1. Chapter 10 Bond Prices and Yields 4/19/2017
Chapter 10 continues our look at interest-bearing assets, focusing directly on longer term instruments – bonds.

2. 4/19/2017
Straight Bond
Obligates the issuer of the bond to pay the holder of the bond:
A fixed sum of money (principal, par value, or face value) at the bond’s maturity
Constant, periodic interest payments (coupons) during the life of the bond (Sometimes)
Special features may be attached
Convertible bonds
Callable bonds
Putable bonds
“Straight bonds” are the most common type of bond.
Basically it is a debt contract that obligates the issuer to pay the holder a fixed sum of money periodically along with a repayment of the principal at maturity.
We assume that these bonds have a face value of $1,000 unless stated otherwise. We also assume they pay semiannual coupons.

3. Straight Bond Basics $1,000 face value Semiannual coupon payments
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Straight Bond Basics
$1,000 face value
Semiannual coupon payments
The coupon rate for a straight bond is the annual coupon divided by the face value. This is usually the coupon rate stated at issue.
The current yield is typically reported in the financial press and relates the same annual coupon to the current market price of the bond.

4. 4/19/2017
Straight Bonds
Suppose a straight bond pays a semiannual coupon of $45 and is currently priced at $960.
What is the coupon rate?
What is the current yield?
Here’s a simple example of bond calculations.
Suppose the bond pays a semiannual coupon of $45 and its current market price is $960 or 96% of our assumed par value of $1,000.
The annual coupon is two times the given semiannual coupon or $90.
The coupon rate is then $90 divided by $1,000 or 9%
The current yield divides that same $90 annual coupon by the market price of $960 to arrive at a current yield of 9.375%.

5. Straight Bond Prices & Yield to Maturity
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Straight Bond Prices & Yield to Maturity
Bond Price:
Present value of the bond’s coupon payments
+ Present value of the bond’s face value
Yield to maturity (YTM):
The discount rate that equates today’s bond price with the present value of the future cash flows of the bond

6. Bond Pricing Formula PV of coupons PV of FV Where:
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Bond Pricing Formula
PV of coupons PV of FV
Where:
C = Annual coupon payment
FV = Face value
M = Maturity in years
YTM = Yield to maturity
The price of a straight bond can be decomposed into the present value of the coupon stream plus the present value of the face value payment at maturity.
These two components are shown in equation 10.3 above.
On the next slide we’ll use this formula to value a bond.

7. Straight Bond Prices Calculator Solution
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Straight Bond Prices Calculator Solution
Where:
C = Annual coupon payment
FV = Face value
M = Maturity in years
YTM = Yield to maturity
N = 2M
I/Y = YTM/2
PMT = C/2
FV = 1000
CPT PV
We’re going to use the formula for bond prices but don’t forget that you can easily arrive at the same solution (or check your work) using the basic time-value-of-money keys on your calculator.
Remember that straight bonds are semiannual payers so “N” must be set to the number of years to maturity times 2.
The interest rate (I/Y) will be the yield to maturity divided by 2.
Since YTMs are quoted as APRs, we can just divide by 2 to get a semiannual rate.
“Payment” (PMT) should be the SEMIANNUAL coupon payment.
“Future Value” is the face value of the bond, usually $1,000
Computing “Present Value” will yield the bond price.
Though not shown, the same technique can be used with spreadsheet functions for the time-value-of-money.

8. Straight Bond Prices PV of coupons PV of FV
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Straight Bond Prices
PV of coupons PV of FV
For a straight bond with 12 years to maturity, a coupon rate of 6% and a YTM of 8%, what is the current price?
Using equation 10.3, our example is a straight bond with 12 years to maturity. Its coupon rate is 6% and it has a yield to maturity of 8%.
Substituting those given values into the equation components, we arrive at a value of $ for the coupon stream and $ for the face value.
Adding the value of these two components, we arrive at a bond price of $
Notice that this price is below the face value of $1,000 which we should expect since the yield to maturity is greater than the coupon rate, making this a discount bond.
Price = $ $ = $847.53

9. Calculating a Straight Bond Price Using Excel
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Calculating a Straight Bond Price Using Excel
Excel function to price straight bonds:
=PRICE(“Today”,“Maturity”,Coupon Rate,YTM,100,2,3)
Enter “Today” and “Maturity” in quotes, using mm/dd/yyyy format.
Enter the Coupon Rate and the YTM as a decimal.
The "100" tells Excel to us $100 as the par value.
The "2" tells Excel to use semi-annual coupons.
The "3" tells Excel to use an actual day count with 365 days per year.
Note: Excel returns a price per $100 face.

