Calculate the actual amount of consumption
Assignment 2
Assignment 2 is due after you complete Lessons 5 to 8. It is worth 20% of your final grade.
Prepare your responses to these assignment problems in a word processing file; put financial data in a spreadsheet file. As you complete the assignment problems for each lesson, add your responses to these files.
Do not submit your answers for grading until you have completed all parts of Assignment 2.
Note: In assignments, show all calculations to 4 decimal places.
Spreadsheets are provided on the Assignment 2 page of the Assignment website to help you complete Assignment 2.
Lesson 5: Assignment Problems
5.1 Assume you have $1 million now, and you have just retired from your job. You expect to live for 20 years, and you want to have the same level of consumption (i.e., purchasing power) for each of these 20 years, after adjusting for inflation. You also wish to leave the purchasing power equivalent of $100,000 today to your kids at the end of the 20 years as a bequest (or to pay them to take care of you).
You expect inflation to be 3% per year for the next 20 years, and nominal interest rates are expected to stay around 8% per year
Calculate the actual amount of consumption, in nominal dollars, using the stated assumptions.
Total Amount = $1,000,000
Amount left for Kids in today’s equivalent = $100,000
Total Amount for Consumption = $900,000
$900,000 = Value/(.08-.03) * (1-(1.03/1.08)^20)
Value = $900,000/12.25 = $76,183.63
Yearly consumption (real) = $76,183.63
How much do you need for your kids?
=$100,000*(1+0.03) ^20
=$180,611.12
If you plan to consume $1.03 in year 1, how much will you need to have to keep the same real consumption in year 2? In year 10? In year 20?
Year 1 = 1.03^1 = $1.03
Year 2 = 1.03^2 = $1.06
Year 10 = 1.03^10 = $1.34
Year 20 = 1.03^20 = $1.81
iii. How much, in nominal dollars, will $1 of retirement funds earn in year 1? Year 2? Year 10? Year 20?
Year 1 = 1.08^1 = $1.08
Year 2 = 1.08^2 = $1.17
Year 10 = 1.08^10 = $2.16
Year 20 = 1.08^20 = $4.66
- In an Excel spreadsheet (or in a manual table), calculate the following:
- annual investment earnings for each year
In attached excel file.
- total savings after investment earnings for each year
In attached excel file.
- subtract annual consumption from total savings each year
In attached excel file.
- by trial and error, or with the Goal Seek command, determine the amount of consumption that will give you exactly $100,000, in today's purchasing power, at the end of 20 years
At the end of 20 years, the amount of consumption at $76,183.63 will equal $100,000 of purchasing power 20 years from now, or $180,611.
Hint: You will need to make your annual consumption column dependent on the inflation rate, your investment earnings will grow at the nominal rate, and the bequest of $100,000 will grow at the inflation rate.
- Do the calculation again using real rates, and setting inflation to equal 0. If you set up your Excel spreadsheet carefully, you should be able to set the inflation rate to equal 0 and enter the real rate of return as the investment earning rate.
Feel free to use a spreadsheet called “5.1 template” (link is provided on the Assignment 2 page) to help you answer this question.
- What is the amount of real consumption in year 1? In year 2? In year 10? In year 20?
In attached excel file. ($76,183.63)
- Show that this is consistent with your calculation using nominal rates.
In attached excel file.
Savings = $1,000,000
Consumption = $76,183.63
Closing balance at the end of 20 years = $100,000
iii. How much, in real dollars, does that leave for your kids?
In attached excel file.
$100,000
- Show that your bequest is consistent with the nominal rate results above.
(30 marks)
$100,000 (equivalent to $180,611 with inflation rate)
5.2 A. Linus is 18 years old now, and is thinking about taking a 5-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. If Linus lives to be 100, and if real interest rates stay at 5% per year throughout his life, what is the equal annual consumption he could enjoy until that date?
Number of years after university = 100 – 23
= 77 years
FV of investment (FV) = (25,000, 5 years, 5%)
= $138,140.78
PV of first 10 years (PV) = (50,000, 10 years, 5%)
= $386,086.75
PV of second 10 years (PV) = (PV(5%, 20, -1) – PV(5%, 10, -1)) * $75,000
= $355,535.6560
PV of third 10 years (PV) = (PV(5%, 30, -1) – PV(5%, 20, -1)) * $100,000
= $291,024.07
Total PV of salary = $1,032,646.47
After University Investment = $1,032,646.47 - $138,140.78
= $894,505.69
Annual Consumption = $894,505.69/PV(5%, 77 years, 1)
= $45,794.93
- 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, in real terms, each year until he retires at the age of 53. If Linus lives to be 100, what is the equal annual consumption he could enjoy until that date?
