Thanks for link, but it really doesn't allow me to (1) delete my earlier comments or (2) submit my retraction. If you can do either or both of these things, please do so with the comments below. Thanks. George.)
I want to revoke and delete the recent criticism I wrote concerning your Bond Investment course. It was excessively negative and unfair.
I think my comments were prompted by my ignorance of Bond Investing. I knew nothing about bonds, and because of this, I expected your course to make me an expert in an hour or two. Not very realistic….
I now realize that the course delivered all it promised to do; I realize also that the real learning will begin, only when I take your principles and begin to research and apply them further. The course offers very clear direction....
No one has contacted me or pressured me to write this. I was simply wrong....
Again, my apologies.
San Diego, CA
---Bond investors laughed all the way to the bank as the stock market broke from 2000 onward. Meanwhile investors in blended mutual funds got walloped. These were clever traps people stepped into because they didn’t understand the choices in their employer sponsored 401(k) plans.
These plans pump dumb money into the market by the billions.
Big drops in interest rates coughed up double digit returns to astute fixed income investors like Bill Gross. I know you are not Bill Gross.
You don’t have his training — yet. But someday you will deal with bonds — one way or another.
Don’t you agree that the bond market is a very real threat to your family wealth if you don’t understand it?
Or perhaps you already have had contact with bonds. Do you have a wealthy relative asking for guidance in managing their large bond portfolio?
This innovative course takes you by the hand and walks you through the process of managing a multi-million-dollar bond portfolio.
I am Dr. Scott Brown. I am a professor of finance of the AACSB Accredited Graduate School of Business of the University of Puerto Rico. I train hundreds of financial managers every semester.
I hold a Ph.D. in finance from the University of South Carolina and a master degree from Thunderbird Graduate School of International Management. Thunderbird is routinely ranked #1 in the U.S. News & World Report.
Here are just a couple of the tremendous benefits you will gain from this training.
Utilize the core knowledge to understand when to speculate, when to go for monthly income and when to sit it out in cash.
Here is what my students say about my courses,
“THERE SIMPLY ISN'T ANYTHING COMPARABLE TO THIS COURSE!” -BILL BAKER, CHARLESTON, SOUTH CAROLINA
It takes time for your brain to absorb and dominate this material. The sooner you start, the better off you will be as steward of your family wealth.
The material is guaranteed in its accuracy and detail.
What’s holding you back? Enroll now.
WARNING: I am serious when I tell you that it takes years to master a subject. The longer you delay the slower your growth.
Don’t forget that this is the only bond management course available today by a major state university professor for public consumption. Enroll now. I look forward to mentoring you in bond portfolio management. -Doc Brown
Hi, I am your guide to bond investing. I am Dr. Scott Brown.
I hold a Ph.D. in finance from the University of South Carolina. I also hold a master degree in international management from Thunderbird, ranked #1 routinely in U.S. and News World Report.
I am teaching this course because I have noticed that very few retail stock investors understand how to buy and sell bonds.
This is troubling considering that most taxpayers who contribute to employer sponsored 401(k) plans blindly pump trillions into the bond market through poorly understood menus of fund selections. Timing is everything when it comes to succeeding in bonds.
In the early 2000s interest rates plummeted. A risk-free bond portfolio would yield you 6% without a thought.
Much higher returns were available in corporate bond portfolios if you wanted to think (a little) to grow rich.
You would have been tipped off by an inverted yield curve. And long durations made the most sense.
But when rates hit bottom rises through come through hike after hike from the federal reserve. During times as interest rates rise short durations make sense; but only if the stock market is dropping. Otherwise moving out of bonds into equities is more prudent in a bullish stock market.
Just get out when you are unsure it is the right moment to be in bonds. Or better yet don’t get in when the bond market goes against you.
Regardless, this course is designed to give you the tools to make such analysis and to take appropriate action through independent thought.
This an intensive program on bond portfolio management. This course is helpful for saving heads of household who invest; as well as finance students and professionals.
At the end of this course, students will be able to manage a portfolio of bonds regardless of the mix of quality, coupon, or maturity. You will also know how to immunize a large bond portfolio from risk of interest rate increases. And, you will know how to re-balance when bonds mature or are called in.
I will also give you clues to show you how to speculate intelligently in bonds. In this course, you will learn,
Learning is reinforced by banks of questions designed to test what you retain. Please ask questions on the forum when it will help other students. Otherwise direct any questions you don’t feel like sharing with other students by sending me a direct message.
The possibilities are endless in bonds. Maybe you will be the next Bill Gross.
Gross is the "king of bonds." He is the world's best known bond fund manager. Mr. Gross founded the PIMCO family of bond funds. He was the first portfolio manager inducted into the Fixed-Income Analyst Society Inc. (FIASI) Hall of Fame. This happened in 1996 for his contributions to bond management.
Bill Gross is known for his ability to change directions without hesitation in response to shifts in the markets.
SmartMoney in 2005 recorded that "Gross doesn't adjust to market conditions – he changes them! His views on the bond market are widely followed by professional investors and the investing public worldwide." Bill left PIMCO for Janus Capital Group in September 2014.
He is a Duke University graduate who finished in psychology in 1966. Gross spent a summer playing professional blackjack in Las Vegas. He served as a naval officer on a destroyer off the coast of Vietnam. After the Navy, Gross obtained his MBA in 1971 from the University of California, Los Angeles.
He became a Certified Financial Analyst (CFA) while working as an investment analyst with Pacific Mutual Life in LA from 71’ to 76’. He was an Assistant Vice President managing fixed income securities.
Then he founded and became the managing director and chief investment officer, for Pacific Investment Management Company (PIMCO). This became the world's largest fixed-income management firm.