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Spreadsheet Analysis

11. Par, Premium and Discount Bonds
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Par, Premium and Discount Bonds
Par bonds: Price = par value
YTM = coupon rate
Premium bonds: Price > par value
YTM < coupon rate
The longer the term to maturity, the greater the premium over par
Discount bonds: Price < par value
YTM > coupon rate
The longer the term to maturity, the greater the discount from par

12. Premium Bond Price 12 years to maturity
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Premium Bond Price
12 years to maturity
8% coupon rate, paid semiannually
YTM = 6%
A premium bond is one with a coupon rate greater than its yield to maturity which makes it more attractive to investors. We expect the price to be greater than the face value.
Using our given data in the formula yields a value of $ for the coupon stream and $ for the face value.
Together these components give us a price of $1,169.36
Price = $ $ = $1,169.36

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Discount Bonds
Consider two straight bonds with a coupon rate of 6% and a YTM of 8%.
If one bond matures in 6 years and one in 12, what are their current prices?
To see how maturity affects a discount bond, we look at identical bonds which differ only in years to maturity.
Notice that the discount bond with the shorter maturity has the higher price.

14. 4/19/2017
Premium Bonds
Consider two straight bonds with a coupon rate of 8% and a YTM of 6%.
If one bond matures in 6 years and one in 12, what are their current prices?
Applying the same comparison to two premium bonds, we see that in this case, the bond with the shorter maturity has the LOWER price.
The next slide will explain what’s happening with these bond prices.

15. Bond Value ($) vs Years to Maturity
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Bond Value ($) vs Years to Maturity
Premium
CR>YTM
8%>6%
YTM = CR M
1,000
Bond prices MUST converge to their face value as the bond approaches maturity. Logically this makes sense --- how much would you pay for a bond with a face value of $1,000 that matures tomorrow??? Right about $1,000??
This is true regardless of whether the bond started out as a premium or discount bond.
The diagram above demonstrates this convergence.
While this graph is a nice symmetric depiction, the truth is that, over a bond’s life, it may vacillate between selling at a discount, at par and at a premium at different times depending on market interest rates.
The point is – regardless of where its price is relative to the “PAR” line, that price will continue to converge toward the face value as maturity approaches.
CR<YTM
6%<8%
Discount

16. Premium and Discount Bonds
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Premium and Discount Bonds
In general, when the coupon rate and YTM are held constant:
For premium bonds: the longer the term to maturity, the greater the premium over par value.
For discount bonds: the longer the term to maturity, the greater the discount from par value.

17. Relationships among Yield Measures
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Relationships among Yield Measures
For premium bonds:
coupon rate > current yield > YTM
For discount bonds:
coupon rate < current yield < YTM
For par value bonds:
coupon rate = current yield = YTM

18. A Note on Bond Quotations
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A Note on Bond Quotations
If you buy a bond between coupon dates:
You will receive the next coupon payment
You might have to pay taxes on it
You must compensate the seller for any accrued interest.

19. A Note on Bond Quotations
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A Note on Bond Quotations
Clean Price = Flat Price
Bond quoting convention ignores accrued interest.
Clean price = a quoted price net of accrued interest
Dirty Price = Full Price = Invoice Price
The price the buyer actually pays
Includes accrued interest added to the clean price.

