Sketch Answers to Economics 3215: Assignment 1
Due: February 2014
1. (a) (i) Say that a financial asset promises to pay $600 per year for six years. If the yield on other similar assets is currently 10% what would you expect the price of this asset to be? Explain and show your calculations.
Price =600 + 600+ 600 + 600+ 600 +600 set i=.10 (10%)
(1+i) (1+i)2 (1+i)3 (1+i)4 (1+i)5 (1+i)6
= $2613.16
(ii) Say that instead of your answer the asset in (i) was currently being sold at a price of $2500. Show how you would find its yield? Given that similar assets pay 10% use a supply-demand story to explain what will happen to this asset.
Solve the following for i:
$2500 = 600 + 600+ 600 + 600+ 600 +600 solve for i=.1153 (11.53%)
(1+i) (1+i)2 (1+i)3 (1+i)4 (1+i)5 (1+i)6
If similar assets yield 10% lenders will buy this asset since it yields 11.53% instead of investing in the assets yielding 10%. The lenders curve (demand curve) for the high-yield asset shifts right leading to a rise in its price and so a fall in its yield. Demand for the 10% assets shifts left leading to a fall in its price (and so a rise in its yield). This continues until the yields on the two assets are the same.
2. Say that you borrow $150,000 from a bank and the average annual yield on the loan is 6%.
(a) If it is repaid in a single installment in 10 years time the installment is A in:
150,000 = A/(1.06)10 gives A = $268,627.20
(b) Repaid in 10 equal installments:
150,000 = A + A + A + A + A + A + A+ A + A + A
(1.06) (1.06)2 (1.06)3 (1.06)4 (1.06)5 (1.06)6 (1.06)7 (1.06)8 (1.06)9 (1.06)10
solving for A gives A= $20,380.19 as the size of the installments.
3.The following table summarizes the results of the January 16th Treasury Bill auction (each row is for
a set of T-Bills with different maturity dates):
Amount soldIssue dateMaturity dateYield (%) Price
$6,200,000,0002014.01.162014.04.24 ? 99.76295
$2,400,000,0002014.01.162014.07.17 ? 99.54681
$2,400,000,0002014.01.162015.01.15 0.952% ?
(i) What is the term-to-maturity ((in days) of each of the three types of T-Bills?
days to maturity:98, 182 and 364 days (so m=365/98, 365/182 and 365/364 in (ii)).
(ii) Use your answers in (i) and the "discount yield" formula to complete the table (show your work).
Price = 100/(1+iA/m) m=365/(days to maturity)
substitute for "m" and use data in the table.
Yield on 98-day bill: .00885 (0.885%), Yield on 182-day bill: .00913 (0.913%)
Price of 364-day bill is: 99.0568
(iii) Now complete the table using the yield-to-maturity formula (show your work). How close are the
two sets of results? Use:
Price = 100/(1+i)1/m
Yield on 98-day: .00888 (0.888%), Yield on 182-day bill: .00915 (0.915%)
3.(a) (i) Terms issued in 2012: 2, 3, 5, 10 and 30 years.
(ii) Most common? 2 year. (iii) Highest yield: 3.179% (30 yr bond issued, Nov. 13. Lowest yield:
0.99% (2 yr bond issued April24 )
(b) If the price was exactly 100 (same as value at maturity rate) the yield and the coupon rate would be
the same (a few bonds in the table come close to this). If the price is greater than the maturity
value of 100 this reduces the return and yield<coupon rate while if price<100m the difference is
an extra source of return and yield>coupon rate.
(c) The 5year bond has a coupon rate of 1.250 so it makes semi-annual coupon payments equal to the coupon rate divided by 2 (.625) and makes a final payment of 100 at maturity. From the overheads (D is the price, m is the number of payments per year (2 here), A is the coupon payment (1.250/2=.625 here), M is the face value (M=100 by convention), N is the number of years to maturity (the bond is issued August, 2013 and matures Sept. 1, 2018 so set N=5 This means you will receive 10 coupon payments (A) of .625 -- every six months -- and then receive the maturity value of 100). iA is the yield (.01957 here):
D = A + A + A + . . . + A + M
(1+ iA/m) (1+ iA/m)2 (1+ iA/m)3 (1+iA/m)Nm (1+ iA/m)Nm
How does it compare to the price actually paid? It is quite close (with Nm=10 I get D=99.5 vs. the actual 99.6.
4. German 10-year bond yield is 2% vs. 34% for a Greek 10-year bond. Say each bond promises a single payment of 20,000 euros at maturity.
(a) What would be the prices of the two bonds? The key equation (abbreviated) is:
Price = 20000 i=.02 for German and .34 for Greece. (1+i)10
German bond price: 16,406.97euros Greek bond price: 1071.48euros
(b) One way to answer this is to assume that the very high 34% yield is just enough to make the expected yield on the Greek bond equal that of the German bond (2%). Using the price of 1071.48 from (a) and iexpected=.02 you can work out what the probability of default might be. Here is the equation for the expected yield where p=probability of default (so (1-p) is probability payment is made):
1071.48= (1-p) 10000
(1.02)10
solving for p gives: p=.9347 (93.47%). (Could get the same result by arguing that:
(1-p)=(Price of Greek Bond)/(Price of German Bond)
assuming both have expected yields of 2%.
(c) The yields two years later (January 2014) are about: 1.8% for the German and 7.9% for the Greek 10-year bond. This implies that the default probability has fallen. The price of the Greek bond would now be: 9350.09euros while the German bond costs: 16732.17 euros. Implying p=.441 if expected yield on both is 1.8%.
