Please Solve Following 10 Questions Show The Detailed Works Each Qu
Please solve following 10 questions (show the detailed works). Each question is accounted for 10 points.
Paper For Above instruction
1. Consider a coupon bond that has a $1,000 par value and a coupon rate of 12%. The bond is currently selling for $1,280 and has 12 years to maturity. What is the bond’s yield to maturity?
2. Consider a bond that promises the following cash flows. Year: Promised Payments: Assuming all market interest rates are 14%, what is the duration of this bond?
3. You are willing to pay $25,000 now to purchase a perpetuity which will pay you and your heirs $2,200 each year, forever, starting at the end of this year. If your required rate of return does not change, how much would you be willing to pay if this were a 15-year, annual payment, ordinary annuity instead of a perpetuity?
4. The demand curve and supply curve for bonds are estimated using the following equations: Demand: P = - (5/6)Q + 1400; Supply: P = (1/3)Q + 700. As the stock market continued to rise, the Federal Reserve felt the need to increase the interest rates. As a result, the new market interest rate increased to 14%, but the equilibrium quantity remained unchanged. What are the new demand and supply equations? Assume parallel shifts in the equations.
5. The one-year interest rate over the next 10 years will be 3%, 4.5%, 6%, 7.5%, 9%, 10.5%, 13%, 14.5%, 16%, 17.5%. Using the pure expectations theory, what will be the interest rates on a 4-year bond, 7-year bond, and 10-year bond?
6. A bank has two, 3-year commercial loans with a present value of $80 million. The first is a $30 million loan that requires a single payment of $37.8 million in 3 years, with no other payments until then. The second is for $50 million. It requires an annual interest payment of $4.5 million. The principal of $50 million is due in 3 years. The general level of interest rates is 7%. What is the duration of the bank’s commercial loan portfolio?
7. One-year T-bill rates are 3% currently. If interest rates are expected to go up after 4 years by 3% every year, what should be the required interest rate on a 10-year bond issued today?
8. Calculate the present value of a $1,000 zero-coupon bond with 8 years to maturity if the required annual
interest rate is 12%.
9. Calculate the duration of a $1,000, 5% coupon bond with three years to maturity. Assume that all market interest rates is 8% for next three years.
10. An economist has estimated that, near the point of equilibrium, the demand curve and supply curve for bonds can be estimated using the following equations: Demand: P =-(2/7)Q + 1000; Supply: P = (1/7)Q + 700.
a. What is the expected equilibrium price and quantity of bonds in this market?
b. Given your answer to part (a), which is the expected interest rate in this market?
Full Solutions to the Questions
Question 1: Yield to Maturity of a Coupon Bond
The bond has a par value of $1,000, a coupon rate of 12%, market price of $1,280, and 12 years to maturity. The annual coupon payment (C) = 12% of 1,000 = $120. The current price (P) = $1,280.
The yield to maturity (YTM) is the rate (r) satisfying the following bond price equation: \[P = \sum_{t=1}^{12} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^{12}}\]
Where F = $1,000. We can approximate the YTM using trial and error or financial calculator.
Using a financial calculator or Excel's RATE function:
RATE = RATE(n=12, pmt=120, pv=-1280, fv=1000)
Approximate calculation yields a YTM of about 8.05%.
Therefore, the YTM of the bond is approximately 8.05%.
Question 2: Duration of a Bond with Given Cash Flows
Suppose the bond's cash flows over the years are as follows:
Year 1: $200
Year 2: $200
Year 3: $200
Year 4: $200
Year 5: $200
Year 6: $200
Year 7: $200
Year 8: $200
Year 9: $200
Year 10: $200
Year 11: $200
Year 12: $1,200
*(Assuming the last payment includes principal and interest.)
For simplicity, assume equal payments with all market interest rates at 14%. The duration is calculated as: \[D = \frac{\sum_{t=1}^{n} t \times PV(CF_t)}{\sum_{t=1}^{n} PV(CF_t)}\] where PV(CF_t) = \(\frac{CF_t}{(1 + r)^t}\)
Calculating the present value of each cash flow and summing products of t and PV(CF_t), then dividing by the total price, yields a duration close to approximately 4.1 years.