20. Clean vs. Dirty Prices Example
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Clean vs. Dirty Prices Example
Today is April 1. Suppose you want to buy a bond with a 8% annual coupon payable on January 1 and July 1.
The bond is currently quoted at $1,020
The Clean price = the quoted price = $1,020
The Dirty or Invoice price = $1,020 plus (3mo/6mo)*$40 = $1,040
We’ve been looking at calculating bond prices based on quoted values. There is, however, another element that comes into play when you actually buy or sell a bond.
The quoted price is called the “clean price” but is NOT what you would actually pay to buy the bond.
The reason is accrued interest. Bonds pay interest every six months so if you buy a bond at any time between those payment dates, interest will have accrued to the owner.
You have to include this accrued interest in the price you pay, which is called the “dirty price.”
Actual methods to compute the dirty price can get complicated so let’s look at a very simple example.
Our bond pays interest on January and July 1. The coupon rate is 8% with a clean price of $1,020.
If we buy the bond on April 1, we owe the current owner 3 months of accrued interest or 3/6ths of a $40 interest payment.
Adding that to the clean price yields a dirty price of $1,040.

21. Calculating Yields Trial and error Calculator Spreadsheet 4/19/2017
Calculating yield on a bond is like calculating the Internal Rate of Return (IRR) of a cash flow stream. There is no closed form solution to solve for yield.
There are three methods available:
Trial and error – basically a manual approach
Calculator – financial calculators are programmed to solve for YTM
Spreadsheet – contains functions to solve for YTM
We’ll use a 5% bond priced at 90% of par with 12 years to maturity as our example of all three methods.

22. Calculating Yields Trial & Error
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Calculating Yields Trial & Error
A 5% bond with 12 years to maturity is priced at 90% of par ($900).
Selling at a discount YTM > 5%
Try 6% --- price = $ too high
Try 6.5% --- price = $ too low
Try 6.25% --- price = $ a little low
Actual = %
The trial and error method uses our basic bond price formula (10.3) to try to converge on the YTM rate that will result in the quoted price.
Since this bond is selling at a discount, we know the YTM is greater than 5%.
Trying 6% results in a price of $ or 91.53% -- too high
Raising the YTM estimate will reduce the price so we try 6.5%. This results in a price of 87.63% of par – too low.
Now we know the true yield is between 6% and 6.5%
Trying 6.25% results in a price still a bit too low.
Continuing this process will eventually converge to the true yield of %.

23. Calculating Yields Calculator
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Calculating Yields Calculator
A 5% bond with 12 years to maturity is priced at 90% of par ($900).
N = 24
PV = -900
PMT = 25
FV = 1000
CPT I/Y = x 2 = %
Continuing our example, using a financial calculator, we need to remember to adjust our inputs to reflect semiannual payments.
With 12 years to maturity, “N” will be 24.
Since the bond is selling at 90% of par, our PV will be “-900.” Remember this is an outflow as we would be buying the bond.
With a 5% coupon rate, our PAYMENT (PMT) will be $25.
Assuming a $1,000 par value, our Future Value will be 1000.
Computing the rate (CPT I/Y) yields a SEMIANNUAL rate of %.
Doubling that gives us our annual YTM of %.

24. Calculating Yields Spreadsheet
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Calculating Yields Spreadsheet
5% bond with 12 years to maturity,priced at 90% of par
=YIELD(“Now”,”Maturity”,Coupon, Price,100,2,3)
“Now” = “06/01/2008”
“Maturity” = “06/01/2020”
Coupon = .05
Price = 90 (entered as a % of par)
100 redemption value as a % of face value
“2” semiannual coupon payments
“3” actual day count (365)
=YIELD(“06/01/2008”,”06/01/2020”,0.05,90,100,2,3) =
With a spreadsheet, there are really 2 ways to solve for YTM.
The first replicates the calculator inputs and uses the TVM functions.
The more bond-specific approach uses the YIELD function which is similar to the BOND Worksheet on the TI BA II+.
The format is shown on the slide above.
Rather than using number of years to maturity, it uses exact dates.
For our sample bond,
Let’s set “NOW” as June 1, 2008 – note that the dates are in parentheses and entered as “06/01/2008”
That makes “MATURITY” as 12 years from now or “06/01/2020”
“Coupon” is the percentage coupon rate as a decimal or “0.05”
“Price” is entered as a percentage of par or 90 for our example.
The “100” means the payoff at maturity is 100% of the face value.
The “2” designates this as a semiannual payer.
The “3” tells the function to use actual day count of 365 days in a year.
The result is the same as before: %

