I. Basic Concepts (Ch. 1-4)
A. Real vs. Financial Assets (Ch 1.2)
Real assets (buildings, machinery, etc.) appear on the asset side of the balance sheet.
Financial assets (bonds, stocks) appear on both sides of the balance sheet. Creating a financial asset creates an offsetting liability.
Sector net worth is defined as the difference between sector assets and liabilities:
Sector Net Worth =
assetsSector Assets – liabilitie
sSector Liabilities. Balance Sheet for a Single Sector Sector Real Assets Sector Net Worth
Sector Financial Assets Sector Financial Liabilities
Since creating a financial asset creates an offsetting liability, the sum across the sectors of the (closed, private) economy yields:
ctors
se Sector Financial Assets = se
ctors Sector Financial Liabilities. Therefore, for the whole (closed, private sector) economy:Real Assets = Net Worth Δ Real Assets = Δ Net Worth
Investment = Savings.
B. Overview of Financial Assets – Bonds, Equities, Derivatives (Ch. 2)
Bonds are fixed payment securities.
- Failure to make payment results in bankruptcy for the bond issuer.
Equity is an ownership claim to the residual value of the firm after payments of bonds and other liabilities.
- Payments to equity holders are called dividends.
- Most types of equity confer voting rights in corporate decisions.
Variants on bonds and equity are numerous.
- e.g. preferred shares, like bonds, pay a fixed income and have no voting rights but, like equity, do not trigger bankruptcy if
payments are not made.
Primary vs. derivative assets:
- Bonds and equities are primary assets.
- Options, futures and forwards are derivative assets because their payoffs depend on the value of underlying primary assets.
C. Taxation (Ref. Ch. 4.1 & 23A)
Interest income is taxed like ordinary income.
Dividend income is given preferential treatment for risk taking. Capital gains income is given preferential treatment for risk taking.
D. Calculating Periodic Compound Interest (Ch. 4.2)
Suppose interest is compounded n ≥ 1 times per year so that the year is broken up into n periods of (equal) length 1/n years.
Let r denote the effective period net interest rate. The effective period gross interest rate is 1+r. The effective annual gross interest rate is (1+r)n.
Let re denote the effective annual net interest rate (EAR),
re = (1+ r)n – 1.
Investors are ultimately interested in effective rates.
Unfortunately, rates re are not quoted in the financial pages.
The rates that are usually quoted correspond to nr. Following the text, we refer to this as the Annual Percentage Rate (APR).
Let ra denote the APR: ra = nr.
If n = 1, then re = ra = r.
If n > 1, then re > ra > r.
e.g. ra = 0.1 and n = 12 re = (1 + 0.1/12)12 – 1 = 0.1047 > 0.1.
ra = 5 and n = 365 re = (1 + 5/365)365 – 1 = 142.461 >> 5.
For r and n “small”, re ≈ ra.
E. Continuous Compounding (Ch. 4.2)
The continuously compounded annual gross rate of interest is the limit of the periodic interest relationship as n → ∞:
1
lim (1
/ )
ne n a
r
r n
.To simplify the analysis, take the natural logarithm 1
ln(1
/ )
ln(1
) lim
a e nr n
r
n
.The limit is ill defined, as the numerator and denominator go to 0. Applying l'Hôpital's rule (the limit is given by the limit of the ratio of the derivative of the numerator to the derivative of the denominator):
2 2
(1
/ )
ln(1
) lim
lim
(1
/ )
a a a e n n a ar
r n n
r
r
r
r n
n
.Take the antilog to solve for re =
e
ra
1
. Continuous compounding yields a finite re.
e.g. ra = 0.1 and n → ∞. Then re = 0.105171.
ra = 5 and n → ∞. Then re = 147.41.
Note: For continuous compounding Ch 4.2 denotes ra = rcc. The
Black-Scholes formula uses the continuously compounded gross rate for T years, (1 + re)T= e . However, it is denoted eraT rT.
F. Nominal vs. Real Interest Rates (Ch. 4.1)
Real interest rates are usually not directly observed. However, we can infer real interest rates using nominal interest rates and inflation rates. i) Periodic Compounding
The ex post period real interest rate can be constructed as follows: Gross Real Rate ≡ Gross Nominal Rate / Gross Inflation Rate 1 + r = (1 + R)/(1 + i)
r = (R – i)/(1 + i) ≈ R – i, where
R is the period nominal net interest rate, and i is the period net inflation rate.
Note. The approximation r ≈ R - i overestimates the (discrete period) real rate for i > 0. It is reasonably accurate only when i is small. ii) Continuous Compounding
When interest is compounded continuously, R and i becomes infinitesimally small.
The ex post gross rate is derived (this analysis uses APRs, Ra and ia):
) ( / a a a a a r R i R i e e e e ra = Ra - ia.
Note. Ch. 4.2 uses the notation rcc = Rcc – icc.
