Thursday, November 15, 2012

Bond Portfolio Management


Numerous interest rates exist and they vary continuously.  The market rate of interest (or nominal rate) for a security can be expressed symbolically as follows:

Equation 1.1
Nominal interest rate (r)
= real interest rate (rr) + expected rate of inflation (q) + various risk premiums (rp)

The real interest rate is the rate at which physical capital is expected to reproduce in an economy.  Since suppliers of funds expect to be compensated for inflation, the expected rate of inflation over the life of the asset should be added to the real rate.  Adjustments to a bond's yield-to-maturity (YTM) are also made for the following risk factors:

(1)  Default risk.  This is the risk of bankruptcy and, therefore, the inability of a corporation to pay the contractual interest on debt.
(2) Liquidity risk.  The risk of not being able to sell an asset quickly at full value.
(3)  Interest-rate risk.  The fluctuation in price, especially for long-term bonds, from changes in the interest rate.  Also included reinvestment rate risk.
(4)  Foreign-exchange risk.  The risk of holding bonds in a foreign currency.

Some other factors affecting YTM are taxes and issue characteristics.  Municipal bond coupons are exempt from federal taxes and, therefore, have lower interest rates than taxable corporate bonds.  Special features in a bond's indenture such as an embedded call or put can influence its yield.

Currently, T-bills with a maturity of 13 weeks have a YTM lower than long-term T-bonds yields.  Using Equation 1.1, explain why these returns differ.
Two factors given in Equation 1.1 could account for the differences.  They are (1) inflationary expectations and (2) interest-rate risk.  Since long-term bonds have several more years to maturity than T-bills, expectations for higher rate of inflation in the future could account for some of the differential.  In addition, long-term bonds have more interest-rate risk than short-term bonds and, therefore, a greater risk premium is required.

Long-term BBB-rated corporate bonds yield more than AAA-rated corporate bonds with the same maturity.  Explain in terms of Equation 1.1.
If we assume both issues have identical characteristics, then the main reason the BBB-rated bonds have a higher return is greater default risk than AAA-rated bonds.

Several times in U.S. financial history short-term securities such as commercial paper have had a higher return than long-term bonds (both T-bonds and corporates).  Explain how this could happen with Equation 1.1.
The main factor must be decreasing inflationary expectations.  If the rate of inflation is expected to fall in the future this makes the expected average rte of inflation smaller for long-term bonds than short-term securities.  For example, the lower rate of inflationary expectations more than offsets the higher default risk and interest-rate risk associated with long-term corporate bonds.


Yield spreads are the differences in yields between risky bonds and default-free U.S. Treasury bonds of similar maturity.  This equation below defines the yield spread for the t th time period.

Equation 1.2
(Yield spread)t
= (yield on risky bond)t - (yield on a U.S Treasury bond)t

Suppose an AAA-grade corporate bond's current YTM is 11% while the YTM on a similar T-bond is 9%.  Equation 1.2 suggests the yield spread for the pair of bonds is 2% or 200 basis points.

Yield spreads tend to vary over the business cycle.  They are larger at business troughs than at peaks, and yield spreads take place in a predictable fashion.  As a result, bond traders and investors who can act quickly when yield spreads appear to be different from normal can make profits.

Tom, an astute bond trader, noted that the yield spread between a BBB-rated corporate bond issue and a similar T-bond issue was 250 basis points.  However, according to Jeff's careful analysis, the normal yield spread for the two bonds for this stage of the business cycle should be 200 basis points.  Jeff feels that he can profit from the abnormal yield spread.

In order for the yield spread to become normal it must decrease.  For this to happen, either the BBB corporate bonds must increase in price (yields fall) or the T-bonds must fall in price (yields increase). Therefore, Tom should purchase the BBB bonds.  If, as is likely, the BBB bonds increase in price, he would profit.


The relationship at a point in time between the YTMs of homogenous bonds and the years to maturity for the bonds, holding other things constant, is called the term structure of interest rates or yield curve.  Three main explanations or theories of the term structure have been given.  An explanation of each one of these hypotheses is presented below.