MarketThoughts reports that Bill Gross "believes that successful investment in the long-run (whether in bonds or equities) rests on two foundations: the ability to formulate and articulate a secular [long-term] outlook and having the correct structural composition within one's portfolio over time." A three- to five-year forecast forces an investor to think long term avoiding the destructive "emotional whipsaws of fear and greed."
Bill Gross says that "such emotions can convince any investor or management firm to do exactly the wrong thing during irrational periods in the market." Gross posits that "those who fail to recognize the structural elements of the investment equation [asset allocation, diversification, risk-return measurements and investing costs] will leave far more chips on the table for other more astute investors to scoop up than they could ever imagine."
In 2016 the net worth of Bill Gross is estimated at 2.4 billion USD.
This course will prepare you to understand the structural elements of the investment equation. Let’s begin.
A bond is a debt security that forces the government or Fortune 500 issuer to make payments to the investor over the time to maturity. Face Value has the same meaning as Par Value.
The face value is the payment to the bondholder at bond maturity.
The Coupon Rate is the annual interest payment to the bondholder. This is per dollar of face (par) value.
A Zero-Coupon Bond pays no coupons, sells at a discount, and makes just one payment at maturity to the amount of face (par) value.
Here is an example.
You save $1,000 and buy a 30-year bond. The bond has a par value of $1,000 and a coupon rate of 8%.
The issuer will pay eight percent of the par value in 60 payments twice every year for three decades. That’s $80 per year in two $40 payments bi-annually.
This doesn’t sound like much. But a family owning $1,000,000 of these bonds would earn $80,000 per year in interest in this scenario.
This represents a solid middle class income. The median household income for the United States was $55,775 in 2015 per the Census ACS survey.
Retail Prices and Yields in U.S. Treasury Bonds
Treasury notes have maturities that range from 1 to 9 years. Treasury bonds have maturities from 9 to 30 years.
The table below shows the prices for a $1,000 par value Treasury bond with a 6.25% coupon that matures in August of 2023.
The bid prices are what buyers of the bond are willing to pay today. The ask price is the level at which sellers hope to unload their bonds.
The bid and ask are quoted as a percentage of par value. The ask price is 137.438.
This translates into a cost of $1,374.38 should you desire to purchase this bond.
The last column is the yield based on the ask price. This average rate of return of 2.598% is the ask yield for an investor who buys on that day at the ask price of 137.438 and holds to maturity.
Bond Pricing with Accrued Interest
The quoted prices in the last table don’t include accrued interest. This builds up in the six months between payments.
Accrued Interest=Coupon/2 X (Days Since Last Coupon)/(Days Between Coupons)
This formula generates the precise bond value on any given day of the year.
Corporate listings give details that allow you to identify the bond by issuer name or symbol. The stated coupon and initial maturity guide investors as to the total return and length of time that the debt will be working. Statistics throughout the day include the high, low, and last (close) price with the range of change. Finally, the yield to maturity allows investors to know what the yield on the bond is if purchased on the date of the listing.
Notice that the yield on the bond is higher the lower the rating.
Characteristics of Bonds
Corporate bonds frequently have call provisions embedded in the indenture. This is a contract between the firm and bond holders. A callable bond allows an issuer to repurchase debt. This may be at a fixed or indexed price. This is done over a specific time known as the call period.
Analysts refer to this as “calling in debt.”
Convertible bonds can be exchanged for shares of common stock. This offers an interesting back door entry to invest in stocks. Value investor Mohnish Pabrai writes of his success in convertible bonds in his book “The Dahndo Investor” available on Amazon.
More Quirks of Bonds
Another corporate bond is puttable. In this case the bond holder can exchange for par value.
He or she can also extend the maturity for years. Floating rate bonds automatically reset pricing on set dates.
This is contractual and is out of the control of managers.
This is different than the make-whole callable bond that resets the pricing based on prevailing interest rates should the firm call the bond. The decision to call a make-whole bond is within the discretion of firm managers.
Even Stranger Bond Realities
Preferred stock pays a dividend like some common stocks. But it is fixed like a coupon payment.
Individual retail investors cannot deduct the dividend payments from taxes. Fortune 500 firms pay less tax on preferred dividends received.
For this reason, the clear majority of investors in preferred stocks are other Fortune 500 firms.
Other Bond Players
Public schools, roads and bridges are financed by states and municipalities by bonds. These are municipal bonds.
Farm Credit agencies also issue bonds. So does the Federal Home Loan Bank Board that presides over Ginnie Mae, Fannie Mae, and Freddie Mac.
Foreign bonds are issued by borrowers selling debt in a different country in its currency.
Eurobonds are denominated in issuing country currency rather than that of the country in which the debt is sold.
Modern Bond Market Innovations
It pays to read the bond covenants. The coupon rate on inverse floaters falls when interest rates rise. Another odd bond is the asset-backed bond. Cash flow from assets specified in the bond indenture service the debt
Issuers can pay interest in cash or via pay-in-kind bonds.
Catastrophe bonds offer higher coupon rates to investors for taking on risks. These perils range from earthquakes to hurricanes.
Indexed bonds fluctuate based on a commodity. Treasury Inflation Protected Securities are TIPS. The par value of the TIP rises with the Consumer Price Index (CPI). The nominal return is calculated as follows.
The real return is calculated with this formula.
TIPS Principal and Interest Payments
This table shows the payment calculations for a TIP issued at 4%.
The inflation runs at 2% in the first, 3% in the second, and 1% in the third as each of three years unfolds. The par value is adjusted by recalculation of a 4% coupon each year. This yields a new coupon payment at the end of the first year of $40.80, then $42.02 at the end of the second, and a final payment of $1,103.55.
