Effective Maturity of Bond - Essay Example

Finance 13 February Assignment Question 11 The duration or the effective maturity of bond is the weighted average of all the maturities of the bond. Since bonds have smaller PV in later periods, each of these maturities is weighted by the proportion of PV of bond, at that maturity, to the current bond value (Mishkin and Eakins 62). Since the maturity given is 3 years, the maturities of the bond are: 3, 2, and 1 years respectively for each year of payment. For payments at each of these maturities, weights will be calculated: Table 1 Calculation of Duration of Bond (At 7% Market Rate) Year Interest (6%) PVi of Interest PVi of Principal PVi of Cash Payments Weight (W) = PVi / Price PV-weighted maturity (t*W) 1 $60 60 / (1.07)1 = $56.07 - $56.07 56.07 / 973.76 = 0.057 1 x 0.057 = 0.057 2 $60 60 / (1.07)2 = $52.41 - $52.41 52.41 / 973.76 = 0.054 2 x 0.054 = 0.108 3 $60 60 / (1.07)3 = $48.98 $816.30 $865.28 865.3 / 973.76 = 0.889 3 x 0.889 = 2.667 Total $973.76 2.83 Hence, the effective (weighted-average) maturity or duration of this bond is 2.83 years. Question 12 Under the duration approximation (Mishkin and Eakins 66): % ΔP = – DUR x Δ i |Δ i = 6.75 – 7 = – 0.25 %| 1 + i % ΔP = – 2.83 x – 0.25 = – 2.83 x – 0.234 = 0.662 % 1.07 Thus, the bond price is expected to be increased by 0.66 percent i.e. it would be $980.21: P (at 6.75%) = P (at 7%) x (1 + 0.00662) = 973.76 x 1.00662 = $980.21 This implies the expected price increase of: $980.21 – $973.76 = $6.45 Under the discounted cash flow method, the price will be calculated as follows: Table 2 Calculation of PV (At 6.75% Market Rate) Interest (6%) PVi of Interest PVi of Principal PVi of Cash Payments Year 1 $60 60 / (1.0675)1 = $56.21 - $56.21 Year 2 $60 60 / (1.0675)2 = $52.65 - $52.65 Year 3 $60 60 / (1.0675)3 = $49.32 $822.04 $871.37 Total $980.23 Under the discounted cash flow method the new price of the bond is $980.23 with the negligible difference of $0.02 from the one duration approximation method. This implies the expected price increase of $980.23 – $973.76 = $6.47 under this method. Question 14 (a) Duration of Portfolio (i) Duration of $30 million The duration of $30 million loan is the weighted average of all the maturities of the bond. But since there is only a single payment for this loan, there would only be a single maturity and hence weighting is not required of its duration. This implies that its duration is also 3 years. (ii) Duration of $40 million Table 3 Calculation of Duration of Bond (At 8% Market Rate) Year Interest (9%) PVi of Interest PVi of Principal PVi of Cash Payments Weight (W) = PVi / Price PV-weighted maturity (t*W) 1 $3.6 3.6 / (1.08)1 = $3.333 - $3.333 3.33 / 41.03 = 0.081 1 x 0.081 = 0.081 2 $3.6 3.6 / (1.08)2 = $3.086 - $3.086 3.09 / 41.03 = 0.075 2 x 0.075 = 0.15 3 $3.6 3.6 / (1.08)3 = $2.858 $31.753 $34.611 34.61 / 41.03 = 0.844 3 x 0.844 = 2.53 Total $41.03 2.76 Hence, the duration of $40 million loan is 2.76 years. (iii) Duration of Portfolio The duration of portfolio is the weighted average duration of individual loans, with the weights being the fraction of PV of each loan out of total PV of portfolio (Mishkin and Eakins 65). The weight for $30 million loan is $30 / $70 = 3/7 and for $40 million loan, it is $40 / $70 = 4/7. Hence, the portfolio duration is: (3/7 x 3) + (4/7 x 2.76) = 1.286 + 1.578 = 2.864 years (b) Under the duration approximation (Mishkin and Eakins 66): % ΔP = – DUR x Δ i |Δ i = 8.5 – 8 = 0.5 %| 1 + i % ΔP = – 2.864 x 0.5 = – 2.864 x 0.463 = – 1.324 % 1.08 Thus, the portfolio value would be decreased by 1.32 percent i.e. it would be $69.084 millions: P (at 8.5%) = P (at 8%) / (1 + 0.01324) = 70 / 1.01324 = $69.084 millions This implies the decrease in portfolio value of $70 – $69.084 = $0.92 millions Question 15 (a) Table 4 Calculation of Cash Received after 2.5 Years (At 12% Market Rate) Year Inflow Period Reinvestment