Homework 1 Solutions Foundations of Finance Laarits PDF

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These are the answers to problem set 1 (homework 1) for Toomas Laarits Foundations of Finance course. They include the answer and explanation....

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Foundations of Finance HW #1 Solutions Prof. Toomas Laarits Last Edit: February 28, 2022 1. 1 point. (a) The price of the bond is: 1000 × 1.02−10 = 820.35 . (b) We have to solve T in 1000 × 1.02−T = 375.44. We can solve it as: 1000 × 1.02−T = 375.44 1.02−T = 0.37544 −T log {1.02} = log {0.37544} log {0.37544} T = − log {1.02} T = 49.47 where the third equation follows from the second by using the property that log {xy } = y log {x}. This property holds for both the 10-log and the natural log (often denoted “ln”), so both can be used to solve this problem. 2. 1 point. (a) The price of the consol bond is its present value: P V1 =

C $100 = $2500.00 = 0.04 r

(b) One year later, the bond is still a perpetuity, so its new price is given by: P V2 =

C $100 = $4000.00 = 0.025 r

(c) The Holding Period Return takes into account both the cash-flows from the securities (the coupon payment) and the resale price: HPR =

$4000.00 + $100 P2 + C − 1 = 0.64 = 64% −1= $2500.00 P1

3. 2 points. To determine whether it is better to get for free asset (a) or (b), we must calculate the respective present values. It is always helpful to show a time line of the cash flows (to simplify the picture I omit the 000s in 10,000). 1

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Asset (b):

Cash flow (a) is just an annuity and so present value is given by:   1 1 PV = C − r r(1 + r)t Given the values of C = $10, 000, t = 10, and r = 6%, the present value is $73, 600.87. If instead r = 13% the PV is 54,262.43. Cash flow (b) is a perpetuity that begins 10 years from now. Just like in the lecture notes, we can value it in two steps. Firstly, we know that in period 10, asset (b) will have a present value of: P V10 =

C r

Thus, with C = $10, 000 and r = 6% the perpetuity is worth 166,666.67 at t = 10. The present value of the asset is given by discounting that value back to the present. Therefore: $166, 666.67 = 93, 065.80 PV = (1.06)10 If interest rates are instead 13% the present value of the perpetuity at t = 10 is $76, 923.08 and discounting back by ten years to today gives: PV =

$76, 923.08 = 22, 660.64 (1.13)10

If interest rates are r = 6% we would prefer asset (b), the perpetuity with a delayed start date. If interest rates are r = 13% we would instead prefer asset (a), the annuity. This result stems from the fact that even though the perpetuity has infinitely many more payments compared to the annuity, those cash flows begin with a ten year delay. At a 13% interest rate these delayed cash flows are penalized heavily, and we instead prefer the annuity. 2

4. 1 point. (a) Allowing you to reinvest every day means that your account balance is compounded every day. The future value formula for an account compounded n times per year is:  r n FV = PV 1 + n where n is the number of compounding periods per year and r/n is the rate per compounding period. Note that the problem indicates that the per period rate is .75% and the number of periods is 252. Therefore the future value will be: F V = $1000 × (1 + .0075)252 = $657.29 The annual holding period return from the investment is therefore: $657.29 $100 =⇒ HPR =557.29% 1 + HPR =

(b) If the fund manager doesn’t allow you to reinvest every day, the amount invested in the investment stays at 100 after the end of each day. Additonally, $100×.0075 = $.75 is deposited into a savings account. By the end of the year you will still have $100 invested in the investment account and 252 × $.75 = $189 in the deposit account. The total value at the end of the year is therefore: $100 + $189 = $289. The annual holding period return from the investment is therefore: $289 $100 =⇒ HPR =189% 1 + HPR =

This example illustrates the power of compounding. .75% per day is a fanastic return, but it’s even better if you’re allowed to reinvest the money every period. 5. 2 points. (a) The annual rate of return is the rate r that solves: $775 =