This is higher than what the bondholder would have received at a flat 4% rate on straight $1,000 par value. The additional amount compensates for inflation over time.
The bond value is the present value of coupons plus the present par value.
Value = PV-Coupon + PV-Par. The mathematical expression is,
Bond value= ∑_(t=1)^T▒〖Coupon/(1+r)^t +(Par value)/(1+r)^t 〗
The maturity date is T. The discount rate is r. The bond price is obtained through a different formula.
Bond price=(Coupon) 1/r [1-1/(1+r)^T ]+(Par value) 1/(1+r)^T
This can be expressed in a time value of money table format,
Bond price = Annuity factor (r,T) + Par value x PV factor (r,T).
These formulas may look daunting but remember that they are hard coded into computer systems that report bond values and prices today. Hence, you will never calculate these.
But these formulas allow you to see the intuition of the inverse relationship between interest rates and bond values. Notice that the interest rate is in the denominator.
Hence, bond values and prices fall as interest rates rise. This explains the inverse relationship between interest rates with bond values and prices.
Bond pricing and valuation formulas show us that prices fall as interest rates rise. This emphasizes that the main source of risk to a bond portfolio is from increasing interest rates. Longer maturity bond values are even more price sensitive to interest rate fluctuations.
Graph of the Inverse Relationship between Bond Prices and Yields
This figure shows the inverse relationship between interest rates and bond prices with a coupon of 8%. Bond prices rise as interest rates fall.
Bond Prices Over Different Interest Rates
This table uses the example in the prior graph of a 30-year bond with an 8% coupon to extend the study of the inverse relation between interest rates and bond prices. Notice that the higher the sensitivity of the price at different market interest rates the longer the maturity. The bond trades at a premium below 8% market or is discounted above.
More Bond Pricing
Bond pricing between coupon dates is calculated as, Invoice price = Flat price + Accrued interest
Bond pricing in excel is easy with this function,
=PRICE (settlement date, maturity date, annual coupon rate, yield to maturity, redemption value as percent of par value, number of coupon payments per year)
Do you know how to use a spreadsheet? Here is how to set one up for valuing bonds.
Here is how Excel can be used to calculate values for the 6.25% coupon January 2012 bond from the last figure.
Make sure that you input the coupon rate and yield to maturity in decimals. You have to use both the maturity and settlement dates in the format DATE(year,month,day). The value of the bond (invoice price) is 137.444.
This is cel B12 in the spreadsheet.
This example also shows how to calculate between coupon payments using a November 2039 4.375% bond. The flat price is 111.819.
Remember that flat price is the cost of a bond without accrued interest. The full price is the price paid by the retail bond investor.
You also see the formula used to calculate the days in the coupon period and the days since last coupon to set up the valuation between coupon payments.
This allows you to find the exact value of the accrued interest at 1.094%. You also see how to calculate the value of a 30-year bond with a 5% coupon rate. This price is 81.07% of face value.
Understanding Bond Yields
Yield to Maturity is the discount rate that equalizes the present value of a bond’s payments to its market price. This is the same estimation as Internal Rate of Return (IRR).
The Current Yield is different than the yield to maturity. It is the annual coupon payment divided by bond price.
Premium Bonds are priced above par value. Discount Bonds sell below par value
Calculating Yield to Maturity
This spreadsheet shows how to calculate yield to maturity with the Excel formula.
Semiannual coupons Annual coupons
Yield to maturity (decimal) 0.06* 0.0599
*The Excel formula entered here is =YIELD(B3,B4,B5,B6,B7,B8) = 0.06
More about Bond Yields
Yield to Call is calculated as is YTM. Here, time to call is replaced by maturity. Then par value is replaced by call price. Premium bonds tend to be called more often than discount bonds. This is because the firm can reissue debt at lower rates rather than continuing to pay the higher than market interest rate of premium bonds. On behalf of shareholders, the firm managers are getting a better deal than market rate when issuing discount bonds. Hence they call premium bonds more frequently and discount bonds less.
Bond Pricing for Callable and Straight Debt
Bond pricing is a function of the prevailing market interest rate. This is illustrated with the blue line. This is a $1,000 par value bond at 8% coupon with 30 years to maturity.
The black line is the pricing relationship of a bond that is callable at 110% of par value at a price of $1,100. Notice that such a bond will not have an accurate yield to maturity if the bond is called before it matures.
For this reason, analysts are more interested in yield to call on callable bonds. This is especially so when interest rates are dropping and the probability of calling the bond is rising. This is most true for premium bonds selling near the call price.
Most callable bonds have several months of call protection where the callable bond cannot be called. Also, deep-discount bonds that sell far below the call price are not likely to be called.
Yield to Maturity and Realized Compound Returns
The realized compound return is the rate of return on bonds with all coupons reinvested to maturity
Horizon analysis is the analysis of bond returns over the years. These are based on forecasts of the bond’s yield to maturity and investment options. This is done to reduce the investment risk.
Reinvestment rate risk represents the future value uncertainty of reinvested coupon payments.
Growth of Capital
Panels A and B show a horizon analysis for two reinvestment rates. At an interest rate lower than 10% the final value of the investment will be less than $1,210. But what if the actual rate earned is 8%? In this case the investment will grow to just $1,208.
Now suppose you purchase a 7.5% annual coupon bond for $980 that matures in 30 years. You will hold it for 20 years when you estimate the yield to maturity will be 8% and coupons will be reinvested at 6%.
The bond will have 10 years remaining and a forecast sales price based on an 8% yield to maturity of $966.45. The futures value of the 20 coupon payments is $2,758.92. Adding these two figures implies that your $980 investment will grow into $3,725.37 in 20 years.