The Expectations Hypothesis
The expectations approach to the term structure of interest rates states that long-term interest rates are the geometric mean of expected forward (or future) short-term interest rates.

The Liquidity Premium Hypothesis
While the expectations approach to the term structure of interest rates implies a flat yield curve if interest rates are not expected to change in the future, the liquidity preference theory, under similar circumstances, implies a rising yield curve.  According to the liquidity preference theory, investors demand a risk premium from long-term securities.  As long-term bonds have more interest-rate risk, investors require higher returns.

The Segmentation Hypothesis
According to the segmented market theory, certain investors and financial institutions prefer to invest in certain segments of the yield curve.  For example, due to the stable long-term nature of their liabilities, pension funds prefer to invest in long-term securities.  On the other hand, commercial banks' short-term liabilities (namely, demand deposits) create a need for short-term assets.  As a result, supply and demand for funds within each segment of the market determine the level and structure of interest rates for that segment.  


The interest-rate risk from coupon-paying bonds is composed of two components:
(1) price risk and
(2) coupon reinvestment rate risk.
These two components of interest-rate risk vary inversely.

If a bond investor wants the promised YTM and the realized YTM to remain equal over the life of a coupon bond, interim interest payments must be invested at the promised YTM.  If interim rates change, the realized YTM can be either greater or less than the promised YTM.

One way to assure the bond investor that the portfolio held will achieve the desired realized yield is to immunize the portfolio.  Immunization takes place when the desired holding period of the portfolio equals the bond portfolio's Macaulay's duration.  For example, immunizing a portfolio for 5 years requires the purchase of a series of bonds with an average Macaulay's duration of 5 years, not an average maturity of 5 years.  Immunization achieves this result because reinvestment risk and price risk exactly offset one another whenever a bond portfolio's desired holding period is equal to its duration.

There are limitations to immunization.  Every time market interest rates change the duration of every bond changes.  Duration tends to diminish more slowly than calendar time for coupon-paying bonds when interest rates do not change.  Therefore, periodic rebalancing will be necessary to continuously immunize a portfolio of bonds.  In addition, yield curves are not necessarily flat and do not shift in a parallel fashion, so the use of duration will not be exact.

Learning Points:

One of the problems with immunization is the cost associated with rebalancing.
According to the expectations approach to the term structure of interest rates, a falling yield curve forecasts falling interest rates in the years ahead.
Yield spreads tend to narrow during a boom period in the economy.
The average duration for a portfolio of zero coupon bonds will change as interest rates change.
If the interest rate is 10%, the weighted average duration for a portfolio of two $40 million face value zero coupon bonds with maturities of 4 and 8 years, respectively, is 5.64 years.
The liquidity preference theory about the term structure of interest rates asserts that if short-term interest rates are expected to remain the same in the future, the yield curve should be a rising yield curve.
Long-term bonds have more price change risk than short-term bonds.

Wednesday, November 14, 2012

Bond Valuation

The value of a bond is simply the present value of all the security's future cash flows, as show below.

Basic Bond Equation:

Present value
= [COUPON1/((1+YTM)^1)] + [COUPON2/((1+YTM)^2)] + .....+ [(COUPONt + face value)/((1+YTM)^t)]

The four terms that appear on the right-hand side of the present-value equation are discussed below:
(1)  Market interest rate:  The discount rate, or interest rate or yield-to-maturity, is the market interest rate that constantly changes.  The bond's yield-to-maturity is represented by the symbol YTM.  The YTM is the discount rate that equates all the bond's future cash flows with the current market price of the bond.

(2)  Face value, F:  The bond's face value (or principal value) is printed on the bond and is invariant throughout the bond's life.

(3)  Coupon:  The dollar amount of a bond's coupon interest payments equals the product of the coupon interest rate, denoted i, and the face value, F.  Symbolically, coupon = iF.

(4)  The number of time periods left until the bond matures and the terminal payment occurs is denoted T.