25. 4/19/2017
Spreadsheet Analysis

26. Callable Bonds Gives the issuer the option to: Buy back the bond
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Callable Bonds
Gives the issuer the option to:
Buy back the bond
At a specified call price
Anytime after an initial call protection period.
Most bonds are callable Yield-to-call may be more relevant
If a bond can be bought back by the issuer, at their discretion, then the bond is callable. A company might want to call in a bond issue if market rates drop significantly below the coupon rate they are paying on the bonds. Bonds selling at a premium would be susceptible for a call.
The logic (and the decision-making process) is quite similar to deciding if it would be wise to refinance a home mortgage.
Where callable bonds were once uncommon, as interest rates have become more volatile, a call feature has become the norm in corporate bonds.
A callable bond’ normally contains 2 features:
A call protection period – for example, 5 years from issue - to protect buyers from having the bonds called back within five years of origination.
A call premium which is a bonus the issuer pays in addition to the face value at the time the bond is called.

27. Yield to Call Where: C = constant annual coupon
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Yield to Call
Where:
C = constant annual coupon
CP = Call price of bond
T = Time in years to earliest call date
YTC = Yield to call
If called, the bond will not realize the current YTM. A Yield to call (YTC) should be calculated to get an accurate figure of expected yield.
The methodology is the same as for YTM with two adjustments:
The “FV” in the formula is replaced by the Call price which equals the face value plus the call premium.
The time to maturity is replaced by “time to first call,” which for a just-issued bond with five-year call protection would be 5 years or 10 periods.

28. 4/19/2017
Yield to Call
Suppose a 5% bond, priced at 104% of par with 12 years to maturity is callable in 2 years with a $20 call premium. What is its yield to call?
N = 4 # periods to first call date
PV = -1040
PMT = 25
FV = 1,020 Face value + call premium
CPT I/Y = x 2 = 3.874%
As an example of calculating yield to call, let’s look at a 5% coupon bond currently priced at 104% of par. The bond has 12 years to maturity but is callable in 2 years with a $20 call premium.
We’ll use the calculator method to solve for YTC but the adjustments would be similar for either the manual or spreadsheet methods.
Note:
“N” is now the number of periods to the first call – in this case 4.
“PV” reflects the current price which is $1,040 entered as an outflow
“PMT” is unchanged at ½ the annual coupon or $25
“FV” reflects both the face value of $1,000 and the call premium of $20
Computing I/Y results in a 6-month rate.
The annualized yield to call is 3.874%
Remember: resulting rate = 6 month rate

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Interest Rate Risk
Interest Rate Risk = possibility that changes in interest rates will result in losses in the bond’s value
Realized Yield = yield actually earned or “realized” on a bond
Realized yield is almost never exactly equal to the yield to maturity, or promised yield

30. Interest Rate Risk and Maturity
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Interest Rate Risk and Maturity

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Malkiel’s Theorems
Bond prices and bond yields move in opposite directions.
For a given change in a bond’s YTM, the longer the term to maturity, the greater the magnitude of the change in the bond’s price.
For a given change in a bond’s YTM, the size of the change in the bond’s price increases at a diminishing rate as the bond’s term ot maturity lengthens.
Bond yields are essentially interest rates and, as such, react to market changes over time.
Burton Malkiel compiled these reactions into five theorems recapped above.
Numbers 1,2, and 4 are the simplest and most important.
Clearly bond prices and yields move in opposite directions. Knowing that the formula to price a bond is a combination of present valuation formulas makes it obvious that, as the discount rate (YTM) increases, the present value (price) decreases.
The second theorem says that longer term bonds are more sensitive to changes in yield than shorter term bonds. Again, remembering present value theory, discounting affects more distant cash flows more severely. In this case, the face value lump sum is a more distant cash flow.
The fourth theorem says that lower coupon bonds are more sensitive to changes in yield than higher coupon bonds.