$980(1+r)^20=$3,725.37 where r = 0.0690 = 6.90%
Bond Pricing Throughout Time
Yield to maturity measures the average rate of return (RoR) if the investment is held until the bond matures. The Holding Period Return is the Rate of Return over the investment period; it depends on the market price of the bond at end of the period.
Price Paths of Coupon Bonds with Constant Market Interest Rates
Notice that the price of a premium or discount bond will equal par on the maturity date. The price of a premium bond will drop.
The discount bond will increase in price until maturity.
How Bond Prices Change Over Time
Zero-Coupon bonds and Treasury STRIPS help us understand bond pricing mechanics.
A Zero-coupon bond pays no coupons. All return comes from price appreciation.
These are created through Separate Trading of Registered Interest and Principal of Securities (STRIPS). Treasury debt is used to create zero-coupon bonds from coupon-bearing notes and bonds.
30-Year Zero-Coupon Bond Pricing over Time at Constant 10% Yield to Maturity
A zero-coupon bond is an original issue discount bond where all return comes from depreciation. There are no coupons. Pricing is depicted in this graph. Notice how dramatically the price of the zero increases approaching maturity.
Bond investors must pay taxes on Zero-coupon implied interest. Original-issue discount bond time to maturity price increases are implicit interest payments to the holder. The IRS views these as built-in price appreciations and thus taxable.
A price appreciation schedule is published by the IRS that references taxable interest income on the zero-coupon bond at https://www.sec.gov/answers/zero.htm. Zero coupon bond appreciation federal, state, and municipal revenues are examples of a phantom tax.
Phantom taxes can also occur in real estate transactions and when buying or selling a business. Financially unsophisticated investors suffer from financial shortfalls from unplanned phantom taxes.
Phantom taxes are common on real estate short sales. The home owner is short the money to cover the cost of the mortgage.
The lender has the right to file a tax return on behalf of the borrower for the unpaid debt. The IRS treats this as income because the borrower did not have to pay the debt.
Taxes are assessed on income that the borrower did not actually earn. This is a tax on phantom income.
It is your job as an investor or business manager to carefully investigate the tax ramification of every transaction you structure. The onus is on you.
Bond Pricing and Default Risk
An investment grade bond is rated BBB and above by S&P. Alternatively investment grade is defined as Baa and above by Moody’s. A speculative grade bond (a.k.a. junk) is rated BB or lower by S&P. It is also defined as Ba or lower by Moody’s. This low level of bond quality is also alternatively termed unrated.
Here’s how ranks are displayed.
Default Risk Affects Bond Pricing
Each differs slightly. Investment grade bonds are rated BBB and above by S&P. These same bonds are Baa and above by Moody’s. Speculative grade or junk bonds are rated BB or lower by S&P, Ba or lower by Moody’s, or unrated.
These same agencies still command the market despite fraudulent CDO ratings as described in the movie, “The Big Short.”
Bond Safety Signals
There are a group of financial ratios that are derived from the income statement and balance sheet known to impact bond ratings. These shift as ratios change each quarter when a public company that issues bonds reports financials.
As bond safety ratios fluctuate so do ratings from these agencies. As the ratings change the actual bond prices fluctuate through the market process at bid and ask.
This is how changes in financial statements impact bond prices in seconds following release of quarterly financial data.
The coverage ratio shows you how company earnings relate to fixed costs. Leverage ratios relate debt to equity.
Liquidity ratios are also important. The current ratio links current assets to current liabilities.
The quick ratio shows assets excluding inventories with respect to liabilities. The profitability ratio measures the rate of return (RoR) on assets or equity. The Cash flow-to-debt ratio shows you the total cash flow to outstanding debt.
Financial Ratios Are Linked to Default Risk
I cover each of these ratios extensively in the value investing course. Notice that strong ratios are associated with strong ratings and vice versa. A Value-Line subscription allows you to filter across thousands of firms with these parameters of these key financial ratios.
The indenture is a contract between bond issuer and holder. The sinking fund is an indenture clause calling for the issuer to periodically repurchase bonds before maturity.
Subordinations are restrictions on additional borrowing. Senior bondholders are paid first in bankruptcy.
Collateral is a specific asset pledged in case of bankruptcy (default). At the opposite end of the fixed income spectrum is a debenture. This is a bond with no collateral.
Default Risk and Yield to Maturity
The stated yield is the maximum potential yield to maturity of bond. The default premium is the increase in promised yield that compensates the investor for default risk. Bond ratings plummet as the risk of default increases.
The risk of default increases with erosion of these key financial ratios.
As bond ratings plummet the default premium rises. Hence, there is an inverse relationship between bond ratings and default premiums.
Corporate versus 10-Year Treasury Bond Yield Spreads
Notice the fast surge in credit spreads from 2008 to 2009. This was when Lehman Brothers went bust in August of 2008.
Credit Default Swaps (CDS)
A Credit Default Swap is a CDS. It was designed as an insurance policy on the default risk of a corporate bond or loan. It allowed lenders to buy protection against losses on large loans. It was later used to speculate on the financial health of companies. The resulting financial debacle is chronicled in the movie, “The Big Short.”
Credit Default Swaps (CDS) Prices
As credit conditions worsened the price of credit default swaps (bond insurance) increased. Notice how prices of 5-year credit default swaps on U.S. banks maxed out in August of 2008 when Lehman collapsed.
German Sovereign Debt
Credit default swaps are also used to insure the sovereign debt of countries. Notice how the 2009 recession in the aftermath of the big short drives up prices of German credit default swaps. Then the German CDS was pummeled again by the Greek debt crisis as prices maxed out.
