mid-term solutions 2022 1

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2411

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Finance

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Apr 3, 2024

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FINM 2411 MID-TERM SOLUTIONS 2022 Question 1: Total 25 points Part A: [5 points] You have just entered into a $1 million 30-year mortgage with a rate of 3% p.a. compounding monthly. What is the amount of your monthly repayments? The repayments are a 360-period annuity with present value equal to the amount borrowed: 1,000,000 = R [ 1 − ( 1.0025 ) − 360 0.0025 ] which implies that the monthly repayment is R=4,216.04. Part B: [5 points] Assume that 10 years has passed – you just made the 120 th payment. How much interest did you pay over the last year? The principal outstanding after 10 years is the present value of all remaining repayments: Balance = 4,216.04 [ 1 − ( 1.0025 ) − 240 0.0025 ] which implies that the current loan balance is $760,198.09. The principal outstanding after 9 years is the present value of all remaining repayments at that time: Balance = 4,216.04 [ 1 − ( 1.0025 ) − 258 0.0025 ] which implies that the current loan balance is $787,538.32 Thus, the amount of principal paid off over the last year is: 787,538.32 − 760,198.09 = 27,340.23 . The total amount paid over the last year is: 12 × 4,216.04 = 50,592.48 . Thus, the amount of interest paid is: 50,592.48 − 27,340.23 = 23,252.26 . - 1 -

Part C: [5 points] Continue to assume that 10 years has passed. You have been promoted at work and decide to increase your monthly payments to $6,000 per month. When will the loan be paid off? Solve for n in the following equation: 760,198.09 = 6,000 [ 1 − ( 1.0025 ) − n 0.0025 ] where n=152.54. That is, you will make 152 full payments and then a partial payment to close out the loan. Part D: [10 points] Ignore part (c) (i.e., go back to the end of part (b) where you have just made the 120 th monthly payment). Suppose you lose your job, so you negotiate with the bank that you will make no payments for the next two years, then you’ll ‘catch up’ and still have the loan paid off at the end of the initial 30-year period. (Your thinking is that it might take you a while to find the right job, but when you do, it will involve a high salary). The bank agrees, but proposes that you should pay a ‘penalty’ rate of interest of 6% p.a. (compounded monthly) during the two-year period while you are making no payments. What is the amount of the new monthly payment – for when you resume payments after the two-year period? First note that, at the end of the two-year period, the loan balance will have increased to: 760,198.09 ( 1.005 ) 24 = 856,864.71 . The repayments are a 216-period annuity (18 years remaining) with present value equal to the amount borrowed: 856.864.71 = R [ 1 − ( 1.0025 ) − 216 0.0025 ] which implies that the monthly repayment is R=5,138.82. Question 2: Total 25 points Part A: [15 points] What is the current value of a $100 4% government bond that matures in 5 years and 2 months from today if the yield to maturity is 3% p.a. (compounding semi-annually)? Measuring time in years, the cash flow stream is as follows: - 2 -

First find the value at the time of the first coupon: 2 + 2 [ 1 − ( 1.015 ) − 10 0.015 ] + 100 ( 1.015 ) 10 = 106.61 . Now discount this back to the present: 106.61 ( 1.015 ) 2 / 6 = 106.08 . Graphically, we are performing the following operation: Part B: [5 points] A one-year $100 zero-coupon bond is currently trading at $97. A two-year zero-coupon bond is currently trading at $94. What rate of interest could you lock in today for a loan that starts one year from now and finishes two years from now? Assume that all of these instruments have the same counterparty, so are of equivalent risk. The current one-year rate can be found by solving: 97 ( 1 + r 0,1 ) 1 = 100 in which case r 0,1 = 3.0928% . The current two-year rate can be found by solving: 94 ( 1 + r 0,2 ) 2 = 100 in which case r 0,2 = 3.1421% . - 3 -

