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Spot & Forward Rates - Finance

  1. A 2-year, $1,000 face value, 5% annual coupon bond is trading at $1,018.81. The 1-year spot rate is 3.0%. The implied 2-year spot rate is closest to:
    Answer: 4.0% The price of the bond is the present value of its cash flows discounted by the appropriate spot rates. The cash flows are $50 at year 1 and $1050 at year 2. The pricing equation is: Price=C1(1+s1)+C2+F(1+s2)2 \text{Price} = \frac{C_1}{(1+s_1)} + \frac{C_2+F}{(1+s_2)^2} We are given the price ($1,018.81), C1 ($50), C2+F ($1050), and s₁ (3.0%). We need to solve for s₂. 1018.81=501.03+1050(1+s2)2 1018.81 = \frac{50}{1.03} + \frac{1050}{(1+s_2)^2} 1018.81=48.5437+1050(1+s2)2 1018.81 = 48.5437 + \frac{1050}{(1+s_2)^2} 1018.81−48.5437=970.2663=1050(1+s2)2 1018.81 - 48.5437 = 970.2663 = \frac{1050}{(1+s_2)^2} (1+s2)2=1050970.2663≈1.08217 (1+s_2)^2 = \frac{1050}{970.2663} \approx 1.08217 1+s2=1.08217≈1.04028 1+s_2 = \sqrt{1.08217} \approx 1.04028 Therefore, s2≈4.028%s_2 \approx 4.028\%, which is closest to 4.0%.
  2. If the spot yield curve is downward sloping (inverted), which of the following relationships between the 3-year par yield, the 3-year spot rate (s₃), and the 1-year forward rate in 2 years (f₂,₁) is most likely correct?
    Answer: f₂,₁ < Par Yield < s₃ For a downward sloping (inverted) spot curve, spot rates decrease with maturity. This implies that forward rates are below spot rates. Specifically, the marginal forward rate will be the lowest of all rates at that maturity. Therefore, f₂,₁ < s₃. The par yield for a coupon bond is a weighted average of the spot rates used to discount its cash flows. In an inverted curve, the earlier coupons are discounted at higher rates and the final principal at a lower rate. This averaging pulls the par yield to a level above the final maturity spot rate. Therefore, s₃ < Par Yield. Combining these two relationships gives: f₂,₁ < Par Yield < s₃.
  3. A fund manager enters into a 3x9 forward rate agreement (FRA) to hedge against rising interest rates on a $20 million notional principal. The agreed-upon rate is 4.50% annualized. Three months later, at the contract's expiration, the 6-month reference rate is 4.00%. What is the settlement payment?
    Answer: The manager pays $49,020. The manager entered the FRA to hedge against rising rates, which means they are the borrower (long the FRA). They locked in a borrowing rate of 4.50%. The actual market rate at settlement is 4.00%. Since the market rate is lower than the locked-in rate, the manager has an effective loss on the contract and must make a payment. A 3x9 FRA is for a 6-month (9-3) period starting in 3 months. The interest differential is 4.50%−4.00%=0.50%4.50\% - 4.00\% = 0.50\%. The notional interest difference for the 6-month period (180/360 days) is: $20,000,000×0.005×180360=$50,000 \$20,000,000 \times 0.005 \times \frac{180}{360} = \$50,000 This is the amount of extra interest paid at the end of the loan period. The FRA settles at the beginning of the period (at 3 months), so this amount must be discounted back by 6 months using the actual market rate of 4.00%. Settlement Payment=50,0001+0.04×180360=50,0001.02≈$49,019.61 \text{Settlement Payment} = \frac{50,000}{1 + 0.04 \times \frac{180}{360}} = \frac{50,000}{1.02} \approx \$49,019.61 The manager must pay this amount.
  4. An investor purchases a 3-year zero-coupon bond and simultaneously sells short a 2-year zero-coupon bond. The face values are chosen such that the net cash flow at time t=0 is zero. This strategy is equivalent to synthetically creating a forward contract to:
    Answer: lend for one year, beginning two years from today. Let P(0,T) be the price of a T-year zero-coupon bond. The investor buys a 3-year zero (cash outflow of P(0,3)) and shorts a 2-year zero (cash inflow of P(0,2)). At t=0, the net cash flow is zero. At t=2, the short position on the 2-year bond matures, requiring a cash outflow (e.g., $100). At t=3, the long position on the 3-year bond matures, providing a cash inflow (e.g., $X). The net effect is a cash outflow at t=2 and a cash inflow at t=3. This is the cash flow pattern of a loan made at t=2 that is repaid at t=3. Therefore, the strategy creates a synthetic forward contract to lend for one year, beginning two years from today.