The Yield Curve
Yield Curve is a graph of yield to maturity as a function of term to maturity. The Term Structure of Interest Rates is the relationship across yields to maturity and terms to maturity across bonds. The Expectations Hypothesis states that yields to maturity are determined solely by expectations of future short-term interest rates.
Returns from Two 2-Year Investment Strategies
What if everybody in an 8% interest rate market believes that interest rates will rise to 10% next year?
Investors earn 8% in the first year and 10% in the second. They buy the one year bond and roll into the second year.
You would expect about 9% as an average return.
The investment will grow by a factor of 1.08 and 1.10 in the first and second year. This is an annual growth rate of 8.995%.
It represents a growth factor of 1.089952 = 1.188. Notice that 8.995% is slightly less than 9%.
Two-year bond investments must return 8.995% to be competitive with the current one year 8% bond.
This is an example of expectations hypothesis that asserts that the slope of the yield curve changes with short term rates. In this paradigm, the expected holding period returns on bonds should be equal.
Notice how the graphic shows that these two strategies are exactly equivalent. Investing in two consecutive one year bonds, one at 8% and the second at 10% yields the same 1.188 cumulative expected return as investing in a two-year bond.
The Forward Rate
Don’t confuse the bond forward rate with that of Forex. This is the Return on Investment (ROI) that equalizes the expected total long term bond return with that from rolling short-term bonds. This formula is useful for estimating the liquidity risk premium.
The liquidity preference theory states that investors require a risk premium on long-term bonds. This liquidity risk premium is a little extra expected return investors require for taking more risk while waiting for the maturity of long-term bonds. This is measured as the spread between the forward Return on Investment and the expected short sale.
Yield Curve Examples
Two possible yield curves are plotted. In panel A the market expects interest rates to rise over time. In panel B rates are expected to fall. But the curve is not inverted. The liquidity premium gives it a hump. Hump shaped curves are not nearly as predictive as inverted yield curves. In the case of an inverted curve public opinion of futures interest rates are so strongly negative that any effect of a liquidity premium is washed out.
Interest rates are thus linked over the years forward.
This theory predicts that an inverted yield curve is highly predictive of an impending decline in interest rates. This has been supported by extensive empirical work in academic economics and finance.
An inverted yield curve is a door to El Dorado for bond investors who pay attention.
The Term Spread
The bottom line measures yield differences between the 10-year and 90-day Treasury Securities. Notice that it has gone negative four times since 1970. After each instance interest rates dropped.
Section 2 Quiz
Section 2 Quiz Mathematical
Interest Rate Risk
A big part of the risk of bond investing involves interest rate sensitivity. First understand that bond prices and yields are inversely related.
Bonds with higher yield to maturity (YTM) undergo smaller price decreases when rates rise as compared to YTM increases when rates fall an equal magnitude. Long-term bond prices are more sensitive to interest rate changes than short-term bonds.
As maturity increases, the sensitivity of bond prices to falling yield, increases at a decreasing rate.
More Interest Rate Sensitivity
Interest rate risk is inversely related to the bond’s coupon rate. Low-coupon bonds are more sensitive to interest rates.
The current yield to maturity is inversely related to the sensitivity of a bond’s price-to-yield.
Work by Princeton economist Burton Malkiel in 1962 shows that all four bonds in this graphic suffer price reductions as yields rise. Furthermore, this relationship is convex. Bond prices and yields are inversely related.
An increase in yield to maturity induces smaller price changes than a decrease of equal magnitude.
The longer maturity bond B is more sensitive to interest rate changes than the shorter maturity bond A. Bond B has 6 times the maturity of bond A but has less than 6 times the interest rate sensitivity.
This shows that the sensitivity of bond prices to fluctuations in yield increases at a decreasing rate as maturity is lengthened.
Bonds B and C are identical except for coupon rate. The lower coupon bond is more price sensitive to interest rates.
Bonds C and D are the same except for yield to maturity. Bond C has the higher yield to maturity and is less sensitive to fluctuations in yield. Economists Homer and Leibowitz in 1972 showed that price sensitivity is inversely related to yield to maturity.
This table gives pricing for 8% Annual Coupon bonds at different yields and times to maturity paying just once a year — to simplify. Short term bonds fall by less than 1% on a rate increase from 8 to 9%. The 10-year bond price falls more than 6%. The 20-year bond falls more than 9%.
Prices fluctuate much more extremely when the asset is a zero-coupon bond. The two bonds are not comparable.
In fact, we can look at each bond as a portfolio of interest payments.
The annual coupon bond makes many payments over 10 or 20 years. The zero pays just once. This dramatically affects pricing.
Here is why.
High coupon bonds have more value tied to payments; rather than final par value. These have lower effective maturities that are some sort of average of all the cash flows.
This is the reason that price sensitivity drops with rising coupon rates as Malkiel explained in his fifth rule.
But a higher yield reduces the present value of all the bond’s payments, and much more so for those most distant in time. At higher yields the bond value is weighted more heavily on earlier payments with lower effective maturity.
This explains why the sensitivity of bond prices to yield fluctuations is lower.
With zeros, the situation is the opposite. Much more value is paid to the final par payment.
The zero-coupon bond is much more interest rate sensitive. Higher yield bonds have more weight on prior payments which have lower maturity.
Duration was formulated by economist Frederick Macaulay. He published a massive study on railroad bonds in 1938 for the National Bureau of Economic Research (NBER).
Macaulay’s duration measures effective bond maturity.
I mentioned this before so that you would understand the importance of correctly measuring effective maturity to gauge bond pricing fluctuations based on the 5 +1 rules of Malkiel, et. al. This represents the weighted average of the times until each payment, with weights proportional to the present value of the coupon.
You can see that each cash flow is discounted and proportional to the bond price. This is multiplied by the time to maturity and summed over coupon and par payments as follows.