Inflows Inflows from DCF Sale Total 1 $160 1.5 (Reinvest) 160 x (1.12)1.5 = $189.65 - 2 $170 0.5 (Reinvest) 170 x (1.12)0.5 = $179.91 - 3 $180 0.5 (Discount) - 180 / (1.12)0.5 = $170.08 4 $230 1.5 (Discount) - 230 / (1.12)1.5 = $194.04 Sum $369.56 $364.12 $733.68 Thus, total cash received after the 2.5 years will be $733.68. (b) Table 5 Calculation of Cash Received after 2.5 Years (At 11% Market Rate) Yr. Inflow Period Reinvestment Inflows Inflows from DCF Sale Total 1 $160 1.5 (Reinvest) 160 x (1.11)1.5 = $187.11 - 2 $170 0.5 (Reinvest) 170 x (1.11)0.5 = $179.11 - 3 $180 0.5 (Discount) - 180 / (1.11)0.5 = $170.85 4 $230 1.5 (Discount) - 230 / (1.11)1.5 = $196.67 Sum $366.22 $367.52 $733.74 The total cash received at market rate of 11% will be $733.74 with only a difference of $0.06. (c) The maturities of the bond are: 4, 3, 2, and 1 years respectively for each year of payment. Table 6 Calculation of Duration of Bond (At 12%) Year Cash Payments PVi of Cash Payments Weight (W = PVi / Price) PV-weighted maturity (t*W) 1 $160 160 / (1.12)1 = $142.86 142.86 / 552.67 = 0.258 1 x 0.258 = 0.258 2 $170 170 / (1.12)2 = $135.52 135.52 / 552.67 = 0.245 2 x 0.245 = 0.49 3 $180 180 / (1.12)3 = $128.12 128.12 / 552.67 = 0.232 3 x 0.232 = 0.696 4 $230 230 / (1.12)4 = $146.17 146.17 / 552.67 = 0.264 4 x 0.264 = 1.056 Total $552.67 2.5 years The duration of the bond is 2.5 years, which is same as its holding period. Question 5: Equilibrium Price and Quantity Shifting of demand curve will increase the intercept of demand equation by $50. The new demand curve will be: P = – 2/5 Q + 990 Solving for the new equilibrium price and quantity: B s = B d (new) Q + 500 = – 2/5 Q + 990 Q + 2/5 Q = 990 – 500 1.4 Q = 490  Q = 350 Substituting it in equation for supply: P = 350 + 500 = 850 New market interest rate will be: (1000 – 850) / 850 = 150 / 850 = 0.1765 = 17.65 % Question 5: Marginal Tax Rate Being indifferent b/w the two bonds implies equating the after-tax corporate bond’s yield with the tax-free municipal bond’s yield (Mishkin and Eakins 251). After-tax Corporate bond yield: 12 % – 12 % (T) = 12 (1 – T) % Municipal bond yield: 8% 12 (1 – T) % = 8 %  1 – T = 8 / 12 = 0.667  T = 1 – 0.667 = 0.333 = 33.33 % Question 6 As per Pure Expectation Theory, rates on longer-term T-bills (3- and 6-year) equal to average of 1 year T-bills rates over the life of long-term T-bills respectively (Mishkin and Eakins 111-112). Year 1: Interest rate = 7.5 % Year 2: Interest rate = 7.5 + 1.5 = 9 % Year 3: Interest rate = 9 + 1.5 = 10.5 % Year 4: Interest rate = 10.5 + 1.5 = 12 % Year 5: Interest rate = 12 + 1.5 = 13.5 % Year 6: Interest rate = 13.5 + 1.5 = 15 % Thus, required interest rate for 3-year T-bill:  (7.5 + 9 + 10.5) / 3 = 9 % Required interest rate for 6-year T-bill:  (7.5 + 9 + 10.5 + 12 + 13.5 + 15) / 3 = 11.25 % Question 14 We can borrow some amount and then lend it to earn a profit with no upfront cash required. Specifically, we can borrow $1000 (or any amount) at two-year rate, 6 %. Then, after two years we would have to repay $1000 x (1.06)2 = $1123.6. Next, we can use this amount to lend for one year and then reinvest the first-year inflows, for one year, in the second year at higher future rate to gain more cash inflows than the cash outflow from loan repayment. That is, in year-1 at 5% rate, we will earn $1000 x (1.05) = $1050 and by reinvesting at 7.25% rate, we will earn $1050 x (1.0725) = $1126.125 in year-2. In this way, net cash inflows will be positive at the end of two years. Yet, in order to make the profit risk-less, the 7.25 % future rate should be guaranteed otherwise cash inflows may not be enough to payoff the cash outflow. If the future rate 7.25% is guaranteed, the profit would be $1126.125 – $1123.6 = $2.525. Works Cited Mishkin Frederic S. and Stanley G. Eakins. Financial Markets and Institutions. 5th ed. India: Dorling Kindersley (India) Pvt. Ltd., 2006. Print. Read More