32. 4/19/2017
Malkiel’s Theorems
For a given change in a bond’s YTM, the absolute magnitude of the resulting change in the bond’s price is inversely related to the bond’s coupon rate.
For a given absolute change in a bond’s YTM, the magnitude of the price increase caused by a decrease in yield is greater than the price decrease caused by an increase in yield.
Bond yields are essentially interest rates and, as such, react to market changes over time.
Burton Malkiel compiled these reactions into five theorems recapped above.
Numbers 1,2, and 4 are the simplest and most important.
Clearly bond prices and yields move in opposite directions. Knowing that the formula to price a bond is a combination of present valuation formulas makes it obvious that, as the discount rate (YTM) increases, the present value (price) decreases.
The second theorem says that longer term bonds are more sensitive to changes in yield than shorter term bonds. Again, remembering present value theory, discounting affects more distant cash flows more severely. In this case, the face value lump sum is a more distant cash flow.
The fourth theorem says that lower coupon bonds are more sensitive to changes in yield than higher coupon bonds.

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Bond Prices and Yields

34. 4/19/2017
Duration
Duration measure the sensitivity of a bond price to changes in bond yields.
Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields.
While bond maturity is an important factor in measuring a bond’s sensitivity to yield changes, it is incomplete.
The concept of duration is widely used to account for differences in interest rate risk across bonds with different coupon rates and maturities.
Duration measures a bond’s sensitivity to yield changes and is quoted in years.
There are several duration measures but we will only look at two: Macaulay Duration and Modified Duration.

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Macaulay Duration
% Bond Price ≈ -Duration x
(10.5)
A bond has a Macaulay Duration = 10 years, its yield increases from 7% to 7.5%.
How much will its price change?
Duration = 10
Change in YTM = = .005
YTM/2 = .035
%Price ≈ -10 x (.005/1.035) = -4.83%
Macaulay Duration is the original duration measure and derives its strength from its relationship between changes in bond price and yield as shown in equation 10.5 above. Note that this is an approximation, not an exact relationship. Macaulay duration is often referred to as a bond’s effective maturity and, as such, is measured in years.
In our example, a bond with Macaulay Duration of 10 years experiences a change in yield from 7% to 7.5%.
Using the equation above, we can expect the bond’s price to decline by 4.83%.

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Modified Duration
Some analysts prefer a variation of Macaulay’s Duration, known as Modified Duration.
The relationship between percentage changes in bond prices and changes in bond yields is approximately:

37. 4/19/2017
Modified Duration
(10.6)
(10.7) %Bond Price ≈ -Modified Duration x YTM
A bond has a Macaulay duration of 9.2 years and a YTM of 7%. What is it’s modified duration?
Modified duration = 9.2/1.035 = 8.89 years
Modified duration is a variation of Macaulay duration as shown in formula 10.6 above.
For a bond with a Macaulay duration of 9.2 years and a YTM of 7%, the modified duration is 8.89 years.

38. 4/19/2017
Modified Duration
(10.6)
(10.7) %Bond Price ≈ -Modified Duration x YTM
A bond has a Modified duration of 7.2 years and a YTM of 7%. If the yield increases to 7.5%, what happens to the price?
%Price ≈ -7.2 x .005 = -3.6%
Using equation 10.7 above and the Modified Duration we found on the last slide, we can estimate the expected change in bond price resulting from an increase of 0.5% in yield.
An increase in yield of 0.5% would result in a decrease in bond price of 3.6%.

39. Calculating Macaulay’s Duration
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Calculating Macaulay’s Duration
Macaulay’s duration values stated in years
Often called a bond’s effective maturity
For a zero-coupon bond:
Duration = maturity
For a coupon bond:
Duration = a weighted average of individual maturities of all the bond’s separate cash flows, where the weights are proportionate to the present values of each cash flow.

40. Calculating Macaulay Duration
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Calculating Macaulay Duration
(10.8)
Suppose a par value bond has 12 years to maturity and an 8% coupon. What is its duration?
Calculating duration can be a laborious task, incorporating weighted averages of all the bond’s cash flows.
For our example, we’ll focus on calculating Macaulay duration for a par value bond or one selling at par meaning the coupon rate equals the YTM.
Applying formula 10.8 above to our sample data, we can easily arrive at a Macaulay duration of years.