The Calculation of the Duration of Two Bonds
This spreadsheet shows the calculation of the durations of two bonds, a zero and an 8% coupon. The assumption is that bond yield to maturity is 10%.
The discount factor is ten percent.
Column E shows the weight and D shows payment present value. Column F is the product of weight and payment. This gives us the values needed to crunch the duration formula by adding the numbers to column F.
Notice that the duration of the zero-coupon bond is 3.0000 years. Duration is the zero-coupon bond’s maturity.
This makes sense since the zero-coupon bond only makes one payment at par.
The three-year coupon bond makes very few payments yet these have an impact on duration. Duration is shorter than the three-year maturity at 2.7774 years. Spreadsheets such as this allow you to change values to make scenario analysis as to how your bond portfolio will fluctuate under different conditions.
Here are the formulas you need to set this spreadsheet up yourself in excel.
The rate of change of the yield to maturity of the bond price can be measured as modified duration.
How to Calculate Duration
Use this spreadsheet to confirm the duration of the 8% bond example. Settlement date is todays date and the maturity are entered in cells B2 and B3 with the Excel date function. DATE(year, month, day). There is no specific date of settlement for this three-year bond.
It is arbitrarily set with a maturity exactly 3 years later.
Coupon rate and yield to maturity are in decimal format. Payment periods go into cell 6. Macaulay and modified duration results are displayed in cells B9 and B10 as 2.7774 and 2.5249.
What Influences Duration?
The easiest way to see what factors influence duration is by noticing that a zero-coupon bond’s duration is the time to maturity. This is not so with a coupon bond.
With time and yield to maturity constant, the bond’s duration and interest-rate sensitivity is higher when the coupon price is lower.
When the coupon rate is held constant, the bond’s duration and interest-rate sensitivity generally increases with time to maturity. Here we see that at or above par bonds have increases in duration that are concomitant with longer maturity.
Both the duration and the interest rate sensitivity of the coupon is higher when the bond yield to maturity is lower. This allows us to set up the formula for calculating the level of duration of a perpetuity.
Perpetuity Duration= (1+y)/y
Now we can use this to graph the duration as a function of maturity to see Malkiel’s bond pricing rules. The plot of the zero-coupon bond shows that the duration is the same as the yield to maturity — this is rule 1.
The fact that the line of the three-year coupon bond is below that of the zero-coupon bond shows that interest rate sensitivity is higher the lower the coupon payment all else equal — this is rule 2. The plot of the 3% coupon is above that of the 15% coupon bond with equal yields to maturity showing that duration and interest rate sensitivity increases with maturity — this is rule 3.
The two 15% coupon bonds with different yields to maturity have different durations. This illustrates rule 4 that duration and interest rate sensitivity of a coupon bond are higher when the bond’s yield to maturity is lower.
In practice, you will see a wide range of durations values for actively traded bonds as an investor. This table show durations for several coupon bonds all paying 6% per year. Duration decreases with decreasing coupon rates and increases with time to maturity. The 20-yer bond would fall 1.15% with a rate increase from 6 to 6.1% (= -12.158 X .1%/1.06) but the 5-year bond duration drops just 0.41% (=-4.342 X .1%/1.06). With a perpetuity duration is independent of coupon rate.
Passive Bond Management
Immunization is a defense strategy to shield net worth from interest rate movements. This involves rebalancing through periodic portfolio realignment.
Terminal Value of a Bond Portfolio after Five Years
Immunization makes a lot of sense for any institution that has bond obligations to pay. The idea was developed by Frank Redington in 1952 who was a practicing insurance actuary.
Immunization allows duration matched assets and liabilities to fully fund an investment portfolio despite adverse interest rate movements.
This table shows that if interest rates stay at 8%, that the funds from the bond will grow to a $14,693.28 obligation for the issuer in five years. Imagine that an insurance company funds this obligation with $10,000 of 8% annual coupon bonds.
These are selling at par value with a 6-year maturity.
If market rates continue at 8% there is no problem. Both bonds will have an equal terminal value.
If interest rates rise, the insurance company faces a shortfall to pay the bond investor. Panels B and C show this scenario under duration matching.
Panel B shows that a small surplus of $0.77 will accrue if rates fall to 7%. Panel C shows that the surplus would be $2.74 should interest rates rise to 9%.
Duration matching balances the accumulated value of the fixed payments with the sale value of the bond.
Growth of Invested Funds
The solid line traces the value of bonds if rates stay at 8%. The dashed curve represents the value if rates increase. Remember that the issuer faces risk from rate increases; not decreases. The initial shock causes a loss. This is recovered by a faster growth rate of reinvested funds.
Inflows and outflows cancel at maturity in 5 years.
Market Value Balance Sheets
This is a balance sheet for the hypothetical insurance company in this example. Both assets and obligations are valued at $10,000 in panel A. This indicates that the plan is fully funded. Panels B and C show that regardless of whether the rate change is positive or negative that assets and liabilities change by virtually identical amounts.
Immunization is displayed in this graph. At 8% the coupon bond fully funds the obligation where the two present value curves are tangent to one another. This is true even for small rate changes. This also shows that immunization requires rebalancing. At 7% the duration is 5.02 years. It is 4.97 at 9%. Without rebalancing the portfolio durations will not be matched. Duration changes with the passage of time. Immunization is not a passive strategy except for the fact that the manager is not trying to identify undervalued bonds.
Cash Flow Matching and Deduction
Cash flow matching involves matching cash flows from a fixed-income portfolio with those of the obligation. This is an immunization strategy. Multi-period cash flow matching is a dedication strategy. This dedicates a zero or coupon bond to each cash flow. This eliminates interest rate risk once and for all since each cash flow is matched. A dedicated portfolio will pay in full no matter the change in interest rates.
