Assignment Part 1: Household Saving and Investment Decisions
Problem 1:
i. Calculate the present value of your bequest of $100,000 in real dollars.
ii. Determine the constant value of consumption that equates the value of your initial wealth on the right side of the equation with the total of consumption and your bequest.
Hint: To do this, you need to recognize that the equation contains an annuity for future consumption. The bequest is a single lump sum value, and the initial wealth is already stated at present value. The present value of labour income is zero.
Use this information to determine
a. the present value of consumption for all 20 years.
b. use that value to determine the annual consumption for each of the 20 years. This part is very similar to the last question on Assignment 1.
Problem 2: Intertemporal Budget Constraint with Nominal Cash Flows
Using the same information as stated in 5.1 and adding the assumption that inflation is expected to be 3% per year over the next 20 years, translate all of the values (i.e., the bequest, the initial wealth, and the consumption, into nominal dollars (i.e., actual dollars rather than purchasing power).
The trickiest part is the fact that the consumption amount will no longer be constant, because inflation will require that you spend 3% more each year to maintain the same consumption. So, if you spent $1 last year to buy a chocolate bar, the same chocolate bar will cost you $1.03 at the end of year 1, $1.0609 at the end of year 2 (i.e., 1.0609 = 1.03 x 1.03 = 1.03^2), and so on.
i. The present value of your bequest will be the same as in part 5.1a, because the present value of the nominal amount must be the same as the real amount at time 0. But in 20 years, the real amount of $100,000 will not be the same as the nominal amount. Calculate the nominal amount you will need to have 20 years from now to satisfy the bequest for your kids.
ii. Calculate the actual amount of consumption, in nominal dollars, for each of the 20 years, using the stated assumptions. This requires you to calculate 20 numbers, so you might want to use an Excel spreadsheet, such as the one provided with this assignment (5.2 template).
Note: If you want to calculate the result algebraically, you could use the formula for the present value of a growing annuity. You can find the formula in thelesson notes at the end of Note 7 in Lesson 4. That formula shows the value of a $1 annuity, which grows to a value of (1+g) at the end of the first period, and then grows at a rate of g every year after that first period.
In your assignment solution, tell us the amount of consumption at the end of the first year and at the end of the 20th year. You can cut and paste the table from your Excel spreadsheet, or you can show your calculations algebraically.
iii. Many financial planners use real rates to plan for their clients' retirements. Given your recent experience with questions 5.1 and 5.2, why might they use real rates rather than nominal rates?
Problem 3:
i. Eighteen-year-old Linus is thinking about taking a five-year university degree. The degree will cost him $25,000 each year. After he's finished, he expects to make $50,000 per year for 10 years, $75,000 per year for another 10 years, and $100,000 per year for the final 10 years of his working career. All these values are stated in real dollars. Assume that Linus lives to be 100 and that real interest rates will stay at 5% per year throughout his life.
a. Calculate the present value of his lifetime earnings.
b. Calculate the present value of the cost of his schooling.
c. Subtract the present value of the schooling cost from his lifetime labour earnings to determine his human capital. Use that value to determine his permanent income, that is, the equal annual consumption Linus could enjoy over the rest of his life.
ii. Linus is also considering another option. If he takes a job at the local grocery store, his starting wage will be $40,000 per year, and he will get a 3% raise each year, in real terms, until he retires at the age of 53. Assume that Linus lives to be 100.
a. Calculate the present value of Linus's lifetime earnings, using a spreadsheet or using the growing annuity formula. You can find the formula in the lesson notes, at the end of Note 7 in Lesson 4.
b. Use that value to determine Linus's permanent income, i.e., how much can Linus spend each year equally over the rest of his life?
iii. Do you think Linus is better off choosing option a. or option b.? Consider both financial and non-financial measures.
Problem 4: In this question, we answer the age-old query, "Are we better off playing the lottery or saving the money?"
Assume you can buy one ticket for $5, draws are made monthly, and a winning ticket correctly matches six different numbers of a total of 49 possible numbers.
a. What would your $400.43 be worth if you invested it at an annual real interest rate of 1% (with monthly compounding)for 47 years?
b. If, instead, you deposited your five dollars per month in a bank at 1% real interest rate with monthly compounding, how much would you have at the end of the first year?
c. If you did this every year for 47 years at the 1% real interest rate with monthly compounding, how much would you have at age 65?
d. How much better off would you be keeping your money, rather than playing the lottery?
e. If, instead of 1%, you earned 5% real interest on your deposits, how much would you have at age 65?
Problem 5: You may use the Excel spreadsheet named LeasevsBuy.xls to answer the following question. If you choose to answer the question without using the spreadsheet, be careful to show all work, so your marker can follow your calculation and award partial marks as needed.