5. (a)Text question 15 p. 78. The $85.5 million is just the promised $2.85 million annual payment times 30 -- this calculation makes no adjustment for the fact that payments received in the future are worth less than payments made now. Instead of reporting $85.5 million it would be more accurate to report the present value of 30 annual payments of $2.85 million each.
Whether you should choose the $46 million now or 30 annual payments of $2.85 million depends on the discount rate you use to calculate the present value of 30 $2.85 million payments -- a good proxy for this is your estimate of a reasonable return on a long-term investment . If you calculate the present value you will find that choosing the $46 million now is a better deal at high discount rates (if you think rates of return are going to be high then you can invest the $46 million received now and get more than $2.85 million per year for 30 years). At low discount rates you are better off taking the 30 annual payments. By my calculations you would be indifferent between the two if the discount rate is 5.134%.
(b) Text question 4 p. 104. You are told the return is 30% with probability .2, 12% with probability .7 and -15% with probability of .1.
(i) The expected value of a $1000 investment is:
.2 x (1000x1.3)+ .7x(1000x.1.12) + .1x(1000x.85) = $1129 so the return is 12.9%
((1129/1000) -1)
You could also get the return by calculating the probability weighted returns:
30%x.2 + 12%x.7 + (-15%)x.1 = 12.9%
(ii) Standard deviation of the return:
variance = .2x(30%-12.9%)2+.7x(12%-12.9%)2+.1x(-15%-12.9%)2 = 136.89
std. deviation = (square root of variance) = 11.7
(iii) If the return on a risk free asset if 7% then the risk premium is (12.9%-7%)=5.9%.
(c) Q. 13 p. 105. Considers whether dividing a $1000 investment between two assets whose returns are completely independent of one another and have the same expected value and std. deviation. Investing in these two assets will spread risk and so lower the variance and standard deviation of the returns. Intuitively to have a very high or very low return will require both assets to do very well or very badly simultaneously -- given that returns on the two assets are independent these extreme outcomes will be unlikely. In terms of the formula for the variance of a portfolio (see overheads or text p.100):
var(ax+by) = a2var(x) + b2var(y) + 2ab cov(x,y)
a, b are the shares of the investment in asset x and asset y. Since the assets are independent cov(x,y)=0 and you are also told that std. deviations are the same so: var(x)=var(y). Then:
var(ax+by)=(a2 + b2)var(y)
but since 0<a<1 and b=1-a (they are shares of the investment in each asset):
(a2 + b2)var(y) < var(y)
so diversifying gives lower risk than holding only asset y.
6. (a) Go to the Bank of Canada website: The index page gives access to the latest Bank of Canada statement, reports and new releases. The column of “Key Indicators” on the right hand side of the page gives the value of the current inflation rate (the Bank’s target variable) as well as its current target for the overnight rate. Click on “Target for the Overnight Rate”. Read through the resulting page and answer the following:
(i) What exactly is the overnight rate? (see website)
(ii) How has the overnight rate changed over the past year? (see website)
(b) Go back to the index page. At the bottom of the “Key Indicators” section of the page you will see “Daily Digest”. Click on it. In addition to data on exchange rates the resulting table reports a variety of recent interest rates (yields). Reproduce the table and based on it answer the following:
(i) Why is the reported yield on the long-term real return bond so much lower than the other rates reported including that for Government of Canada long-term bonds +10 yr.?
The real return bond is a real yield, i.e. it is unaffected by inflation. The yield on
Government of Canada long-term bonds +10yr. is a nominal yield and so part of it will be
compensation for expected inflation. This inflation premium explains why the nominal yield is higher.
(ii) How close is the actual overnight rate to the Bank of Canada’s target rate? Very close!
(iii) What is the difference between the yield on 1 month corporate paper and a 1 month Treasury Bill? Which is higher? Why is this likely the case?
Corporate paper was higher. This is likely a risk premium.
(iv) Which are the highest rates reported in the Daily Digest? How much higher are they than the overnight rate? See data sources (highest: Conventional 5 yr. mortgage, Prime Business rate, Lowest: 1 month T-Bill and the real return bond).
7. Statistics Canada’s CANSIM database. Go to the library webpage and choose “CANSIM II via E-STAT”. Click “Accept and Enter” (if you are accessing the database from off-campus you will first have to log onto the Proxy server: you will receive instructions from the webpage). On the next page Click “Search CANSIM”. In the resulting search box type in ‘yields’ and click search. Pick the table providing monthly financial statistics (click the table number). In the rates box on the next page select (by holding down CTRL while clicking the relevant data series names):
Chartered bank consumer loan rate
Government of Canada 5 year bond yield
Chartered bank 5 year conventional mortgage
Chartered bank 5 year GIC
Prime corporate paper rate (3-month)
Treasury bill (3-month) (average yield)
Set the time period as: January 1981 to the most recent period available. Click the “Retrieve as a Table” button at the bottom of the page. You will now be asked about format. It is probably easiest to choose worksheet with time as rows (but suit yourself). Once you have chosen your format click the “Retrieve Now” button. If the download does not start click on “Click here to fetch results”. Open and save the resulting file. Note that these series are reported to two decimal places (you may have to change the display format in the spreadsheet to see this).
(a) Plot the resulting yields over time. Do the various yields and interest rates tend to move together? Generally yes.
(b) What is the highest yield (rate) for any asset in the time period? What is the lowest?
Some of the short-term rates are over 20% others are under 2%.
(c) Which asset consistently has the highest yield? Which ones have the lowest?
Consumer loan rate is typically the highest.
T-Bill is often the lowest but not consistently (when the yield curve slopes down
it is usually one of the highest).
(d) Prime Business rate and Long-term Government of Canada bond 1955-present. 1955-81 yields began rising in the 1970s to a peak in the early 1980s.
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