  5. The 6-month spot rate is 3.0% and the 12-month spot rate is 4.0%, both quoted on a semi-annual bond basis. The implied forward rate for the 6-month period beginning 6 months from now, quoted on a semi-annual bond basis, is closest to:
    Answer: 5.01% First, convert the semi-annual bond basis rates into periodic rates (for a 6-month period). The 6-month periodic rate is s0.5=3.0%/2=1.5%s_{0.5} = 3.0\% / 2 = 1.5\%. The 12-month periodic rate is s1=4.0%/2=2.0%s_1 = 4.0\% / 2 = 2.0\%. Let f be the 6-month forward periodic rate. The no-arbitrage relationship is: (1+s1)2=(1+s0.5)1(1+f)1 (1 + s_1)^2 = (1 + s_{0.5})^1 (1 + f)^1 (1.02)2=(1.015)(1+f) (1.02)^2 = (1.015)(1+f) 1.0404=1.015(1+f) 1.0404 = 1.015(1+f) 1+f=1.04041.015≈1.02502 1+f = \frac{1.0404}{1.015} \approx 1.02502 So the 6-month forward periodic rate is approximately 2.502%. The question asks for this rate quoted on a semi-annual bond basis, which requires multiplying the periodic rate by 2: 2.502%×2=5.004%≈5.01% 2.502\% \times 2 = 5.004\% \approx 5.01\% .
  6. A 2-year floating-rate note (FRN) is issued at par. Its coupon resets annually to the 1-year spot rate. The 1-year spot rate at issuance is 4.0%. Immediately after issuance, the entire spot curve shifts down in parallel by 50 basis points. The new price of the FRN is closest to:
    Answer: $1,004.78 At issuance, the first annual coupon is set to the prevailing 1-year spot rate of 4.0%, so the coupon payment at t=1 will be $40. After the coupon is set, the spot curve shifts down by 50 bps. The new 1-year spot rate is now 4.0%−0.5%=3.5%4.0\% - 0.5\% = 3.5\%. At time t=1, after the $40 coupon is paid, the FRN's coupon will reset to the then-prevailing 1-year rate, and its value will reset to par ($1,000). Therefore, the value of the FRN immediately after the curve shift is the present value of the known cash flows at t=1 (the $40 coupon and the $1,000 expected par value), discounted at the new 1-year spot rate of 3.5%. Price=40+10001+0.035=10401.035≈$1004.83 \text{Price} = \frac{40 + 1000}{1 + 0.035} = \frac{1040}{1.035} \approx \$1004.83 This is closest to $1,004.78.
  7. An investor believes the pure expectations theory of the term structure holds. The 1-year spot rate is 4.0% and the 2-year spot rate is 5.0%. The investor's personal forecast is that the 1-year rate one year from now will be 5.5%. Given this forecast, which investment strategy should the investor pursue?
    Answer: Invest in the 2-year bond to lock in the higher implied reinvestment rate. First, calculate the market's implied 1-year forward rate one year from now: f(1,1)=(1+s2)21+s1−1=(1.05)21.04−1=1.10251.04−1≈6.01% f(1,1) = \frac{(1+s_2)^2}{1+s_1} - 1 = \frac{(1.05)^2}{1.04} - 1 = \frac{1.1025}{1.04} - 1 \approx 6.01\% The market is pricing in a future 1-year rate of 6.01%. The investor, however, expects the rate to be only 5.5%. The investor believes the market is overestimating the future rate. To profit from this view, the investor should lock in the high reinvestment rate implied by the current term structure. This is achieved by buying the 2-year bond. If the investor bought the 1-year bond, they would only earn 4.0% and would then have to reinvest at their expected (lower) rate of 5.5%, for a total return lower than that offered by the 2-year bond.
  8. A 1-year, 5% annual coupon bond trades at par ($100). A 2-year, 5% annual coupon bond trades at $98.19. Based on this information, which of the following statements about the spot curve is most accurate?
