain how a currency swap differs from an interest rate swap and from an FX swap. (6 marks)
Question 13 (10 marks). (a) A UK company issues a seven-year $180m bond and enters a currency swap at £1 = $1.80. Show its interest and principal flows. (6 marks) (b) State two reasons for a currency swap. (4 marks)
Model answers
Show the model answer to Question 11
(a) Derivation (3 marks).
- A two-year loan should cost the same as a one-year loan rolled over at the forward rate. (1)
- So (1 + r2)² = (1 + r1)(1 + rf). (1)
- Rearrange for rf. (1)
(b) Forwards (4 marks).
- Year 2: 1.036² ÷ 1.03 − 1 = 4.2%. (2)
- Year 3: 1.043³ ÷ 1.036² − 1 = 5.7%. (2)
(c) Zero value (3 marks). Neither party should pay to enter the swap. (1.5) So the PV of the expected floating payments equals the PV of the fixed payments. (1.5)
Show the model answer to Question 12
(a) Forwards (6 marks).
- Year 2: 4.2%; year 3: 5.7%; year 4: 1.051⁴ ÷ 1.043³ − 1 = 7.5%; year 5: 1.058⁵ ÷ 1.051⁴ − 1 = 8.7%.
- (1.5 each)
(b) Fixed rate (8 marks).
- Expected bank payments: 2.6%, 3.8%, 5.3%, 7.1%, 8.3% on $100m. (2)
- Discount factors: 0.971, 0.932, 0.881, 0.820, 0.754. (2)
- PV = 2.52 + 3.54 + 4.67 + 5.82 + 6.26 = $22.81m. (2)
- Sum of factors.
4.358, so x = 22.81 ÷ 4.358 = $5.23m, a rate of 5.23%. (2)
(c) Differences (6 marks). An interest rate swap is in one currency with interest on a notional principal and no exchange of principal. (2) A currency swap is in two currencies, can be fixed against fixed and has an actual exchange of principal at the end at a rate fixed at the start. (2) An FX swap exchanges principal at the start and re-exchanges at the end, with no interest payments between. (2)
Show the model answer to Question 13
(a) Flows (6 marks). Sterling principal = 180 ÷ 1.80 = £100m. (1) During the swap the company receives dollar interest on $180m, used to pay the bond interest. (2) It pays sterling interest on £100m. (1) At the end it receives $180m to redeem the bond and pays £100m. (2)
(b) Reasons (4 marks). Hedging long-term currency risk; borrowing cheaply in the international market and swapping into the preferred currency; swapping fixed for floating across currencies to hedge interest rate risk. Any two. (2 each)
Pathfinder on Forward rates from spot rates, swap pricing, currency swaps and FX swaps
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 swap pricing or currency swaps. Practise the forward rate and swap pricing steps in the worked sections.
Revise first: Pricing an interest rate swap
Questions give a spot yield curve and ask for forward rates. See the section on forward rates from the yield curve.
Revise first: Forward rates from the yield curve
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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