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1.3 The risk and return from conventional assets

[ SLIDE saving - deposit - funds - bonds - stock || VIDEO modem ( 1/2) - LAN 1/2) - DSL ]


Before we look into more advanced securities called derivatives because their value can be derived from others, it is useful to review some of the conventional assets held in a portfolio.

* Bank savings account.
Investors who want keep the possibility to quickly withdraw a limited amount of cash usually make a deposit in a bank savings account (for example, check UBS, Handelsbanken, Deutsche Bank). Depending on the total amount invested and the seasonal variations in the interest rates, deposits are rewarded with 0-2% interest excluding fees and taxes. The bank will of course invest the money further for its own profit, but tough regulations ensure that the risk of a bank defaulting on savings account is tiny and the governments often protect deposits to an upper limit around EUR 50,000.

* Bank certificates of deposit.
Investors willing to lock up their money for a couple of years until a certificate of deposit reaches the maturity date can expect larger returns around 3-5% (for example, check UBS). For this first type of longer term investment, it becomes important to distinguish the simply compounded annual percentage rate (APR) from the discretely compounded annual percentage yield (APY) that includes the interest on interest rates.



Masters: simple, discrete and continuous compounding of interest rates
Consider an amount A   invested for an annual interest rate R during n   years. If the money earned once a year is not reinvested, the terminal value from a simply compounded calculation leads to

$\displaystyle W=\underbrace{AR_s}_\textrm{1st year} + \underbrace{AR_s}_\textrm{2nd year} + \hdots + \underbrace{AR_s}_\textrm{n-th year}+A = A(1+nR_s)$ (1.3#eq.1)


If the money is compounded m   times a year and immediately re-invested at the same rate, the terminal value from a discretely compounded calculation becomes

$\displaystyle W=A\left(1+\frac{R_m}{m}\right)\times\left(1+\frac{R_m}{m}\right)... ...\hdots \times \left(1+\frac{R_m}{m}\right) = A\left(1+\frac{R_m}{m}\right)^{mn}$ (1.3#eq.2)

Increasing the compounding frequency to infinity mܥ,   the terminal value from the continuously compounded calculation tends to

$\displaystyle W=A\exp(Yn)$ (1.3#eq.3)

where the yield Y   can be understood as the annual growth rate of the investment. Continuous compounding is often used for simplicity instead of the more realistic discrete compounding; both are in any case simply connected via

$\displaystyle R_m=m(\exp[Y/m]-1), \hspace{1cm} R=\exp[Y]-1$ (1.3#eq.4)

where Rm is the discretely and R   the continuously compounded annual rate.


For example, take a 1% monthly rate of a unit investment: depending on the compounding, this translates into WAPR=1.01 x 12=1.12,   WAPY=1.0112=1.126825,   which approaches the continuous annual value of W=exp(0.01 x 12)=1.1275.   In other words, the return has an APR of 12% and an APY of 12.68%. Brokerage houses often insure a single certificate of deposit for up to EUR 500,000. The main disadvantage of a certificate of deposit is that it locks up the money for a long time unless a steep penalty is payed... up to half of the return from the interest rate!

* Money market funds.
With similar interest rates, around 3-5%, and no maturity constraints, money market funds are only slightly more risky (for example, check CS First Boston, SEB, Credit Lyonnais). A fund manager collects money from a pool of investors and distributes it into a large number of bonds to spread out the risk of defaulting on a debt. Good managers will pick bonds with high return to risk ratios, for a managing commission of up to one fifth of the fund's return, which is directly deducted from the investors profit. Specialists argue that no money market fund has ``broken the buck'' (i.e. returned less than the original investment) in the last 15 years; it is not easy to verify even such a strong statement, but our readers from around the world may want to comment in our World Forum.

* Mutual and hedge funds.
By combining holdings in cash, bonds and stock, fund managers can produce larger returns with an increased amount of risk. Every day, the manager counts up the value of all the fund's holdings and, by dividing by the number of shares that have been purchased by the investors, calculates the Net Asset Value (NAV) per share of that fund. New investors send their money to the fund manager, who will issue new shares from that fund for the latest quoted value. Holdings are continuously sold and reinvested, which is why mutual funds are sometimes called open-end funds. If the fund manager is doing a good job, the net asset value increases and the investors make a profit when they eventually sell their shares. Nevertheless, management commission of around 1-2% of the NAV eat away a considerable fraction of the average 5-10% growth that can be expected in the long term.