Question 3: Valuation of Annuity vs Perpetuity
Perpetuity value (PV) = \$25,000, paying \$2,200 annually. The rate of return r is derived from: r = \( \frac{\text{Payment}}{\text{PV}} \) = \( \frac{2200}{25000} \) ≈ 8.8%
The perpetuity's yield is roughly 8.8%. The present value of a 15-year annuity (ordinary) paying \$2,200 annually at the same rate is:
PV_{15} = 2200 \times \left[\frac{1 - (1 + r)^{-15}}{r}\right]
Calculating:
PV_{15} = 2200 \times \left[\frac{1 - (1 + 0.088)^{-15}}{0.088}\right] ≈ 2200 \times 9.51 ≈ \$20,922
Thus, you would be willing to pay approximately \$20,922 for a 15-year, \$2,200 annual payment annuity
Question 4: New Demand and Supply Equations after Interest Rate Change
Original demand: P = - (5/6)Q + 1400
Original supply: P = (1/3)Q + 700
At new interest rate of 14%, equivalent to a higher price level, the demand curve shifts inward (less quantity demanded at any price), and supply shifts outward (more quantity supplied at any price). Assuming parallel shifts, the new equations are:
Demand: P = - (5/6)Q + 1388
(shifted downward with increased interest rates)
Supply: P = (1/3)Q + 712 (shifted upward)
Question 5: Interest Rates Using Pure Expectations
Given forward rates:
Year 1-2: 3%
Year 2-3: 4.5%
Year 3-4: 6%
Year 4-5: 7.5%
Year 5-6: 9%
Year 6-7: 10.5%
Year 7-8: 13%
Year 8-9: 14.5%
Year 9-10: 16%
Year 10: 17.5%
Interest rate on a 4-year bond:
r_4 = \left[\frac{(1 + 3\%) \times (1 + 4.5\%) \times (1 + 6\%) \times (1 + 7.5\%)
\right]^{1/4} - 1 \approx 5.55\%
Similarly, for 7-year bond: r_7 ≈ 8.15\%
For 10-year bond:
r_{10} ≈ 10.27\%
This involves calculating geometric average of the forward rates over each period.
Question 6: Duration of Bank’s Loan Portfolio
The first loan: Present value (PV) = \$30 million; single payment \$37.8 million in 3 years.
The second loan: PV = \$50 million; annual interest \$4.5 million, principal \$50 million due in 3 years.
The total cash flows:
Year 1 & 2: \$4.5 million interest each year
Year 3: \$4.5 million interest + \$50 million principal
The duration of the portfolio is weighted average of individual durations:
Loan 1 duration ≈ 2.86 years; Loan 2 duration ≈ 2.97 years; weighted average ≈ 2.93 years.
Question 7: Required Interest Rate on 10-year Bond
Current 1-year T-bill rate: 3%. Expected increases: +3% each subsequent year for 4 years, then staying at 15%.
The required rate for the 10-year bond is approximately the average expected rate over 10 years:
r_{10} = \frac{1}{10} \left(3\% + 3\% + 3\% + 3\% + 15\% \times 6 \right) / 10 ≈ 8.4\%
Question 8: Present Value of Zero-Coupon Bond
FV = \$1,000; n=8 years; r=12%
PV = \(\frac{FV}{(1 + r)^n}\) = \(\frac{1000}{(1.12)^8}\) ≈ \$405.99
Question 9: Duration of a 3-year Coupon Bond
Coupon = 5% of 1000 = \$50 annually; face value = \$1,000; market interest rate = 8%. The present value of cash flows using discount rates:
Duration ≈ 2.78 years.
Question 10: Equilibrium in Bond Market
Demand: P = -(2/7)Q + 1000; Supply: P = (1/7)Q + 700
Set demand = supply:
-(2/7)Q + 1000 = (1/7)Q + 700
Solving for Q:
Q = 210
Price:
P = (1/7)*210 + 700 = 730
Interest rate = \(\frac{Coupon Payment}{Price}\) or similar logic applies, giving approximately 6.85%.
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