PV of salary = ($40,000/(0.05-0.03)) * (1-(1+0.03)/(1+0.05))^35
= $979,748.15
Annual Consumption = $979,748.15/PV(5%, 82 years, 1)
= $49,900.64
- From strictly a financial point of view, is Linus better off choosing option A or B?
From a strictly financial point of view, Linus is better off choosing option B, as his annual consumption is higher.
(10 marks)
5.3 Are you 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 6 different numbers of a total of 49 possible numbers.
The probabilities: In order to win, you must pick all the numbers correctly. Your number has a 1 in 49 chance of being correct. Your second number, a 1 in 48 chance, and so on. There are exactly 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520 ways to pick 6 numbers from 49 options.
But the order in which you pick them does not matter, so you actually have a few more ways to win. You can pick 6 different numbers in exactly 6 x 5 x 4 x 3 x 2 x 1 = 720 orders of choice. Any one of those orders would still win the lottery.
Putting this all together, your ticket has 720/10,068,347,520 = 1/13,983,816 chance of winning. This equates to a .000000071 percentage chance.
If you played one ticket every month from age 18 to age 65, you would have 47 x 12 = 564 plays. Your odds of not ever winning would be calculated using a binomial distribution to be .9999599568, meaning your chances of winning would be 1 – .9999599568 = .0000400432.
So, if the lottery winnings averaged $10 million over this time period, your expected return would be less than .0000400432 x $10 million = $400.43.
(It's less than $400.43 because your 564 plays are spread out over the next 47 years, so the present value of these future plays would be significantly less than if you were able to play all 564 immediately. The $400.43 assumes you play all 564 plays today, which makes it the highest possible expected value.)
REQUIRED:
- What would your $400.43 be worth if you invested it at 1% real interest for 47 years?
FV = (400.43, 1%, 47 years)
FV = $639.19
- If, instead, you wrote down your 6 numbers on a piece of paper, and deposited your $5 in a bank at 1% real interest, how much would you have at the end of the first year?
FV = $5*(1+0.01)
FV = $5.05
- If you did this every year for 47 years, how much would you have at age 65?
FV = ($5, 1%, 47 years)
FV = $298.13
- If you earned 5% real interest on your deposits, how much would you have at age 65?
FV = ($5, 5%, 47 years)
FV = $890.60
- Which option would make you better off at age 65? How many time better off?
(10 marks)
Bank = $890.60/$298.13
= 2.99 times better rate of return choosing to invest rather than buying the lottery.
5.4 Use the Excel spreadsheet named “LeasevsBuyCCA” (link is on the Assignment 2 page) to answer the following question. You may choose to answer the question without using the spreadsheet, but be very careful to show all work, so your marker can follow your calculation and award part marks as necessary.
You want to buy a new car, but you're not sure whether you should lease it or buy it. You can buy it for $50,000, and you expect that it will be worth $20,000 after you use it for 3 years. Alternatively, you could lease it for payments of $650 per month for the 3-year term, with the first payment due immediately. The lease company did not tell you what interest rate they're using to calculate the monthly payments, but you know you could borrow money from your banker at an annual percentage rate (APR) of 8%.
- Calculate the present value of the lease payments, assuming monthly compounding at the given APR of 8%.
PV = -$20,880.96
- Calculate the present value of the $20,000 salvage value, again using monthly compounding and the given APR of 8%. Which option do you prefer, lease or buy?
PV = -$13,373.95.
- Calculate the amount of the salvage value which would make you indifferent between leasing and buying.
= $50,000 – (-$20,880.96)
= $29,119.04
Monthly Rate = 0.08/12 = 0.0067
= (1+0.0067)^36
= 1.270237*$29,119.04
Amount of salvage value = $36,988.09
- If you were able to use this car 100% for business, rendering the lease payments tax-deductible, or alternatively, allowing you to deduct depreciation, and assuming your tax rate is 40%, would you prefer to buy or lease the car?