A large variety of funds pursue different investment strategies (countries, industries, risk levels, ethical factors): on Feb 23, 2002, the Financial Times newspaper listed more than seven pages with funds... more than shares! In this context, it is good to remember that market indices (such as FTSE-100, Russel 1000, NASDAQ-100) are by definition an arithmetic average; since funds now represent a larger fraction of the market, it is clear that roughly half of the funds under-perform that index. Pursuing a variety of often contradicting strategies can nevertheless be exploited by the marketing departments of the management firms, who simply highlight even few funds that outperformed the index to advertise the skills of all the managers.

* Bonds.
Investors aiming for the 5.2% long term average return produced by the US bond market have to minimize the fees and commissions payed in transaction coses to the managers... but then have to manage the investment risk by themselves. As a matter of fact, individuals can lend money both to the government and to corporations: both borrow capital from the public by issuing bonds and other fixed income instruments that are traded on the bond markets. The price of a bond evolves from its initial nominal principal or face value and the issuer pays a regular predetermined amount of cash called coupon until the bond reaches the maturity or redemption date, when the principal is finally payed back to the investor.



Masters: fixed stream of payments of a bond / discount factor.
Throughout its life, a bond generates a predetermined stream of payments

$\displaystyle A + \left(\sum_{i=1}^n A \tau_i X_i \right)$ (1.3#eq.5)


where the amount A is the principal value outstanding at maturity (usually normalized to 1 or 100), $ \tau_i$ is the tenor or the frequency (in fractions of years, e.g. 30/360 for a monthly coupon in the LIBOR convention) and $ X_i$ the fixed annual interest rate used to calculate the coupon per unit investment (e.g. 0.05 for a 5% coupon).
As the name suggests, a zero-coupon bond does not pay any coupon and simply returns the contractual value AP(T,T)=A on the maturity date T. Its present value AP(t,T) measured at time t<T can be calculated from a no-arbitrage argument, provided that the interest rates are fixed and that the issuer is certain to pay the loan back on time. Indeed, investing an equivalent amount of cash on the money market for a yield Y should result in the same final value as when the bond matures; if this were not true and the price lower (alt. higher), it would be possible to buy (alt. sell) bonds in exchange of cash on the money market and generate a risk-free profit at the maturity date. Arbitrageurs would immediately take advantage of such opportunities until the demand (alt. offer) moves the price back to the equilibrium value
$\displaystyle P(t,T)\exp(Y[T-t]) = P(T,T) = 1$      
$\displaystyle \Rightarrow P(t,T)=e^{-Y[T-t]}$     (1.3#eq.6)


In a risk-free economy, the present value of an asset can always be calculated from a price known in the future by multiplication by the discount factor $ \exp(-Y[T-t])$ . This is also true for coupons payed at $ t_1=t+\tau_1, t_2=t_1+\tau_2, \hdots, t_n=t_{n-1}+\tau_n\equiv T$ , which can be discounted back in time as

\begin{displaymath}\begin{array}{rcl} Bnd(t,\{t_i\},T) &=&\displaystyle A e^{-Y[... ...laystyle AP(t,T) + \sum_{i=1}^n X_i\tau_i AP(t,t_i) \end{array}\end{displaymath} (1.3#eq.7)


showing that the present value of a coupon-bearing instrument can always be reduced to a linear combination of zero-coupon bonds (exercise 1.07).


In the real world, the spot price of a bond is determined by the offer and demand from investors and depends also on the credit worthiness of the issuer. Rating agencies such as Standard & Poor, Moody's or KMV use different criteria to judge issuers who are labeled from the safest ``investment grade'' (AAA, AA, A, BBB, of which 2.95% American corporate bonds defaulted in 2002) down to ``speculative'' (BB, B, CCC, CC), ``junk'' or ``default'' (C,D). The price of a bond drops sharply when the risk of defaulting on a debt rises: check the historical value of the Argentinian government bonds as its credit worthiness was finally downgraded from C to D in December 2001.

The spot price quoted for a variety of bonds can be accessed with a dozen minutes delay free of charge over the Internet (take e.g. Yahoo, Bloomberg, etc) and the closing prices are reported one day later in the press: for example, on Feb 23, 2002, the Financial Times printed the values in 1.3#tab.1.

Table 1.3#tab.1: Bonds traded in London quoted on Feb 23, 2002 by the press
Issuer Red date Coupon S&P rating Bid Price Bid Yield
Sweden 01/09 5.000 AA+ 100.154 4.97
Ford 06/10 7.875 BBB 102.241 7.50
Marconi 03/10 6.375 B+ 35.000 26.73


The first row shows a Swedish government bond that matures Jan 2009 and pays a 5% annual coupon: with a good investment grade AA+ and a yield in line with the market's expectations, the price (given as a percentage of the principal value of EUR 1,000) is 0.154% higher than the principal. If you bought this bond on Feb 22, you would now earn 0.03% less than the original coupon.