$1000 (1 + r)3

Solving for r gives: r=



$1000 $775

1/3 3

− 1 = 0.0887 = 8.87%

(b) Recall that the IRR is defined as the interest rate that makes the present value of the payments equal to the price. A bond with 1000 face value and 9% annual coupon makes a 90 coupon payment every year. At maturity it also pays back the face value of 1000. Hence, the IRR solves: $850 =

$90 $1090 $90 + + (1 + I RR) (1 + I RR)2 (1 + I RR)3

You can solve this equation using Excel Solver or Yield function, or with a financial calculator. The calculation will yield in I RR = 15.64% which is larger than the rate found in part (a). Another way to see the result is to plug in the rate r from part (a) to the equation that defines the I RR. Plugging in I RR = 8.87%, we find: $90 $1090 $90 + + = $1003.30 2 1.0887 1.0887 1.08873 To make the present value equal to the price of 850, the IRR must be greater than 8.87%. 6. Excel Question. 3 points. (a) First, we need to calculate the annual HPRs. HPR is defined as: HPR + 1 =

Pt+1 + Dt+1 Pt

where Pt+1 is the price of the SP500 at the end of year t + 1 and Dt+1 are the dividends paid out over the year (we are assuming all the dividends are paid out all at once right at the end of the year). See spreadsheet on the course website for calculations. (b) Our best guess of next year’s HPR is given by the arithmetic average return as it describes what an average year looks like. The average return in the current sample is E[HPR] = 11.90%. This is best calculated in Excel using the AVERAGE function. (c) This replicates the calculation we did for GameStock Inc in the lecture notes. Recall that in the lecture notes we started with 1000 and calculated the ending value of an investment with all the interim dividends reinvested. Each annual HPR we have calculated in (a) captures the return from a year of investing, reflecting both the capital gains, and cash-flows. To find the total return for a long horizon investment we need to calculate how one dollar invested at the end of 1960 would have grown by the end of 2021. Denote by Vt the value of the portfolio with dividends reinvested at year t. Like we saw in the lecture notes (Multiple Period Returns slides), we just need to multiply together the individual period HPRs: HPR1960,2021 + 1 =

V2021 = (1 + HPR1961 )(1 + HPR1962 )...(1 + HPR2021 ) V1960 4

Another way to see this calcualtion is to note that V1961 = V1960 × (1 + HPR1961 ) =⇒ V1962 = V1960 × (1 + HPR1961 )(1 + HPR1962 ) and so on. You can also work through the GameStock example to see how we calculated Vt there. To find the annualized rate we need to take the above expression to the 1/T power where T is the number of years in the period in question: HPRA,1960,2021 + 1 = [(1 + HPR1961 )(1 + HPR1962 )...(1 + HPR2021 )]1/61 which yields in HPRA,1960,2021 = 10.60% Note that the last formula is equivalent to the geometric mean of the individual HPRs. Therefore you can use the GEOMEAN function to calculate the geometric average of the whole sample. But in the Excel sheet I also show a calculation without the use of GEOMEAN. (d) The lowest HPR was in 2008 that saw HPR of -36.55%. This was the first year of the Global Financial Crisis and saw the failure of Lehman Brothers on September 15th. (e) Here we are looking for the total return over a five year investment. Just like in part (c), this requires us to apply the formula for HPR of a multiperiod investment: HPR2008,2013 = (1 + HPR2009 ) × (1 + HPR2010 )... × (1 + HPR2013 ) − 1 = 125.84% Note the notation convention: we are using HPR2009 to denote the HPR for an investment that bought at the closing price of 2008, and held all the way to the end of 2009 (and includes any dividends paid out over the calendar year). For that reason the first HPR we need to include in the calculation is HPR2009 as it captures the first calendar year after the worst year on record. (f) We already found in (e) that the five year HPR after 2008 was 125.84%. Just like in part (c), to find the annualized rate we need to take this to the 1/T power, where T is the number of years in the period: 1

HPRA,2008,2013 = [(1 + HPR2009 ) × (1 + HPR2010 )... × (1 + HPR2013 )]5 −1 = 17.70% Again, you can use the GEOMEAN function in Excel for this calculation.

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