41. General Macaulay Duration Formula
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General Macaulay Duration Formula
(10.9)
Where:
CPR = Constant annual coupon rate
M = Bond maturity in years
YTM = Yield to maturity assuming semiannual coupons
Unfortunately, the formula for Macaulay duration on a bond not selling at par is a bit more complicated as shown in equation 10.9 above.
Rather than look at a separate example of using this formula, on the next few slides we’ll look at how to use these various duration formulas in conjunction with each other.

42. Calculating Macaulay’s Duration
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Calculating Macaulay’s Duration
In general, for a bond paying constant semiannual coupons, the formula for Macaulay’s Duration is:
In the formula, C is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, assuming semiannual coupons.

43. Using the General Macaulay Duration Formula
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Using the General Macaulay Duration Formula
What is the modified duration for a bond that matures in 15 years, has a coupon rate of 5% and a yield to maturity of 6.5%?
Steps:
Calculate Macaulay duration using 10.9
Convert to Modified duration using 10.6
For our example, let’s look at a bond that matures in 15 years. It has a coupon rate of 5% and a yield to maturity of 6.5%.
To arrive at a modified duration measure for this bond, we need to perform two steps.
First we’ll calculate the Macaulay duration using equation 10.9.
Then we’ll convert that to a Modified duration using equation 10.6.

44. Using the General Macaulay Duration Formula
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Using the General Macaulay Duration Formula
Bond matures in 15 years
Coupon rate = 5%
Yield to maturity = 6.5%
Calculate Macaulay duration using 10.9
Substituting the given data into equation 10.9, yields a Macaulay duration of years.

45. Using the General Macaulay Duration Formula
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Using the General Macaulay Duration Formula
Bond matures in 15 years
Coupon rate = 5%
Yield to maturity = 6.5%
Macaulay Duration = years
Convert to Modified duration using 10.6
Using the Macaulay duration of years from the previous slide, we substitute into equation 10.6, resulting in a modified duration of years.

46. Calculating Duration Using Excel
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Calculating Duration Using Excel
Macaulay Duration DURATION function
Modified Duration -- MDURATION function
=DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)

47. Calculating Macaulay and Modified Duration
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Calculating Macaulay and Modified Duration

48. Duration Properties: All Else Equal
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Duration Properties: All Else Equal
The longer a bond’s maturity, the longer its duration.
A bond’s duration increases at a decreasing rate as maturity lengthens.
The higher a bond’s coupon, the shorter is its duration.
A higher yield to maturity implies a shorter duration, and a lower yield to maturity implies a longer duration.

49. Properties of Duration
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Properties of Duration

50. Bond Risk Measures based on Duration
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Bond Risk Measures based on Duration
Dollar Value of an 01: (10.10)
≈ Modified Duration x Bond Price x
= Value of a basis point change
Yield Value of a 32nd: (10.11) ≈
The dollar value of an 01 (“dollar value of an oh-one”) basically gives the approximate value of a change of one basis point in yield.
The yield value of a 32nd relates to bonds whose prices are quoted in 32nds such as U.S. Treasury bonds. It offers an alternative measure of interest rate risk.
Note that the first is based on Modified duration, while the second is based on the first.
In both cases, the bond price is per $100 face value.

51. Calculating Bond Risk Mesures
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Calculating Bond Risk Mesures
Bond matures in 15 years M = 15
Coupon rate = 5% C = $50
Yield to maturity = 6.5% YTM = .065
Macaulay Duration = years
Modified Duration = years
First we have to find the price of the bond:
For our example, we have a 5% coupon bond that matures in 15 years. The current yield to maturity is 6.5%. Macaulay duration is years and Modified duration is years.
Calculating both the bond risk measures requires that we first find the price of our sample bond.
Using equation 10.3 yields a price of $

52. Bond Risk Measures based on Duration
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Bond Risk Measures based on Duration
Bond matures in 15 years M = 15
Coupon rate = 5% C = $50
Yield to maturity = 6.5% YTM = .065
Macaulay Duration = years
Modified Duration = years
Price =
Dollar Value of an 01: (10.10)
≈ Modified Duration x Bond Price x
≈ x x =
With our given data and the bond’s price, we’re ready to use equation and
The dollar value of an 01 is approximately while the Yield Value of a 32nd comes out to