The curvature in the bond price to yield is expressed by …
The last term arises from a Taylor expansion of the modified duration equation. The additional term represents convexity. Investors love convexity because bonds with greater curvature gain more when yields fall but lose less when rates rise. But the market prices convexity since it is desirable. Investor pay more and get lower yields on bonds with greater convexity.
Bond Price Convexity
This graph shows that the percentage change in the bond price is a convex function of the change in yield to maturity. The straight line is the percentage price change predicted by the duration rule for a 30 year, 8% coupon bond selling at an initial yield of 8%. The modified duration of the bond at its initial yield is 11.26 years.
The straight line is a plot of -D*Δy = -11.26 X Δy.
This indicates that small changes in interest rates have little effect on duration. But the duration approximation (straight line) underestimates duration. The true price and yield relationship is the upward curved (convex) line.
This shows that the duration approximation can become very inaccurate for large interest rate changes.
Why Do Investors Like Convexity?
More convexity means greater price increases when the market is good. But the bond investor endures smaller losses when interest rates movements induce a reduction in bond portfolio value.
Active Bond Management Potential Profit Sources
The Substitution swap is an exchange of one bond for a bond with similar attributes but better pricing.
An inter-market swap switches from one segment of the bond market to another. Rate anticipation swap switching is made in response to forecasts of interest rate changes.
More Active Strategies
Pure yield pickup swaps move to higher yield bonds, usually with longer maturities. Tax swaps entail swapping two similar bonds to receive a tax benefit.
Horizon analysis is a forecast of bond returns based largely on the prediction of the yield curve slope at the end of an investment horizon.
Active Fixed-Income Investment Strategy
For the strategy to work …
information cannot already be embedded in bond prices to be of value.
Making big bets on interest rate movements is unwise except during the rare periods when the yield curve slope is highly negative.
If the manager believes markets are efficient they will use substitution swaps and inter-market spreads rather than try to identify undervalued securities.
Active Bond Speculative Strategies.
Bill Gross made a fortune for his investors betting on the inverted yield curve in bonds. Other bond portfolio managers attempt to identify undervalued securities.
This was explained to me by successful bond trader Ben Eiler. Ben is a regular on CNBC.
An example he gave was a bond series in a state. He knew that the person in charge of the municipal bonds was not financially sophisticated.
Ben bought the bonds at a cheap discount.
He knew that the price would soar when the state governor announced that he had assembled a task force to clean up the administration of the municipal bonds. That is because such task forces are composed of the brightest and the best in finance from top business schools.
They came in and cleaned house. Ben made a sizeable profit for his investors as the price rose. Hence, one way you can use what you have learned from this course is to attempt to find underpriced bonds.
This course gives you the tools to manage your own bond portfolio as a passive or active investor.
Keep in touch with me if you do. I want to know what happens. -Doc Brown
Section 3 Quiz Conceptual
Section 3 Quiz Mathematical
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My Curriculum Vitae:
Investment Writing and Speaking:
I am an internationalspeaker oninvestments. In 2010 I gave a series of lectures onboard Brilliance of the Seas as a guest speaker on their Mediterranean cruise. Financial topics are normally forbidden for cruise speakers. But with me they make an exception because of my financial pedigree.
On day 6 the topic I discussed was “Free and Clear: Secrets of Safely Investing in Real Estate!“ The day 7 topic was “Investment Style and Category: How the Stock Market Really Works!” Then on day 8 I spoke about “The 20% Solution: How to Survive and Thrive Financially in any Market!” The final talk on day 11 was “Value Investing for Dummies: When Dumb Money is Smart!”
Gina Verteouris is the Cruise Programs Administrator of the Brilliance of the Seas of Royal Caribbean Cruise Lines. Regarding my on-board teachings she writes on June 19th, “You have really gone above and beyond expectations with your lectures and we have received many positive comments from our Guests.”
I sponsored and organized an investing conference at Caesars Palace in Las Vegas in 2011 under my Wallet Doctor brand. This intimate conference was attended by 14 paying attendees.
As such many strides were made in financial education that week. For instance I met a woman who is a retired engineer from the Reno, Nevada area.
She made a fortune on deep in the money calls during the bull markets of the 90s.
This humble and retired engineer inspired me to look more seriously at deep in the money calls with far expiration. She also gave me an important clue regarding trading volume.
Her call option and volume insights have been confirmed in the Journal of Finance.
In 2012 I gave a workshop at the FreedomFest Global Financial Summit on stock investing at the Atlantis Bahamas Resort. I was also a panelist on a discussion of capital markets.
My course “How to Build a Million Dollar Portfolio from Scratch" at the Oxford Club is an international bestseller. In 2014 I co-authored “Tax Advantaged Wealth” with leading IRS expert Jack Cohen, CPA. This was the crown jewel of the Oxford Club Wealth Survival Summit.
I have been a regular speaker at the Investment U Conferences.
In 2012 I gave a workshop entitled “How to Increase Oxford Club Newsletter Returns by 10 Fold!” The conference was held at the Grand Del Mar Resort in San Diego, California. This resort destination is rated #1 on TripAdvisor.
In 2013 I spoke at the Oxford Club’s Investment U Conference in San Diego California. The talk was entitled “The Best Buy Signal in 103 Years!” Later in the summer I spoke at the Oxford Club Private Wealth Conference at the Ojai Valley Inn.
This was at the same time that Jimmy Kimmel married Molly McNearney in the posh California celebrity resort. It was fun to watch some of the celebrities who lingered.
I also operate a live weekly investment mentorship subscription service under the Bullet-Proof brand every Monday night by GoToWebinar.