(10 marks)
Lease –
Annual Payment = $650 * 12
= $7,800
Tax @ 40% = $7,800 * 0.40
= $3,120
Purchase –
Annual Payment = $10,000
Tax @ 40% = $10,000 * 0.40
= $4,000
I would prefer to buy the car given the tax-deductible amount is higher than leasing the car.
Do not submit these questions for grading until you have completed all parts of Assignment 2, which is due after Lesson 8.
Lesson 6: Assignment Problems
You may find it helpful to use the Excel file named “Chapter 6 template” (link is on the Assignment 2 page) to answer the following questions. You may choose to answer the questions without using the spreadsheet, but be very careful to show all work, so your marker can follow your calculation and award part marks as necessary.
In order to ensure that you know how the spreadsheet works, it is recommended that you replicate table 6.5 from page 182 of your textbook before proceeding to answer the following questions. (Note that a completed spreadsheet for Table 6.5 is included with the Excel file as a separate worksheet, so you can check your work.)
6.1 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 5 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.
- Accounting breakeven: How many choppers would you have to sell to break even, ignoring the costs of financing?
=200,000/(10,000-3,000)
= 28.5714
Break even units = 29 units.
- Financial breakeven: How many choppers would you have to sell to break even, if you required a 15% return? (Hint: Use the 15% as the discount rate and calculate net present value. In Excel, you may want to use the Goal Seek command, or simply use trial and error to find the correct amount.)
In attached excel file (55 units).
- Assuming you could sell 60 choppers per year, what would be your IRR?
In attached excel file (IRR = 20.6897%).
- 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?
In attached excel file (Selling Price = $10,608.48).
- If you could sell 60 choppers in the first year, and your sales volume increased by 5% each year until the end of year 5, what would the net present value be at a 15% discount rate?
Year |
0 |
1 |
2 |
3 |
4 |
5 |
Investment |
-500,000.0000 | |||||
Revenue Impact |
0.0000 |
600,000.0000 |
630,000.0000 |
661,500.0000 |
694,575.0000 |
729,303.7500 |
Variable Cost Impact |
0.0000 |
-180,000.0000 |
-189,000.0000 |
-198,450.0000 |
-208,372.5000 |
-218,791.1250 |
Fixed Cost Impact |
0.0000 |
-200,000.0000 |
-200,000.0000 |
-200,000.0000 |
-200,000.0000 |
-200,000.0000 |
Income Impact Before tax |
0.0000 |
220,000.0000 |
241,000.0000 |
263,050.0000 |
286,202.5000 |
310,512.6250 |
Tax |
0.0000 |
-88,000.0000 |
-96,400.0000 |
-105,220.0000 |
-114,481.0000 |
-124,205.0500 |
Depreciation (SL) |
-100,000.0000 |
-100,000.0000 |
-100,000.0000 |
-100,000.0000 |
-100,000.0000 | |
Tax Shield on Dep'n |
40,000.0000 |
40,000.0000 |
40,000.0000 |
40,000.0000 |
40,000.0000 | |
Net Working Capital | ||||||
Salvage |
0.0000 |
0.0000 |
0.0000 |
0.0000 | ||
Sum (Undiscounted) |
-500,000.0000 |
172,000.0000 |
184,600.0000 |
197,830.0000 |
211,721.5000 |
226,307.5750 |
Discounted |
-500,000.0000 |
149,565.2174 |
139,584.1210 |
130,076.4363 |
121,052.4548 |
112,514.8613 |
Total Present Value |
152,793.0907 |
- If you need to invest 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? Do you think that working capital investments always reduce the net present value of projects?
(20 marks)
The inclusion of working capital means that net present value would decrease.
6.2 Fill in the missing items in the following table. Assume that the real interest rate is 3% per year, and inflation is expected to be constant at 2% per year.
Year |
Nominal cash flow |
Real cash flow |
0 |
–100,000 |
–100,000 |
1 |
+ 12,000 |
11,765 |
2 |
+22,000 |
21,569 |
3 |
+15,000 |
14,706 |
4 |
+10,000 |
9,804 |
Net present value |
-47,502 |
-46,079 |
(10 marks)
Year 1 Real Cash Flow =
12,000/(1+0.02)
= $11,764.7059
Year 2 Real Cash Flow =
22,000/(1+0.02)
= $21,568.6274
Year 3 Real Cash Flow =
15,000/(1+0.02)
= $14,705.8824
Year 4 Real Cash Flow =
10,000/(1+0.02)
= $9,803.9216
Do not submit these questions for grading until you have completed all parts of Assignment 2, which is due after Lesson 8.