You want to acquire a new car, but you're not sure whether you should lease it or buy it. You can buy your chosen model for $50,000.You expect it to be worth $20,000 after you use it for three years. Alternatively, you could lease it for $650 per month for athree-year term, with the first payment due immediately. The lease company has not told you what interest rate they're using to calculate the monthly payments, but you know you could borrow money at an annual percentage rate (APR) of 8%.
a. Calculate the present value of the lease payments, assuming monthly compounding at the given APR of 8%.
b. Calculate the present value of the $20,000 salvage value, again using monthly compounding and the given APR of 8%. Deduct the salvage value from the purchase price to determine the present value of the cost of buying the vehicle.
c. Based on your calculations, which option do you prefer: lease or buy?
d. Calculate the salvage value at which you should be indifferent between leasing and buying.
e. Assume your tax rate is 40%. If you were to use this car 100% for business, rendering the lease payments tax-deductible, or alternatively, allowing you to deduct depreciation straight-line at $10,000 per year for three years, would you prefer to buy or lease the car? (Hint: Use the after-tax borrowing rate to discount the cash flows.)
Assignment Part 2: The Analysis of Investment Projects
You may use the Excel spreadsheet named Chapter6_template.xls to answer the following question. If you choose to answer the question without using the spreadsheet, be careful to show all work, so your marker can follow your calculation and award partial marks as needed.
To ensure that you know how the spreadsheet works, it is recommended that you replicate Table 6.5 (textbook p. 182). A completed spreadsheet for Table 6.5 is included with the template file as a separate worksheet, so you can check your work.
Question 1: You and your friends are thinking about starting a motorcycle company named Apple Valley Choppers. Your initial investment would be $500,000 for depreciable equipment, which should last five years, and your tax rate would be 40%. You could sell a chopper for $10,000, assuming your average variable cost per chopper is $3000, and assuming fixed costs, such as rent, utilities, and salaries, would be $200,000 per year.
a. Accounting breakeven: How many choppers would you have to sell fornet income to equal zero, ignoring the costs of financing?
b. Financial breakeven: How many choppers would you have to sell to generate NPV of zero, if you required a 15% return?
c. Assuming you could sell 60 choppers per year, what would be your IRR?
d. Assuming you could sell 60 choppers per year, what would your selling price have to be to generate a net present value of $150,000 at a 15% discount rate?
e. If you could sell 60 choppers in the first year, and your sales volume increased by 5% each year until the end of year five, what would the net present value be at a 15% discount rate?
f. If, at the beginning of each year, you expect to need working capital equal to 10% of the next (coming) year's sales revenue, what would be the effect on the net present value of the project? Only changes in the amount of working capital require cash flows. Assume a sales price of $10,000 per chopper and a sales quantity of 60 choppers.
g. Do you think that the need for working capital always reduces the net present value of projects? Can you think of circumstances where working capital could increase the NPV of a project? (Hint: Think of airline tickets purchased in advance.)
Question 2: Complete the following table. Assume that the real interest rate is 3% per year, and inflation is expected to be constant at 2% per year. Recall that nominal cash flows must be discounted using nominal rates, and real cash flows must be discounted using real rates.
|
Year
|
Nominal cash flow
|
Real cash flow
|
PV of Real CF
|
|
0
|
-100,000
|
-100,000
|
-100,000
|
|
1
|
12,000
|
11764.7
|
11,422.00
|
|
2
|
22,000
|
21,145.70
|
19,931.90
|
|
3
|
15,000
|
14,134.80
|
12,935.40
|
|
4
|
10,000
|
9,238.50
|
8,208.20
|
|
Net present value
|
|
|
-47,502.47
|
|
Discount rate
|
|
3%
|
|
Note that present values are taken at time 0, at which point real cash flows and nominal cash flows are the same, because there is no time for inflation to affect the cash flows.
Assignment Part 3: Principles of Market Valuation
Question 1: Find a website that shows exchange rates for all major international currencies. One such site is XE website. Another is oanda website.
a. What is the current exchange rate between the Canadian dollar and the US dollar?
b. XE.com allows you to see current exchange rates for gold ounces (type "gold" into the From or To box).
i. What is 1 ounce of gold worth in Canadian dollars?
ii. In US dollars?
iii. What does this imply is the exchange rate between Canadian dollars andUS dollars?
iv. Is this the same as your answer to part a?
c. If you saw that the US dollar price of gold was one dollar less than the price you quoted in b.ii., how could you use that information to make an arbitrage profit?
To get full marks, you have to show a full "round turn." That is, you need to start and end with the same currency to show the amount of your profit.
