This document provides information about bond valuation and modeling bond prices using Excel functions. It includes examples of using the PRICE and PV functions to value a bond given its coupon rate, par value, maturity date, and market yield. It also shows how bond prices change over time as market interest rates change, rising to 15% or falling to 5% from the initial 10% rate. The document discusses modifications needed to the model for bonds that pay interest semiannually, such as dividing the coupon payment, years to maturity, and market yield by two.
Ch05 P24 Build a Model Spring 1, 201372212Chapter 5. Ch 05 P24 B.docx
1. Ch05 P24 Build a Model Spring 1, 20137/22/12Chapter 5. Ch 05
P24 Build a ModelExcept for charts and answers that must be
written, only Excel formulas that use cell references or
functions will be accepted for credit. Numeric answers in cells
will not be accepted.A 20-year, 8% semiannual coupon bond
with a par value of $1,000 may be called in 5 years at a call
price of $1,040. The bond sells for $1,100. (Assume that the
bond has just been issued.)Basic Input Data:Years to
maturity:20Periods per year:2Periods to maturity:Coupon
rate:8%Par value:$1,000Periodic payment:Current
price$1,100Call price:$1,040Years till callable:5Periods till
callable:a. What is the bond's yield to maturity?Periodic YTM
=Annualized Nominal YTM = Hint: This is a nominal rate, not
the effective rate. Nominal rates are generally quoted.b. What
is the bond's current yield?Current yield = Hint: Write
formula in words.Current yield =/ Hint: Cell formulas should
refer to Input SectionCurrent yield =(Answer)c. What is the
bond's capital gain or loss yield?Cap. Gain/loss yield =- Hint:
Write formula in words.Cap. Gain/loss yield =- Hint: Cell
formulas should refer to Input SectionCap. Gain/loss yield
=(Answer)Note that this is an economic loss, not a loss for tax
purposes.d. What is the bond's yield to call?Here we can again
use the Rate function, but with data related to the call.Peridodic
YTC =Annualized Nominal YTC =This is a nominal rate, not
the effective rate. Nominal rates are generally quoted.The YTC
is lower than the YTM because if the bond is called, the buyer
will lose the difference between the call price and the current
price in just 4 years, and that loss will offset much of the
interest imcome. Note too that the bond is likely to be called
and replaced, hence that the YTC will probably be earned.NOW
ANSWER THE FOLLOWING NEW QUESTIONS:e. How
would the price of the bond be affected by changing the going
market interest rate? (Hint: Conduct a sensitivity analysis of
price to changes in the going market interest rate for the bond.
2. Assume that the bond will be called if and only if the going rate
of interest falls below the coupon rate. That is an
oversimplification, but assume it anyway for purposes of this
problem.)Nominal market rate, r:8%Value of bond if it's not
called:Value of bond if it's called: The bond would not be
called unless r<coupon.We can use the two valuation formulas
to find values under different r's, in a 2-output data table, and
then use an IFstatement to determine which value is
appropriate:Value of Bond If:Actual value,Not
calledCalledconsideringRate, r$0.00$0.00call
likehood:0%$0.00$0.00$0.002%$0.00$0.00$0.004%$0.00$0.00$
0.006%$0.00$0.00$0.008%$0.00$0.00$0.0010%$0.00$0.00$0.0
012%$0.00$0.00$0.0014%$0.00$0.00$0.0016%$0.00$0.00$0.00
f. Now assume the date is 10/25/2010. Assume further that a
12%, 10-year bond was issued on 7/1/2010, pays interest
semiannually (January 1 and July 1), and sells for $1,100. Use
your spreadsheet to find the bond’s yield.Refer to this chapter's
Tool Kit for information about how to use Excel's bond
valuation functions. The model finds the price of a bond, but the
procedures for finding the yield are similar. Begin by setting
up the input data as shown below:Basic info:Settlement
(today)MaturityCoupon rateCurrent price (% of par)Redemption
(% of par value)Frequency (for semiannual)Basis (360 or 365
day year)Yield to Maturity:Hint: Use the Yield function. For
dates, either refer to cells in Basic Info above, or enter the date
in quotes, such as "10/25/2010".