What is the value of a $1,000 bond with an 8% coupon rate, 3 years before maturity?  The YTM is 10%.

iF = (0.08)($1,000) = $80
T = 3
YTM = 10

Present value
= [$80/((1+0.10)^1)] + [$80/((1+0.10)^2)] + [($80 + $1000)/((1+0.10)^3)]
= $72.727 + $66.116 + $811.420
= $ 950.26

Semiannual Coupon Payments

Semiannual coupon payments are more common than annual coupons.  Therefore, the present value of a bond model must be modified slightly to accommodate the semiannual coupon payments.

A $1,000 bond with an 8% coupon rate, 3 years before maturity.  The YTM is 10%.  What is the bond's present value if the coupons are paid semiannually?

Converting annual compounding to the semiannual compounding model for the same bond.  The bond's annual coupon payments must be halved to get the semiannual coupon payments.  The bond's compounding period is 6 months.  The appropriate market rate of return or discount rate for semiannual coupons is half of the bond's YTM.

Coupon Payment iF = (0.08)($1,000) = $ 80
Semiannually Coupon Payment = 0.5 x iF = (0.08/2)($1,000) = $ 40

T = 3 years
2T = 6 semiannual periods

YTM = 10%
Semiannual discount rate = YTM/2  = 10%/2 = 5%.

The present value of the bond will be the asked price for the 8% coupon bond maturing in 6 six-month periods.  Note that using the more frequent compounding interval reduced the present value of the bond when compared with the bond with annual coupon payments.

Accrued Coupon Interest

Accrued interest is interest that has been earned by an investor but that has not yet been paid to that investor.  Bond buyers pay bond sellers accrued interest whenever a bond is purchased on a date that is not a scheduled coupon interest payment date.  Thus, if a bond were sold between its semiannual interest payment dates, the purchaser should pay the market price of the bond plus the appropriate fraction of the accrued coupon interest earned but not yet received by the party selling the bond.

What is the purchase price for a bond that is paying a 6% annual coupon rate in semiannual payments if its YTM=10.0% and it has 2 years and 10 months from its purchase date until its maturity?  What is the accrued interest?  Assume the bond is traded in a year of 366 days when calculating the accrued interest.

Since this bond has 2 months short of 3 full years until it matures, we must first
-  (Step one) calculate  its present value on the next interest payment date, which is 2.5 years until maturity.  After the present value of a 2.5 year bond has been calculated,
- (Step two) we will add the interest payment received on that date to the price of the bond.
- (Step three)  Next, the value determined in Step two should be discounted back 4 months to the purchase date.


Sometimes the issuer of a bond has the option to call (or redeem) the bond before it reaches maturity.  This is likely to occur when the coupon interest rate on similar new bonds is substantially below the coupon interest on existing bonds because the corporation can save money on future interest payments.  When a bond has an excellent chance of being called, an investor may want to calculate the yield-to-call for the bond.    This can be accomplished by modifying the basic bond equation, in the following manner:

  • T would now equal the number of time periods until the bond is called, and
  • Face value would now be the call value.


Macaulay's duration (MD) is the average time it takes to receive the cash flows expected from a bond or another asset.

The table below shows the relationship between Macaulay's duration (MD) and years-to-maturity for semiannual bonds of various coupon rates that have YTM of 9%.
A bond's duration is less than its year-to-maturity for coupon bonds.
MD equals T for zero coupon bonds.
MD also increases at a decreasing rate as maturity increases for bonds selling at par or above.
For discount bonds, note that duration will eventually decline.

Table: Macaulay's Duration (in Years) for Bond Yielding 9 Percent (Semiannual Compounding)

Rate 3% 6% 9% 12%
Year to 
5 4.622 4.345 4.134 3.968
10 8.249 7.347 6.797 6.426
20 12.272 10.437 9.615 9.148
50 12.391 11.703 11.469 11.35
70 11.832 11.648 11.587 11.586
100 11.366 11.616 11.609 11.606
Infinity 12.111 12.111 12.111 12.111

The relationship between a change in the price of a bond relative to a change in its YTM is usually referred to as modified duration (MMD) for a bond.