Question 2:
a. Complete the following table using XE.com or a similar foreign-currency quote website. In each cell, record the number of units of the currency stated in the column heading that you could buy with one unit of the currency shown at the start of the row. For example, if you sold US$1, how many European Euros could you buy? Enter that amount in the third column, second row. Continue until you have completed all sections of the table.
|
sell/buy
|
US $1
|
Euros €1
|
Canadian $1
|
Japanese ¥1
|
|
US $1
|
1
|
0.9026
|
1.31996
|
108.845
|
|
Euros €1
|
1.10795
|
1
|
1.46275
|
120.591
|
|
Canadian $1
|
0.757419
|
0.693645
|
1
|
82.4412
|
|
Japanese ¥1
|
0.00918729
|
0.00829226
|
0.0121297
|
1
|
Assignment Part 4: Valuation of Known Cash Flows: Bonds
Question 1:
a. Look up the U.S. Treasury yield curve online.
i. What is the promised yield for a one-month T-bill?
ii. For a six-month T-bill?
iii. A 20-year T-bond?
iv. A 30-year T-bond?
v. What date did you look up these yields?
vi. On what website did you find these yields?
b. Is this yield curve flat, rising, or inverted?
Question 2 The Law of One Price implies that financial instruments with the same risk and the same cash flows at the same time should have the same price.
You are given the following table containing incomplete information on four different bonds. Assume that all these bonds have the same risk, and any coupon payments are paid annually.
Note that you can find an optional webcastthat shows calculations for a similar example with three strip bonds and three-year coupon bonds. SeeAbout FNCE/ECON 300: Optional Webcasts: "Law of One Price" or link from the Lesson 8 Reading and Learning Objectives.
|
Bond #
|
1
|
2
|
3
|
4
|
|
|
1-year strip bond
|
2-year strip bond
|
2-year 6% coupon bond
|
2-year 7% coupon bond
|
|
Purchase price ($xxxx.xx)
|
-950
|
-902.5
|
-1009.23
|
-1027.69
|
|
Time 1 cash flow
|
1000
|
0
|
60
|
70
|
|
Time 2 cash flow
|
0
|
1000
|
1060
|
1070
|
|
Yield to maturity (xx.xx%)
|
5.26%
|
5.26%
|
5.50%
|
5.50%
|
a. What is the yield to maturity on Bond #1?
b. What is the price of Bond #3?
c. You are considering two investments from the bonds listed in the table.
Portfolio 1: 60 units of Bond #1 + 1060 units of Bond #2
Portfolio 2: 1000 units of Bond #3.
Show that the future cash flows from these two portfolios would be identical, in amount and timing.
d. Based on the information in the given table,
i. What would it cost to buy 1000 units of Bond #3?
ii. What would it cost to buy 60 units of Bond #1?
iii. From part c. above and your answers in part d.i and ii, infer the value of 1060 units of Bond #2.
iv. What is the value of one unit of Bond #2?
v. What is the implied yield of Bond #2?
e. How many units of Bond #1 and #2 would you need to replicate the future cash flows of 1000 units of Bond #4?
f. Using your answer to part eabove, determine the following
i. What's the value of 1000 units of Bond #4?
ii. What's the yield of Bond 4?
g. Fill in the missing information in the given table:
Question 3 Assume that you are thinking of buying a default-free bond. Specifically, you're thinking of buying a bond issued by Risklessco, a company thatis considered default-free by all major bond rating firms.You will select one of the following three bonds, which are identical except for the special features listed.
|
Bond
|
Face value
|
Maturity
|
Coupon rate (paid annually)
|
Yield to maturity*
|
Special features
|
Price
|
|
A
|
1000
|
20 years
|
5.50%
|
5%
|
None
|
?
|
|
B
|
1000
|
20 years
|
5.50%
|
5.50%
|
Callable
|
Par
|
|
C
|
1000
|
20 years
|
5.50%
|
3.50%
|
Callable and Convertible into Risklessco Stock
|
?
|
* Yield to maturity represents the market's required rate of return. It is calculated using only stated coupon payments and face value, without regard forthe special features.
a. Bond B is considered identical to Bond A except for the callability provision.
i. What would be the price of Bond A?
ii. What is the implied value of the callability provision?
iii. Who does the callability provision benefit: the issuer or the purchaser?
iv. Is this consistent with the price you calculated for Bond A relative to the price of Bond B? Briefly explain.
b. Bond C is considered identical to bond B except for the conversion privilege.
i. What would be the price of Bond C?
ii. What is the implied value of the conversion privilege?
iii. Does the conversion privilege benefit the issuer or the purchaser of the bond?
iv. Is this consistent with the price you calculated for Bond C relative to the price of Bond B? Briefly explain.