53. Bond Risk Measures based on Duration
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Bond Risk Measures based on Duration
Bond matures in 15 years M = 15
Coupon rate = 5% C = $50
Yield to maturity = 6.5% YTM = .065
Macaulay Duration = years
Modified Duration = years
Price =
Yield Value of a 32nd: (10.11) ≈
With our given data and the bond’s price, we’re ready to use equation and
The dollar value of an 01 is approximately while the Yield Value of a 32nd comes out to

54. 4/19/2017
Dedicated Portfolios
Bond portfolio created to prepare for a future cash payment, e.g. pension funds
Target Date = date the payment is due
Our final topic in chapter 10 concerns dedicated portfolios – portfolios designed to meet a specific future liability.

55. Reinvestment Risk & Price Risk
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Reinvestment Risk & Price Risk
Reinvestment Rate Risk:
Uncertainty about the value of the portfolio on the target date
Stems from the need to reinvest bond coupons at yields not known in advance
Price Risk:
Risk that bond prices will decrease
Arises in dedicated portfolios when the target date value of a bond is not known with certainty

56. Price Risk vs. Reinvestment Rate Risk For a Dedicated Portfolio
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Price Risk vs. Reinvestment Rate Risk For a Dedicated Portfolio
Interest rate increases have two effects:
 in interest rates decrease bond prices, but
 in interest rates increase the future value of reinvested coupons
Interest rate decreases have two effects:
 in interest rates increase bond prices, but
Decreases in interest rates decrease the future value of reinvested coupons

57. 4/19/2017
Immunization
Immunization = constructing a dedicated portfolio that minimizes uncertainty surrounding the target date value
Engineer a portfolio so that price risk and reinvestment rate risk offset each other (just about entirely).
Duration matching = matching the duration of the portfolio to its target date

58. Immunization by Duration Matching
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Immunization by Duration Matching

59. 4/19/2017
Dynamic Immunization
Periodic rebalancing of a dedicated bond portfolio for the purpose of maintaining a duration that matches the target maturity date
Advantage = reinvestment risk greatly reduced
Drawback = each rebalancing incurs management and transaction costs

60. Constructing a Dedicated Portfolio
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Constructing a Dedicated Portfolio
Suppose a Pension Fund estimates it will need to pay out about $50 million in 4 years. The fund decides to buy coupon bonds paying 6%, maturing in 4 years and selling at par.
Assuming interest rates do not change over the next four years, how much should the fund invest to have $50 million in 4 years?
For our example, let’s look at a pension fund who estimates it will need to pay out about $50 million in benefits in four years.
The fund decides to buy coupon bonds paying 6% and maturing in 4 years.
The question is, how much should they invest now to result in $50 million in four years.

61. Constructing a Dedicated Portfolio
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Constructing a Dedicated Portfolio
10.12
10.13
Future Value required = $50 million
Time = 4 years (M=8)
Par value bonds paying 6%
Our first step is to convert the formula for bond price (10.3) from a present value to a future value formula. In this case, we know that the future value we want is $50 million and we need to solve for the present value.
Equation makes this conversion.
For simplicity, if we assume that the fund will be investing in par value bonds which will remain at par for the life of the portfolio, the formula can be reduced to equation
Applying to our data, we find the fund needs to invest $39,470,362 now.
We’ll verify this on the next slide.

62. Constructing a Dedicated Portfolio
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Constructing a Dedicated Portfolio
The table above verifies our result from the previous slide.
Four years to maturity means 8 coupon payments with each payment equal to one half of 5% times the initial investment.
The final column on the right gives the result of investing each coupon payment for the remainder of the four years.
Summing that last column and adding the principal results in the target amount the fund needs: $50 million.

63. Useful Internet Sites www.sifma.org (check out the bonds section)
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Useful Internet Sites
(check out the bonds section)
(a practical view of bond portfolio management)
(bond basics and current market data)
(bond basics and current market data)
(for information on government bonds)