Learning Points:

Duration for a zero coupon bond is the same as its term to maturity.
Shorter-term bonds are almost always more volatile in terms of price than longer-term bonds for a given change in interest rates.
Bond price volatility is inversely related to the bond's coupon.
Duration of a coupon-paying bond is always less than its term to maturity.
For any given maturity, bond price movements that result from an equal absolute decrease or increase in the yield-to-maturity are asymmetrical.
There is an inverse relationship between a bond's coupon and duration.
As a bond's YTM increases, if other things are held constant, its duration decreases.
The duration for a perpetual bond is finite.
When a bond is selling at a discount, its YTM exceeds the coupon rate.
When a bond's YTM equals its coupon rate, the bond's price is more than par value.

Tuesday, November 13, 2012

Portfolio Analysis

Total risk of an individual asset is measured by the standard deviation (or variance) of returns of the asset.

Let's look at the relationship between the risk and return for a portfolio of assets.  For a portfolio of assets, the return on a portfolio is simply the sum of the individual assets, its risk is not simply the sum of the risk of the single assets.  When a portfolio's standard deviation is calculated, attention must be given to the covariance of the returns of the assets.  Harry Markowitz has shown that portfolios dominate individual assets from a risk and return standpoint.  He also showed how the optimal portfolio could be determined.

The covariance is a statistical measure of how the returns of two assets move together.

The correlation is also a measure of the relationship between two assets.  The correlation coefficient can take on a value from -1 to +1.  Correlation and covariance are related by the following equation:

COVij = DiDjPij

Di and Dj are the standard deviations of returns for assets i and j
Pij is the correlation coefficient for assets i and j.

The return on a portfolio of assets is simply the weighted average return.

A portfolio of 4 common stocks with the following market values and returns:

X             $10,000                          10%
Y               20,000                           14%
Z               30,000                            16%
M              40,000                           15%

The return on this portfolio is determined thus:

= ($10,000/$100,000)(10) + ($20,000/$100,000)(14) + $30,000/$100,000)(16) + ($40,000/$100,000)(15) 
=   (0.1)(10) + (0.2)(14) + (0.3)(16) + (0.4)(15)
=   1 + 2.8 + 4.8 + 6)
=  14.6%

The variance of returns for a portfolio of assets can be calculated using a general formula.

A two-asset portfolio can be used to illustrate some of the principles of diversification.

Portfolio theory suggests that as the correlation coefficient declines risk should decline, but the portfolio's expected return should not vary, if the weights of the assets are held constant.  The important point is that if assets are less than perfectly positively correlated (that is, a correlation coefficient below +1), diversification can reduce risk.

The A and B Corporations have the following expected risk and return inputs for next year:
Er(A) = 18%         Er(B) = 22%
SD(A) = 22%       SD(B) = 30%

The portfolio risk (standard deviation) for a portfolio of 50% in each asset is 21.8632 percent.  Determine the correlation coefficient that will be necessary to reduce the level of portfolio risk by 25%. 

A 25% reduction in risk is (0.25)(21.8632%) = 5.4658%.  Therefore, the new level of risk will be 21.8632% - 5.4658% = 16.3974%.
Correlation coefficient = -0.2337

What is the expected return of the equally weighted portfolio?

Expected return = 20%

If we consider the infinite number of portfolios that could be formed from two or more securities and plotted these portfolios' expected return and risk, we would create graphs like so.

The efficient frontier is represented by the curve line.  Portfolios along this line dominate all other investment possibilities.   The highest return portfolio(on the far right of the curve) is the only one that is likely a one-asset portfolio.  The curvature of the efficient frontier depends upon the correlation of the asset's returns.  The efficient frontier curve is convex toward the expected return axis because all assets have correlation coefficients that are less than positive unity and greater than negative unity.

  1. Markowitz portfolio analysis makes the following assumptions:
  2. The investor seeks to maximise the expected utility of total wealth.
  3. All investors have the same expected single period investment horizon.
  4. Investors are risk-averse.
  5. Investors base their investment decisions on the expected return and standard deviation of returns from a possible investment.
  6. Perfect markets are assumed (e.g. no taxes and no transaction costs).

Given the above assumptions, an investor will want to hold a portfolio somewhere along the efficient frontier. The exact location depends on the investor's risk-return preferences.  A set of indifference curves for each investor will show his or her risk-return tradeoff.  Those investors with more risk-aversion require more compensation for assuming risk and will choose a portfolio along the lower end of the efficient frontier.  The portfolio will be optimal because no other portfolio along the efficient frontier can dominate another in terms of risk and return.

