Questions: Spot Rates, Forward Rates, and No-Arbitrage Relationships

Question 1

The 1-year spot rate is 4% and the 2-year spot rate is 5%. What is the implied 1-year forward rate starting in one year, f(1,2)?

• A. Approximately 6%, derived from (1.05)² / (1.04) - 1
• B. 4.5%, which is the simple average of the two spot rates
• C. 5%, because the 2-year rate already reflects expectations for year 2
• D. 1%, which is the spread between the two spot rates

Answer: (A)

The no-arbitrage condition requires (1 + s₂)² = (1 + s₁)(1 + f(1,2)). Solving: f(1,2) = (1.05)²/(1.04) - 1 = 1.1025/1.04 - 1 ≈ 6.01%. The intuition is that lending for two years at 5% must produce the same terminal value as lending for one year at 4% and rolling over at the forward rate. If f were only 4.5% (the average), an arbitrageur could exploit the gap by borrowing at the low side and lending at the high side. Option B is the most tempting wrong answer — averaging spot rates is a natural impulse but incorrect because compounding is multiplicative, not additive.

Question 2

An investor observes that the implied 1-year forward rate starting in year 1 is 7%, but a broker is offering a one-year loan starting in one year at 8%. Assuming no transaction costs, what should the investor do?

• A. Borrow at the 8% forward contract and lend synthetically using spot rates to lock in a riskless profit
• B. Lend at the 8% forward contract and borrow synthetically using spot rates to lock in a riskless profit
• C. Do nothing — forward rates and spot rates can diverge without creating arbitrage
• D. Invest in the 2-year spot bond since it implies higher total returns

Answer: (B)

When a forward rate in the market (8%) exceeds the no-arbitrage forward rate (7%), you should lend at the high rate and borrow synthetically. Strategy: lend at the offered 8% forward; simultaneously borrow for 2 years at the 2-year spot rate and lend for 1 year at the 1-year spot rate — this synthetic short position costs 7% implied. The difference (8% - 7%) is riskless profit. No-arbitrage logic says this gap cannot persist: arbitrageurs will lend at 8%, driving that rate down until parity is restored. Option C is wrong precisely because the no-arbitrage condition is not merely a tendency — it is an enforced equality in liquid markets.

Question 3

Forward rates represent the financial market's unbiased prediction of what future spot rates will actually be.

• T. True
• F. False

Answer: False

False. Forward rates embed expected future spot rates *plus* term premia (compensation for bearing the risk of holding longer-duration instruments) plus convexity adjustments. An upward-sloping yield curve does imply that forward rates exceed current short rates, but this does not mean rates are expected to rise by that full amount — part of the difference compensates for duration risk. Disentangling pure expectations from term premia requires a term structure model with additional assumptions. Using forward rates as pure forecasts systematically overpredicts future short rates in most historical periods precisely because term premia are persistently positive.

Question 4

If the 2-year spot rate exceeds the 1-year spot rate, the implied 1-year forward rate starting in year 1 must exceed the 2-year spot rate.

• T. True
• F. False

Answer: True

True. From (1 + s₂)² = (1 + s₁)(1 + f(1,2)), if s₂ > s₁ then the left side grows faster than (1 + s₁)², which forces f(1,2) > s₂. Intuitively: if you earn more by lending for 2 years than 1 year, the implied rate for the second year must be even higher than the 2-year average — otherwise the 2-year advantage wouldn't exist. An upward-sloping spot curve therefore implies forward rates that are above and more steeply rising than the spot curve itself. This is why forward curves always 'overshoot' the spot curve when the yield curve is upward sloping.

Question 5

Explain the no-arbitrage condition that ties forward rates to spot rates, and why this relationship must hold in liquid markets.

Model answer: The no-arbitrage condition states that lending for n years at the n-year spot rate must produce the same terminal value as a sequence of shorter loans at the corresponding spot and forward rates. For example, lending for 2 years at s₂ must equal lending for 1 year at s₁, then rolling over for a second year at f(1,2): (1+s₂)² = (1+s₁)(1+f(1,2)). If this equality fails, two strategies with identical payoffs have different costs — a free lunch that arbitrageurs would immediately exploit.

In practice, if the forward rate implied by spot rates were lower than an available forward contract, traders would borrow synthetically (via spot rates) at the cheap implied rate and lend at the expensive contract rate, earning riskless profit. This trading pressure pushes prices back toward parity. The key insight is that forward rates are not independently determined — they are fully pinned by spot rates through this compounding relationship. The relationship is mathematically exact, not approximate, which is what makes it a no-arbitrage condition rather than a tendency or approximation.