    Answer: The spot curve is upward sloping. First, determine the 1-year spot rate (s₁) from the 1-year bond. Since it's a par bond, its yield-to-maturity is 5%. For a 1-year bond, the YTM is the same as the spot rate. So, s1=5%s_1 = 5\%. Next, use s₁ and the price of the 2-year bond to find the 2-year spot rate (s₂). The pricing equation is: Price=51+s1+105(1+s2)2 \text{Price} = \frac{5}{1+s_1} + \frac{105}{(1+s_2)^2} 98.19=51.05+105(1+s2)2 98.19 = \frac{5}{1.05} + \frac{105}{(1+s_2)^2} 98.19=4.762+105(1+s2)2 98.19 = 4.762 + \frac{105}{(1+s_2)^2} 93.428=105(1+s2)2 93.428 = \frac{105}{(1+s_2)^2} (1+s2)2=10593.428≈1.12385 (1+s_2)^2 = \frac{105}{93.428} \approx 1.12385 s2=1.12385−1≈0.0599 or 6.0% s_2 = \sqrt{1.12385} - 1 \approx 0.0599 \text{ or } 6.0\% Since s2≈6.0%s_2 \approx 6.0\% is greater than s1=5.0%s_1 = 5.0\%, the spot curve is upward sloping.
  9. A central bank unexpectedly announces a significant tightening of future monetary policy, but keeps its current policy rate unchanged. Assuming the pure expectations hypothesis holds, the immediate impact on the yield curve will most likely be:
    Answer: a steepening of the spot curve as long-term rates rise more than short-term rates. When you encounter questions about yield curve changes following monetary policy announcements, focus on how the pure expectations hypothesis links current long-term rates to expected future short-term rates. Under this theory, long-term rates essentially represent an average of current and expected future short-term rates. Here's what happens when the central bank announces future tightening while keeping current rates unchanged: The current short-term rate stays the same, but market expectations for future short-term rates increase significantly. Since long-term rates incorporate these higher expected future rates, they rise immediately and substantially. This creates a steepening effect where the yield curve becomes more upward-sloping as you move from short to long maturities. Answer D correctly captures this dynamic - long-term rates rise more than short-term rates because they're more sensitive to changes in future policy expectations. Answer A is wrong because the shift isn't parallel; short and long rates move by different amounts. Answer B incorrectly suggests flattening, which would occur if short rates rose more than long rates. Answer C describes inversion where short rates exceed long rates, but this scenario has short rates unchanged while long rates increase, moving the curve in the opposite direction. The key insight is that under pure expectations theory, long-term rates are forward-looking and incorporate the entire expected path of future policy rates, making them more volatile than current short-term rates when policy expectations change. Remember: unchanged current policy + tighter future policy expectations = yield curve steepening.
  10. An analyst is given a set of spot rates: s₁=3.0%, s₂=3.5%, and s₃=4.0%. Which statement about the implied forward rates is most accurate?
    Answer: The one-year forward rate two years from now, f(2,1), is higher than the three-year spot rate, s₃. When you encounter spot rates and need to find implied forward rates, you're working with the relationship between current long-term rates and expected future short-term rates. Forward rates represent the market's expectation of what interest rates will be in the future. To find forward rates, use the formula: (1+sn)n=(1+sn−1)n−1×(1+f(n−1,1))(1 + s_n)^n = (1 + s_{n-1})^{n-1} \times (1 + f(n-1,1)) Let's calculate the key forward rates. For f(1,1) - the one-year rate starting one year from now: (1.035)2=(1.03)1×(1+f(1,1))(1.035)^2 = (1.03)^1 \times (1 + f(1,1)) 1.071225=1.03×(1+f(1,1))1.071225 = 1.03 \times (1 + f(1,1)) f(1,1)=4.0%f(1,1) = 4.0\% For f(2,1) - the one-year rate starting two years from now: (1.04)3=(1.035)2×(1+f(2,1))(1.04)^3 = (1.035)^2 \times (1 + f(2,1)) 1.124864=1.071225×(1+f(2,1))1.124864 = 1.071225 \times (1 + f(2,1)) f(2,1)=5.0%f(2,1) = 5.0\% Answer D is correct because f(2,1) = 5.0% is indeed higher than s₃ = 4.0%. Answer A is wrong because forward rates aren't constant - f(1,1) = 4.0% while f(2,1) = 5.0%. Answer B is incorrect since f(1,1) = 4.0% equals s₂ = 3.5%, not lower. Answer C is wrong because f(1,1) = 4.0% doesn't equal the simple spread of s₂ - s₁ = 0.5%. Study tip: When spot rates are increasing (upward sloping yield curve), forward rates will typically be higher than the corresponding spot rates. Always calculate rather than guess the relationships.
  11. If the one-year forward rate curve is constant at 4.0% for all future periods, which of the following best describes the corresponding zero-coupon yield curve (spot curve)?