International Investing - "Why not diversify Internationally rather than Domestically?"

With over half of the market value of stocks and bonds outside the U.S., international investing is important as a way to improve returns and lower risk.  It is fairly easy to invest internationally today through open-end international mutual funds.

It is possible to reduce risk below the level in domestic markets by diversifying internationally.
There are tables showing the correlation matrix of returns for the stock indexes of various countries.  These correlations are in most cases lower than the correlation between large portfolios of domestic stocks.  The lower international correlation of stock returns encourages multinational investing because it allows an investor to reduce risk even further.

Why are the international correlation coefficients lower than the correlation between domestic stocks?
Countries have different political systems, different customs, different regulations governing trade and business, and cultural differences.  In addition, some countries are experiencing internal conflicts.  Furthermore, some countries may have high inflation while other countries may be experiencing low inflation, or some may be having a booming economy, while others may be experiencing a recession.  All these factors tend to cause the correlation of security returns between countries to be lower than the correlation of returns between securities (or groups of securities) in the domestic economy.

While international investing can reduce risk below domestic levels, one important risk factor was not considered - foreign exchange risk.  When investing in another country the investor must convert domestic currency into a foreign currency.  Then, when the foreign investment is terminated, the process must be reversed.  In essence, two investments are made.  One investment is in the foreign security and the other is in that foreign country's currency.

Example 1:
Bought US $1,000 worth of Australian stock and hold it for 1 year.
Current exchange rate is US $1 dollar for 1 Australian dollar (AD).
Therefore, this purchased stock worth AD 1,000.

Australian stock return 10%.
Exchange rate at end of the year was US$0.90 /AD 1/

Returns from a U.S. perspective was as follows:
1.  Convert the Australian investment into US dollars.
1,000 x 1.1 = AD 1,100 (ending investment with 10% growth) x US$90 =  US$990

The positive 10% return from the Australian investment, but the loss on the foreign currency investment more than offset the gain from the Australian security.

The net return can be calculated with the following equation:

Rn = (1+Rd) (!+Rc) - 1
Net return = (1 + foreign security's return) (1 + currency return)  - 1

The investors can do something about exchange rate.  An investor could lock in a fixed return by purchasing the appropriate forward contract on the foreign currency.  For the hedge to work properly, however, the investor needs to know exactly how much foreign currency to sell in the forward market.

Example 2
Mr. A is planning to purchase a gilt, a British government bond, with an annual yield-to-maturity of 9%.  Currently, the exchange rate is $1.80 per pound sterling, and the 1-year forward exchange rate is $1.90 per pound.  If Mr. A hedges his position, what annual return can he lock in with the British bills (ignoring transaction costs)?

Mr. A makes two investments, one in a British government bond and the other in the British pound.  The British bill will earn a riskless 9%.  Her currency investment will yield 5.56%.

$1.90/$1.80 - 1 = 1.056 - 1 = 0.0556 = 5.56%

The combined net return that is locked in, is,

Rn = (1 + Rd) (1+Rc) -1
= (1.09) (1.0556) - 1
= 1.1506 -1
= 0.1506
= 15.06%.

This type of hedge will work because Mr. A knows exactly how many pounds sterling he will have from the riskless British government bond at the end of the year.  If his investment had been in the risky British equity market, he would have been uncertain about the investment's outcome and, as a result, uncertain about the number of pounds he would have at the end of the year.

Learning Points:

  • International investing offers the individual investor the potential to reduce risk below the level in the domestic market.
  • Exchange rate risk cannot be eliminated by hedging in the forward foreign exchange market.
  • An efficient frontier that is constructed with international equities usually dominates one constructed only with domestic equity securities.
  • Exchange rate risk is a problem when investing in international equities and international bonds.
  • The majority of international equity indexes have betas less than 1 when their returns are regressed with the returns of the S&P500 index.
  • If an investor buys a British security, the investor's return will not be the same whether the investment is made in dollars or pound sterling.
  • If you invest in a British corporation and earn a 12% rate of return on your British investment, this return will be enhanced from the Malaysian perspective if the Malaysian ringgit decreases in value relative to the British pound.
  • An American investing in Japan would be happy if the dollar depreciated in value relative to the yen, but a Japanese investing in the U.S. would not.  
  • Even though a U.S. and British investor may experience identical returns from an investment in France, from the perspective of their particular currencies, their returns will most likely differ.