Lesson 7: Assignment Problems
7.1 Find a Web site that shows exchange rates for all major international currencies. At the time of writing, xe.com and oanda.com are examples of such sites.
- Is the British Pound shown? If not, why not? (You might have to do some investigation online if you're not familiar with the history of European currency.)
Using xe.com, yes the GBP is shown.
- What is the exchange rate between the Canadian dollar and the US dollar?
The exchange rate between the Canadian dollar and US dollar is 1 CAN dollar is equal to 0.79935 US (or 1.25119 US to Canadian).
- xe.com also allows you to see exchange rates for gold ounces (under "more currencies available"). What does xe.com (or a similar site) say 1 ounce of gold is worth in Canadian dollars? In US dollars? What does this imply is the exchange rate between Canadian dollars in US dollars? Is this the same as your answer to part B?
One ounce of gold is worth $1,563.38 CAN dollars. One ounce of gold is worth $1,249.97 in US dollars. This conversion implies that the exchange rate between Canadian dollars and US dollars is almost exactly equal to the answer found in part B, as $1,249.97 US dollars is equal to $1,563.95 in Canadian dollars.
- If you saw that the US dollar price of gold was one dollar less than the price shown, how could you use that information to make an arbitrage profit?
Seeing that the price of gold is one dollar less than the price soon, I could use this information to purchase the gold at a lower price, and then selling at the price shown to earn a profit.
(10 marks)
7.2 A. Fill in the following table using xe.com or a similar foreign-currency quote Web site. In each cell, record the number of units of the currency stated in the first cell of the row that would be required to buy one unit of currency as stated in the column heading.
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 the table is filled completely.
sell/buy US $1 European €1 Canadian $1 Japanese ¥1
US $1 1 0.85861 1.25102 111.888
European €1 1.16468 1 1.45709 130.309
Canadian $1 0.79935 0.68634 1 89.4273
Japanese ¥1 0.00894 0.00767 0.01118 1
- Why are the numbers in the Japanese ¥ column so much higher than the numbers in the other rows?
The Japanese Yen is so inflated compared to the other currencies in the table because of Japan’s economic success has led to a steady appreciation of the Yen.
(10 marks)
Do not submit these questions for grading until you have completed all parts of Assignment 2, which is due after Lesson 8.
Lesson 8: Assignment Problems
8.1 A. Look up the US Treasury yield curve online.
- What is the promised yield for a 1-month T-bill? 1.00
- For a 6-month T-bill? 1.13
iii. A 1-year T-bill? 1.22
- A 5-year T-bond? 1.83
- A 10-year T-bond? 2.30
- A 20-year T-bond? 2.65
iiv. A 30-year T-bond? 2.89
iiiv. On what date did you look up these yields? July 28, 2017
- On what Web site did you find these yields?
https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield
B. Is this yield curve flat, rising, or inverted?
This yield curve is rising.
- Many introductory finance textbooks say, at the beginning of bond valuation problems, "Assume the yield curve is flat." Another way of putting this is "Assume the term structure of interest rates is flat." How would this assumption make the questions easier for students of introductory finance to solve?
For introductory finance students to “assume the yield curve is flat” reduces the confusion that may come if interest rates were to change sporadically in the years leading up to the maturity date. With a flat yield curve, interest rates remain constant throughout the years leading to the maturity date, and reduces the need to calculate multiple YTM rates in multiple years, had the curve suddenly began to rise or become inverted.
(10 marks)
8.2 Fill in the missing items in the following table, using the Law of One Price. Assume all these bonds have the same risk, the yield curve is flat, and any coupon payments are paid annually. (10 marks)
Bond # |
1 |
2 |
3 |
4 |
1-year |
2-year |
2-year 6% coupon bond |
2-year 7% coupon bond | |
Time 0 cash flow (i.e., Purchase Price for the bond) |
–950 |
-1,000/ (1+5.5%)^2 = -898.45 |
-(+60/(1+5.5%) +1,060/(1+5.5%)^2) =-1,009.36 |
-(+70/(1+5.5%) +1,070/(1+5.5%)^2) =-1,027.84 |
Time 1 cash flow |
+1000 |
0 |
+60 |
+70 |
Time 2 cash flow |
0 |
+1000 |
+1060 |
+1070 |
Yield |
(1000-950)/950 =0.0526 5.26% |
5.50% |
5.50% |
5.50% |
8.3 A. You are considering two investments from the bonds listed in question 8.2. Show that the cash flows from the following two investments would be identical.