G. Stock, Indexes, and Listings (Ch. 2.3-2.4, 3.1-3.3) i) Primary vs. Secondary Markets
New securities are issued in primary markets (e.g. IPOs), whereas trade of already-issued securities takes place in secondary markets. Secondary markets include:
- formal stock exchanges; e.g. TSX, NYSE, Nikkei. - less formal over-the-counter markets; e.g. NASDAQ. ii) Market Indexes
Market-valued indexes are directly related to the total value of stock on the exchange, e.g. S&P/TSX Composite, S&P/TSX 60. The Dow Jones is an archaic price-weighted index.
iii) Stock Listings e.g. from the Globe and Mail website:
Canadian Tire Corporation CTC.A-T Updated: Dec 31, 2014 4:00PM EST Close: C$ 122.74 Net Change: C$ .67 % Change: .55%
Open 122.46 Beta 0.53
High 123.13 Trailing P/E 16.39
Low 121.51 P/E 1-year forward 16.27 Bid/Ask 122.70/123.05 Forward PEG 1.99
Volume 89,934 Indicated Annual Div. 2.10 52-Week Range 93.20 to 130.36 Dividend Yield 1.71 “Over the last five days, shares have gained 0.30% and outperformed the S&P TSX index over the last 52 weeks with a 23.37% increase in price.”
Exercise. The full listing also gives the “Previous Close” and “Trailing EPS”. Calculate these values from the above.
Exercise. Canadian Tire Co. also lists CTC-T. It closed at $250. Why are they priced so differently?
H. Buying Stock and Holding Period Rate of Return
Time Action Cash Flow
0 Buy share - P0
1 Receive dividend,
sell share
P1 + Dividend
Buying stock at time 0 and selling it at time 1 yields: Accounting Profit = (P1 + Dividend) - P0.
The one-period holding period rate of return (HPR1) is the accounting
profit over the initial investment. For stocks:
1 0 1 0 (P Dividend) P HPR P .
The t-period HPR depends on how the dividends are reinvested. If they are reinvested at the going interest rate then
0 0 ( t t) t P CD P HPR P ,
where CDt is the future value of the cumulative dividends.
Exercise. How would you calculate the HPR on CTC.A-T from the close at the end of 2013 to the close at the end of 2014? What
information would you need? Estimate HPR. What is the return expressed as an annual rate? Ignore taxes assuming investment in RRSP. For the dividend history (and share reinvestment plan) see:
I. Trading with Margin (Ch. 3.5)
Trading with margin involves some form of borrowing from a broker. This practice provides the investor with “leverage.”
i) Buying Stock on Margin
Buying stock on margin involves borrowing part of a stock purchase from a broker.
The margin ratio is:
MarketValueofAssets LoanDebt M
MarketValueofAssets
.
The maintenance margin is denoted by M. If M < M, the broker issues a margin call.
When a margin call is issued, the investor must add cash to their
account or else the broker will sell some of the stock to satisfy M > M. Exercise. You’re bullish on ABC stock and want to use margin to maximize your leverage. You have $10,000, the stock price is
currently P0 = 10 per share, the “initial margin requirement” is 50% of
the market value of assets (i.e. MI = .5), M = 0.3, and the effective borrowing rate is re = .1052. ABC does not pay dividends.
a) How much stock can you initially buy?
b) What is HPR1 if P1 = 12.5 at the end of the year?
c) What is HPR1 if P1 = 7.5?
d) What is HPR1 if P1 = 10?
e) What is the Pt at which you receive a margin call at an
Proposition. In the long run, to avoid a margin call on a long position in a stock that pays not dividends, the stock price must rise at least at the rate of interest.
Demonstration. The margin ratio at t is:
(1 )t e t t P N Loan r M P N .
Let Pt = P0(1+p)tand Loan = θ(P0N), where p is the geometric
average rate of return on the stock and θ is the proportion of the original market value that was borrowed.
Substituting yields: 0 0 . [(1 ) (1 ) ] (1 ) 1 1 1 t t e t t e P N p r M P N p r M p
If re < p, then M increases over time and converges to 1.
If re = p, then M is constant.
If re > p, then M decreases and eventually hits zero.
To avoid a margin call in the long run, re ≤ p.
Exercise. In each case (re < p, re = p, re > p), determine whether the
“average” HPRt on the stock is above or below re and p; i.e. compare
ii) Short Sale of Stock
Short-selling involves borrowing the stock and immediately selling it in the hope that the stock can be repurchased cheaply in the future to repay the loan and earn a profit.
Time Action Cash Flow
0 Borrow share; sell it. P0
1 Buy share and repay the dividend to replace the share originally borrowed.
- (P1 + Dividend)
Proceeds from short sale are kept with a broker and earn interest rb.
Often on small accounts, brokers do not pay interest, so rb = 0.
Accounting Profit = P0(1+ rb) - (P1 + Dividend).
The borrower must put up other assets as collateral (e.g. cash, bonds, stock) in a margin account and satisfy M s M s . Unlike equation (3.2) in the text which omits the “–1” term, we calculate margin as follows:
1 S MarketValueofAssets
M
ValueofStockOwed
.
Exercise. You’re bearish on ABC stock and want to short sell the stock using maximum leverage. You have $10,000 cash in a margin account, the stock price is P0 = 10, the initial margin requirement is
50% of the value of the short position, M s = .3, the lending rate is re=.1052, and rb = 0.
f) How much stock can you short initially?
g) What is HPR1 if P1 = 12.5 at the end of the year?
h) What is HPR1 if P1 = 7.5?
i) What is the Pt at which you receive a margin call for an