Portfolio Performance Evaluation

Investment companies come in two forms.  The open-end investment company, or as it is usually called a mutual fund, is the most common type of investment fund.  Closed-end funds, the second type of investment company, are not as popular as the open-end funds.  Unlike the open-end funds, closed-end funds cannot sell more shares after the initial public offering.

Stocks and bond mutual funds can be classified into various categories such as growth funds, balanced funds, income funds, government bond funds, junk bond funds, and international funds. Mutual funds offer small investors the ability to diversify for lower commission costs and lower management fees than most alternatives.

Most of the empirical evidence to date indicates that most mutual funds do not beat the market, after adjustments are made in their returns for transaction costs and risk.

Net Asset Value per Share
A mutual fund's net asset value per share (NAV) is equal to the total market value of all the mutual fund's holdings minus liabilities divided by the fund's total number of outstanding shares on a particular day.

One-Period Rate of Return
A mutual fund's one-period rate of return can be calculated with the following equation:

Rt = [Ct + Dt + (NAVt - NAVt-1)] / NAVt-1

Rt = The fund's return for time period t
Ct = The capital gains disbursement during time period t
Dt = The cash dividend or interest disbursement during time period t
NAVt = The fund's net asset value per share at the end of time period t
NAVt-1 = The fund's net asset value per share at the end of time period t+1
(NAVt - NAVt-1) = The change in the fund's net asset value per share from the beginning of time period t until the end of time period t [or, equivalently, the beginning of period (t-1)], as the result of capital gains and cash dividends that were not distributed to the owners during time period t.

There are two important aspects of any investment that must be evaluated over the investment's holding period - the return and the risk. The multiperiod (or compounded) rate of return is called the geometric mean return.  The geometric mean rate of return for an investment can be calculated with the following equation:

GMR = [(1+R1)(1+R2)(1+R3) ...(1+Rt) -1]^(1/t)

GMR = The geometric mean return
R1 = The return for the time period 1
R2 = The return for the time period 2
Rt = The return for time period T
T = The total number of time periods

The annual returns over the past 5 years are given below for the company XYZ:

Year    Return
1991    10%
1992    12%
1993    15%
1994    13%
1995    16%

The geometric mean rate of return for the company XYZ over the past 5 years is:

= [(1.10)*(1.12)*(1.15)*(1.13)*(1.16) - 1 ] ^ (1/5)
= 1.131798
= 13.18%

Other Performance Measures

The Sharpe portfolio performance measure is based on the capital market line (CML) and total risk, which makes it more suitable for evaluating portfolios rather than individual assets.  On the other hand, both the Jensen and Treynor performance measures are based on the capital asset pricing model (CAPM) and are more flexible because by using systematic risk (beta) they can be used to evaluate the performance of both portfolios and individual assets.  All three performance measures tend to rank a group of diversified portfolios similarly.

Learning points:

  • Most growth-oriented mutual funds do not outperform the S&P 500 on a risk-adjusted basis.
  • The Treynor performance measure is better than the Sharpe performance measure for evaluating the performance of individual common stocks.
  • The compounded rate of return is another name for the geometric mean rate of return.
  • The geometric mean rate of return for an asset will always be smaller than its arithmetic mean rate of return.
  • Most empirical studies of mutual funds report that the funds did not consistently do better than the market average.
  • One good way for small investors to diversify their holdings is to purchase mutual funds.
  • After its initial offering, a closed-end mutual fund cannot sell more shares.
  • The internal rate of return (IRR) should be used only with investments for which the size and timing of the cashflows are determined endogenously (for example, cashflows from an investment in a manufacturing machine).
  • The geometric mean return (GMR) should be used for investments in which the cashflows are exogenously determined (for example, investor deposits and withdrawals from a mutual fund).