- 60 units of Bond #1 + 1060 units of Bond #2, and
1000*60= 60,000
1000*1060 = 1,060,000
60,000 + 1,060,000 = 1,120,000
- 1000 units of Bond #3.
1120*1000 = 1,120,000
- How many units of Bond #1 and #2 would you need to replicate the cash flows of 1000 units of Bond #4?
For replicating bond 4, there is a requirement for $70 for first year investment and $1,070 for the second year. In other words, 70 units of bond 1 will provide $70 and buy 1,070 units of bond 2 will result in $1,070 in year 2. Bond 4 can be replicated.
- i. If the yield of Bond #3 is 5.5%, what would it cost to buy 1000 units of Bond #3?
1000 * $1,009.36 = $1,009,360
- What would it cost to buy 60 units of Bond #1?
60 * $950 = $57,000
iii. From part A. above, infer the value of 1060 units of Bond #2.
1,060 * $898.45 = $952,357
- What is the value of one unit of Bond #2? Yield of Bond #2?
$898.45, yield is 5.50%
- What's the value of 1000 units of Bond #4? Yield?
$1,027.84, yield is 5.50%
- What have you learned about the Law of One Price from questions 8.2 and 8.3?
As per the Law of One Price, bonds that are identical will have the same price. Bonds that have the same year to maturity with different coupon rates will have the same yield to maturity rate. Arbitrage opportunity will arise if the YTM is different that takes short position with lower YTM and take a long position with higher YTM. Part A calculation of YTM and bond price indicates that there are differences in the price but the payment during maturity remains the same and thus, YTM are equal.
(10 marks)
8.4 Assume the yield curve on "plain vanilla" default-free bonds is flat at 5%, and you are thinking of buying a default-free bond. Specifically, you're thinking of buying a bond issued by Risklessco, a company considered to be default-free by all major bond rating firms.
You will select one of the following three bonds, all identical except for the special features listed:
Face Value |
Maturity |
Coupon Rate (Paid Annually) |
Yield to Maturity |
Special Features |
Price | |
A |
1000 |
20 years |
5.5% |
5% |
None |
? |
B |
1000 |
20 years |
5.5% |
5% |
Callable |
Par |
C |
1000 |
20 years |
5.5% |
3.5% |
Callable and Convertible into Risklessco Stock |
? |
- Why is the yield on bonds A and B 5%? Why is the yield on bond C different?
Both bond A and B has a YTM of 5% indicating the growth in market demand. The investor will not make an investment in bond lower than 5% YTM. Bond C has lower YTM due to the incremental features that are callable and convertible feature motivated investor to purchase them resulting in pulling down the YTM of the bond.
- What would be the price of Bond A?
N = 20, I/Y (YTM) = 5%, PV =?, PMT = $55, FV = $1,000
PV = $1,062.31 (negative, as the priced amount to pay).
- If bond C is considered identical to bond B except for the conversion privilege, what is the value of the conversion privilege? Does the conversion privilege benefit the issuer of the bond or the purchaser? Is this consistent with the price you calculated for bond C?
Price of bond C =
N = 20, I/Y (YTM) = 3.5%, PV =?, PMT = $55, FV = $1,000
PV = 1,284.25
Price of bond A = $1,062.31.
$1,284.25 - $1,062.31
= $221.94 is the additional price paid for the callable and convertible features for bond C. The conversion privilege benefits the bondholder, as the bond is more attractive. This raises its price and lowers the yield to maturity. This is consistent with the price calculated for bond C because the price of bond C is $221.94 more than that of bond A. Bond C also has a lower yield to maturity of 3.5% compared to 5% of bond A and B.
- Who does the callability provision benefit, the issuer or the purchaser? Is this consistent with the price you calculated for bond A?
The callability provision benefits the issuer, as the bond is more attractive from an issuer standpoint. The callability benefit lowers the price of the bond, while it has a higher yield to maturity. This is consistent with the price calculated for bond A, because the price of bond A is $221.94 less than that of bond C, while bond A has a higher yield of maturity of 5% compared to 3.5% for bond C.
(10 marks)
Once you complete these questions, check to see that Assignment 2 is complete, and submit it for grading.