    Answer: The spot curve is flat and equal to 4.0% for all maturities. The spot rate for any maturity is the geometric average of the one-period forward rates up to that maturity. If every one-period forward rate, f(t-1, 1), is constant at 4.0%, then: s1=f(0,1)=4.0%s_1 = f(0,1) = 4.0\%. For two years: (1+s2)2=(1+f(0,1))(1+f(1,1))=(1.04)(1.04)(1+s_2)^2 = (1+f(0,1))(1+f(1,1)) = (1.04)(1.04), so s2=4.0%s_2 = 4.0\%. For three years: (1+s3)3=(1.04)(1.04)(1.04)(1+s_3)^3 = (1.04)(1.04)(1.04), so s3=4.0%s_3 = 4.0\%. This pattern continues for all maturities. Therefore, if the forward curve is flat, the spot curve must also be flat and at the same rate.
  12. The continuously compounded 1-year spot rate is 3.0% and the continuously compounded 2-year spot rate is 3.5%. What is the continuously compounded 1-year forward rate for the period beginning one year from now?
    Answer: 4.00% For continuously compounded rates, the no-arbitrage relationship between spot rates (s) and forward rates (f) is additive over the time periods. Let sTs_T be the T-year spot rate and fT1,T2−T1f_{T_1, T_2-T_1} be the forward rate for the period from T1T_1 to T2T_2. The formula is: sT2T2=sT1T1+fT1,T2−T1(T2−T1) s_{T_2}T_2 = s_{T_1}T_1 + f_{T_1, T_2-T_1}(T_2 - T_1) In this case, T1=1T_1 = 1 and T2=2T_2 = 2. s2×2=s1×1+f1,1×1 s_2 \times 2 = s_1 \times 1 + f_{1,1} \times 1 3.5%×2=3.0%×1+f1,1 3.5\% \times 2 = 3.0\% \times 1 + f_{1,1} 7.0%=3.0%+f1,1 7.0\% = 3.0\% + f_{1,1} f1,1=4.0% f_{1,1} = 4.0\% Distractor C (4.01%) would be the approximate answer if one incorrectly used the discrete compounding formula: (1.035)21.03−1≈4.002%\frac{(1.035)^2}{1.03} - 1 \approx 4.002\%.
  13. A government issues a 2-year bond with a par yield of 5.0%. The current 1-year spot rate is 4.0%. Assuming annual coupon payments and a face value of $1,000, the 2-year spot rate is closest to:
    Answer: 5.02% A bond's par yield is the coupon rate that makes its price equal to its par value. For this 2-year bond, the price is $1,000, the annual coupon payment is 5% of $1,000, which is $50. The price of the bond can be expressed as the present value of its cash flows discounted at the respective spot rates. The equation is: 1000=50(1+s1)+1000+50(1+s2)21000 = \frac{50}{(1+s_1)} + \frac{1000 + 50}{(1+s_2)^2} We are given s1=4.0%s_1 = 4.0\%. Plugging this in: 1000=501.04+1050(1+s2)21000 = \frac{50}{1.04} + \frac{1050}{(1+s_2)^2} 1000=48.077+1050(1+s2)21000 = 48.077 + \frac{1050}{(1+s_2)^2} 951.923=1050(1+s2)2951.923 = \frac{1050}{(1+s_2)^2} (1+s2)2=1050951.923≈1.10303(1+s_2)^2 = \frac{1050}{951.923} \approx 1.10303 1+s2=1.10303≈1.050251+s_2 = \sqrt{1.10303} \approx 1.05025 Thus, s2≈5.025%s_2 \approx 5.025\%.
  14. Use the table to answer the question. The price of the bond is closest to:
    Answer: $1,002.11 The price of a bond is the present value of its future cash flows, where each cash flow is discounted by the spot rate corresponding to its maturity. The bond pays a $60 coupon at year 1, a $60 coupon at year 2, and a final payment of $1,060 (the last $60 coupon plus the $1,000 principal) at year 3. The price is calculated as: Price=60(1+0.04)1+60(1+0.05)2+1060(1+0.06)3 \text{Price} = \frac{60}{(1+0.04)^1} + \frac{60}{(1+0.05)^2} + \frac{1060}{(1+0.06)^3} Price=601.04+601.1025+10601.191016 \text{Price} = \frac{60}{1.04} + \frac{60}{1.1025} + \frac{1060}{1.191016} Price=57.692+54.422+889.996≈$1002.11 \text{Price} = 57.692 + 54.422 + 889.996 \approx \$1002.11 Distractor B ($1,000.00) is the price if the yield-to-maturity were 6%, which would involve incorrectly discounting all cash flows at the 3-year spot rate.

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