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This implies that the forward rate from time 1 to time 2 must be such that: ( 1 + r 0,1 )( 1 + f 1,2 ) = ( 1 + r 0,2 ) 2 ( 1.030928 ) ( 1 + f 1,2 ) = ( 1.031421 ) 2 where f 1,2 = 3.1915% . Part C: [5 points] Consider a collateralised debt obligation based on a reference set of 100 corporate bonds. Assume that all of the bonds are equally weighted and that the recovery rate is set to 40%. You are considering a product that loses all of its value once 4.8% of the capital is subject to a credit event. What is the probability of a total loss of value if we assume that credit events are independent across bonds and over time, such that each bond has a 3% probability of experiencing a credit event during the life of the product you are considering? How does that probability change if we assume that there is a 50/50 chance of the default probability being 0% or 6% over the relevant period? First, note that each bond represents 1% of the total capital. Since each credit event results in an assumed 60% loss of value for that bond, each credit event results in an assumed loss of 0.6% of the total capital. Hence a ‘detachment point’ of 4.8% pertains to 8 credit events. The relevant probabilities can be computed from the binomial distribution as follows: Probability of default 3.00% 6.00% Number of bonds 100 100 Required number of defaults 8 8 Probability of >= X defaults 1.06% 25.17% Thus, the probability of total loss, assuming a flat 3% chance of a credit event is 1.06%, and the probability assuming a 50/50 chance of 0% or 6% is 12.58%. Question 3: Total 25 points Part A: [15 points] XYZ pays dividends twice a year at the end of June and December. It is currently the end of March and the next dividend (paid in June) is expected to be $1 per share. The following dividend (paid in December) is expected to be 10% higher. Dividend growth is then expected to decline linearly to 3% over the following 10 years. That is, the dividend in June next year will be somewhat less than 10% bigger than the December dividend. The growth in the next dividend will be smaller again, and so on. Once we get to the June dividend 10 years from now, growth thereafter will be 3% in perpetuity. (Note: This is 3% growth from one dividend to the next – it is not annual growth.) If the appropriate discount rate is 10%, what is the present value of this dividend stream? The present value of this series of cash flows is $101.88. See attached spreadsheet. - 4 -

Part B: [10 points] You are about to start a new job. You will be paid monthly and $2,500 per month will be paid into your retirement account. The first payment will be made one year from now. You expect this amount to increase by 3% on each annual anniversary of your employment. You plan to work for 40 years at which point you will retire. You expect the fund to generate a return of 6% p.a. (Note: This is an annual rate). During your retirement, you plan to make annual withdrawals at the beginning of each year where each withdrawal should have the same purchasing power as $200,000 today. Assume that inflation is expected to be 2.5% p.a.  At what point during your retirement will the funds be exhausted?  What is the most that you could afford to withdraw each year (expressed in current purchasing power terms like the $200,000 figure above) such that you would have sufficient funds to cover your expected 30-year retirement? See the attached spreadsheet. When withdrawals are $200,000 per year real, the funds will be sufficient to cover 17 full years (given the first withdrawal starts at t=0) and part of the amount for the end of the 17 th year. Use Goalseek to set the 30-year closing balance to 0, by changing cell N6. You can afford to withdraw $137,160 real each year. Alternatively, by hand: ( 1 + r mth ) = 1.06 Therefore, r mth = 0.487 % FV of annual contribution i.e. annual contribution at end of year 1 FV = 2500 [ ( 1.00487 ) 12 − 1 0.00487 ] = $ 30,816.32 Growing this annual contribution at 3%, we have a growing annuity PV = 30,816.32 0.06 − 0.03 [ 1 − ( 1.03 1.06 ) 40 ] = $ 701,438.58 Thus, the future value at retirement is FV = 701,438.58 × ( 1 + 6% ) 40 = $ 7,214,799.36 The future value of the first withdrawal, given that it grows with inflation is 200,000 × ( 1 + 2.5% ) 40 = 537,012.77 The first cashflow in the annuity formula (i.e. the second withdrawal) is - 5 -

537,012.77 × ( 1 + 2.5% ) = 550,438.09 At what point the funds will be exhausted requires solving for n in a growing annuity formula 7,214,799.36 − 537,012.77 = 550,438.09 0.06 − 0.025 [ 1 − ( 1.025 1.06 ) n ] Using logs, we get n = ln ( 0.5754 ) ln ( 0.9670 ) = 16.46 Reaching the same conclusion that the funds will be able to cover withdrawals up until the end of the 16 th year. To calculate the amount we could afford to draw each year to cover 30 years of withdrawals, need to solve for P 7,214,799.36 1.06 = P 0.06 − 0.25 [ 1 − ( 1.025 1.06 ) 31 ] Note we need to use n=31 to include the first payment that is paid at t=0 of retirement. So that the annuity formula includes this first cashflow, we need to discount the FV of total funds at retirement back by one period. The ‘P’ used here refers to the cashflow at t=0 Solving the above, we get P = 368,283.44 Discounting this back to present value, we have 368,283.44 1.025 40 = $ 137,160 This reconciles with the Goalseek excel method. - 6 -

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