Marconi's corporate bond expires March 2010 and pays a coupon of 6.375%; after the downgrade of telecom operators and speculations about the company's financial fitness, the coupon is now well below what the market expects for the speculative B+ rating. This explains why the bond lost 65% of its principal value and was now only worth 35.00. If you bought this bond on Feb 22, you could earn a very high yield of 26.73% during the next 10 years, provided that the company does not go bankrupt in the mean time.

Small systematic costs have a large impact on the long term return of a portfolio: investors should never neglect the possibility of tax reductions or outright exemptions when buying municipal, state or government bonds.

* Stock.
Encouraged by the average 7-11% long term average growth of the stock market, investors often add company shares to their portfolio. By doing so, they become co-owners and link the fate of their investment to the future earnings of these companies. Every quarter of a year, the management appointed by the shareholders assembly reports on the profits or losses and sometimes distributes a fixed dividend for every share to reward the investors.

In many countries, the tax on dividend income is higher than the tax on the gain in capital - although recent modifications of the taxation in the US may revert this trend. Shareholders therefore prefer to keep the dividend yield low and let the value of shares grow with the company as long as growth remains possible. The valuation of the company's assets, together with the latest results and the expectation of future earnings directly impact on the offer and demand from investors on the stock market (such as NYSE, NASDAQ) which ultimately determines the price of shares.

The spot prices quoted for every share can be read free of charge on the Internet after only a dozen minutes delay (take e.g. Yahoo, Bloomberg, etc) and the closing prices are reported one day later in the press: for example, on Feb 23, 2002, the Financial Times printed the values in 1.3#tab.2 below.

Table 1.3#tab.2: Stocks traded in London quoted on Feb 23, 2002 by the press
Company Price +/- High Low Volume Yield P/E
Hilton 215 1/2 -4 3/4 259 3/4 152 8,081 4.0 11.9
AstraZeneca 3519 +64 3564 2724 3,874 1.4 29.9
Marconi 16 3/4 -2 3/4 800 12 1/4 103,986 - -


The first row shows that the share from Hilton hotels fell GBP 4.75 to 215.50, in a liquid market with more than 8 million shares exchanged during the trading day. This price is somewhere in the middle of the range over which the share was trading during the last 12 months, as indicated by the high and low ends of the price interval. The price-to-earning ratio shows that it approximatively takes P/E=11.9 years for the company earnings to add up to the original purchase price (``paying back your investment'') if the earnings remain fixed. Assuming a small one percent growth G=0.01 for a mature industry, you can show (exercise 1.06) that this corresponds to an expected return on investment of G+E/P=0.01 +1/11.9=0.094 or 9.4%, which is indeed much larger than the 4% dividend yield payed in cash to the shareholders; investors should therefore expect Hilton's share price to rise by an annual 5.4%.

AstraZenca pharmaceuticals have been growing very fast during the last years and, expecting that this will continue into the future, investors are willing to pay a much larger price-to-earning ratio of 29.9. For the sake of simplicity, assume that the exponential growth reached from the Low to the quoted Price during exactly one year, so that the growth rate can be estimated from G=ln(P2/P1)/(t2-t1)=ln(3519/2724)=0.25.   This translates into astronomical returns G+E/P=0.25 +0.033=0.28 which exceed by far the 3.3% justified by the earnings and the 1.4% payed as dividends.

Finally, note the near collapse of Marconi's share from GBP 800 to 16.75, which shows that the company has large financial difficulties and may go bankrupt, i.e. the share value drop to zero forever. This is consistent with the low credit worthiness perceived for its corporate bond (1.3#tab.1) and underlines the fact that a high investment risk can lead to large losses.

Intra-day prices can in general not be accessed free of charge; daily values adjusted for occasional splits can, however, be downloaded using the MKTSolution applet below.

MKTSolution applet:  select market data for shares in General Motors Co. and press Draw for a plot as a function of the trading days during the year 2001. You can perform measurements by clicking inside the plot area and access up-to-date market data for a broad range of symbols under the previous link.


Market data: historical values from closing prices
  1. Study the price history and the market volume for shares in General Motors traded during one year. How large is the drop that can be associated with the WTC attack on Sep 11, 2001?
  2. Have a closer look at the daily price increments and compare them with the random walk described in the previous section.
  3. Follow the link to the Market data applet and identify the market symbol for an update of your favorite companies.

From this overview, it should be clear that a higher return can be expected if the investor accepts a larger risk. The comming sections describe simple methods to maximize the return from a portfolio and determine the risk from historical data. But how much risk should an investor take anyway? A mountaineer says this a matter of taste, while common sense tells you not to wake up in the middle of the night to worry about a portfolio!

SYLLABUS  Previous: 1.2 Capital and markets  Up: 1 INTRODUCTION  Next: 1.4 Modern portfolio theory