I am an associate professor of finance of the AACSB Accredited Graduate School of Business at the University of Puerto Rico. My research appears in some of the most prestigious academic journals in the field of investments including the Journal of Financial Research and Financial Management. This work is highly regarded on both Main Street and Wall Street. My research on investment newsletter returns was considered so important to investors that it was featured in the CFA Digest.
The Certified Financial Analyst (CFA)is the most prestigious practitioner credential in investments on Wall Street.
Prestigious finance professor Bill Christie of the Owen School of Business of Vanderbilt University and then editor of Financial Management felt that our study was valuable to financial society. We showed that the average investment newsletter is not worth the cost of subscription.
I am the lead researcher on the Puerto Rico Act 20 and 22 job impact study. This was signed between DDEC secretary Alberto Bacó and Chancellor Severino of the University of Puerto Rico.
(See Brown, S., Cao-Alvira, J. & Powers, E. (2013). Do Investment Newsletters Move Markets? Financial Management, Vol. XXXXII, (2), 315-338. And see Brown, S., Powers, E., & Koch, T. (2009). Slippage and the Choice of Market or Limit orders in Futures Trading. Journal of Financial Research, Vol. XXXII (3), 305-309)
I hold a Ph.D. in Finance from the AACSB Accredited Darla Moore School of Business of the University of South Carolina. My dissertation on futures market slippage was sponsored by The Chicago Board of Trade. Eric Powers, Tim Koch, and Glenn Harrison composed my dissertation committee. Professor Powers holds his Ph.D. in finance from the Sloan School of Business at the Massachusetts Institute of Technology [MIT]. Eric is a leading researcher in corporate finance and is a thought leader in spin offs and carve outs.
Dr. Harrison is the C.V. Starr economics professor at the J. Mack Robinson School of Business at Georgia State University.
He holds his doctorate in economics from the University of California at Los Angeles. Glenn is a thought leader in experimental economics and is the director of the Center for the Economic Analysis of Risk.
Tim Koch is a professor of banking. Dr. Koch holds his Ph.D. in finance from Purdue University and is a major influence in the industry.
My dissertation proved that under normal conditions traders and investors are better off entering on market while protectingwith stop limit orders. The subsequent article was published in the prestigious Journal of Financial Research now domiciled at Texas Tech University — a leading research institution.
I earned a masters in international financial management from the Thunderbird American Graduate School of International Business. Thunderbird consistently ranks as the #1 international business school in the U.S. News & World Report, and BloombergBusinessWeek.
I spoke at the 2010 annual conference of the International Association of Business and Economics (IABE) conference in Las Vegas, Nevada. The research presented facts regarding price changes as orders flow increases in the stock market by advisory services.
I spoke at the 2010 Financial Management Association [FMA] annual conference in New York on investment newsletters. The paper was later published in the prestigious journal “Financial Management.”
I presented an important study named “Do Investment Newsletters Move Markets?” at the XLVI Annual Meeting of the Consejo Latinoamericano de Escuelas de Administración (CLADEA) in 2011 in San Juan, Puerto Rico. The year before that I presented my futures slippage research at a major renewable energy conference in Ubatuba, Brazil.
I spoke at the Clute International Conferences in 2011 in Las Vegas, Nevada. The research dealt with the price impact of newsletter recommendations in the stock market.
I presented a working paper entitled “The Life Cycle of Make-whole Call Provisions” at the 2013 Annual Meeting of the Southern Finance Association in Fajardo, Puerto Rico in session B.2 Debt Issues chaired by Professor LeRoy D. Brooks of John Carroll University. Luis Garcia-Feijoo of Florida Atlantic University was the discussant. I chaired the session entitled “Credit And Default Risk: Origins And Resolution.” Then I was the discussant for research entitled "NPL Resolution: Bank-Level Evidence From A Low Income Country" by finance professor Lucy Chernykh of Clemson University and Abu S Amin of Sacred Heart University and Mahmood Osman Imam of the University of Dhaka in Bangladesh.
That same year I presented the same study to the Annual Meeting of the Financial Management Association in Chicago, Illinois. I did so in session 183 – Topics in Mergers and Acquisitions chaired by James Conover of the University of North Texas with Teresa Conover as discussant. I chaired session 075 – Financial Crisis: Bank Debt Issuance and Fund Allocation. Then I was the discussant for TARP Funds Distribution: Evidence from Bank Internal Capital Markets by Elisabeta Pana of Illinois Wesleyan University and Tarun Mukherjee of the University of New Orleans.
I am a member of the MBA Curriculum Review Committee, the MBA Admissions Committee, The Doctoral Finance Admissions Committee, the Graduate School Personnel Committee, and the Doctoral Program Committee of the School of Business of the University of Puerto Rico.
I am the editor of Momentum Investor Magazine. I co-founded the magazine with publisher Daniel Hall, J.D. We have published three issues so far. Momentum Investor Magazine allows me to interview very important people in the finance industry. I interview sub director Suarez of the DDEC responsible for the assignment of Puerto Rico act 20 and 22 licenses for corporate and portfolio tax reduction in the third edition. Then I interview renowned value investor Mohnish Prabia in the upcoming fourth edition — to be made available via Udemy. Valuable stock market information will be taught throughout.
In October of 2010 I arranged for the donation to The Graduate School of Business of the University of Puerto Rico of $67,248 worth of financial software to the department that has been used in different courses. This was graciously awarded by Gecko Software.
I have guided thousands of investors to superior returns. I very much look forward to mentoring you as to managing your investments to your optima! –Scott
Dr. Scott Brown, Associate Professor of Finance of the AACSB Accredited Graduate School of Business of the University of Puerto Rico.