Sheet27/22/12
Chapter4/11/10Chapter 5. Tool Kit for Bonds, Bond Valuation,
and Interest RatesThe value of any financial asset is the present
value of the asset's expected future cash flows. The key inputs
are (1) the expected cash flows and (2) the appropriate discount
rate, given the bond's risk, maturity, and other characteristics.
The model developed here analyzes bonds in various
ways.BOND VALUATION (Section 5.3)A bond has a 15-year
maturity, a 10% annual coupon, and a $1,000 par value. The
3. required rate of return (or the yield to maturity) on the bond is
10%, given its risk, maturity, liquidity, and other rates in the
economy. What is a fair value for the bond, i.e., its market
price?First, we list the key features of the bond as "model
inputs":Years to Mat:15Coupon rate:10%Annual Pmt:$100Par
value = FV:$1,000Required return, rd:10%The easiest way to
solve this problem is to use Excel's PV function. Click fx, then
financial, then PV. Then fill inthe menu items as shown in our
snapshot in the screen shown just below.Value of bond
=$1,000.00Thus, this bond sells at its par value. That situation
always exists if the goingrate is equal to the coupon rate.The
PV function can only be used if the payments are constant, but
that is normally the case for bonds.Bond Prices on Actual
DatesThus far we have evaluated bonds assuming that we are at
the beginning of an interest payment period. This is correct for
new issues, but it is generally not correct for outstanding bonds.
However, Excel has several date and time functions, and a bond
valuation function that uses the calendar, so we can get exact
valuations on any given date.Here is the data for MicroDrive's
bond as of the day it was issued.Settlement date (day on which
you find bond price) =1/5/11Maturity date =1/5/26Coupon rate
=10.00%Required return, rd =10.00%Redemption (100 means
the bond pays 100% of its face value at maturity)
=100Frequency (# payments per year) =1Basis (1 is for actual
number of days in month and year)1Click on fx on the formula
bar (or click Insert and then Function). This gives you the
"Insert Function" dialog box. To find a bond's price, use the
PRICE function (found in the "Financial" category of the "Insert
Function dialog box). The PRICE function returns the price per
$100 dollars of face value.Using PRICE function with inputs
that are cell references:Value of bond based on $100 face value
=$100.00Value of bond in dollars based on $1,000 face value
=$1,000.00Using the PRICE function with inputs that are not
cell references:Value of bond based on $100 face value
==PRICE(DATE(2011,1,5),DATE(2026,1,5),10%,10%,100,1,1)
Value of bond based on $100 face value =100.000Value of
4. bond in dollars based on $1,000 face value =$1,000.00Interest
Rate Changes and Bond PricesSuppose the going interest rate
changed from 10%, falling to 5% or rising to 15%. How would
those changes affect the value of the bond?We could simply go
to the input data section shown above, change the value for r
from 10% to 5% and then 15%, and observe the changed values.
An alternative is to set up a data table to show the bond's value
at a range of rates, i.e., to show the bond's sensitivity to
changes in interest rates. This is done below, and the values at
5% and 15% are boldfaced.Bond ValueTo make the data table,
first type the headings, then type the rates in cells A89:A93,
and then put the formula =B41 in cell B88, then select the range
A88:B93. Then click Data, What-IF-Analysis, and then Table to
get the menu. The input data are in a column, so put the cursor
on column and enter C20 the place where the going rate is
inputted. Click OK to complete the operation and get the
table.Going rate,
r:$1,0000%$2,500.005%$1,518.9810%$1,000.0015%$707.6320
%$532.45We can use the data table to construct a graph that
shows the bond's sensitivity to changing rates.CHANGES IN
BOND VALUES OVER TIME (Section 5.4)What happens to a
bond price over time? To set up this problem, we will enter the
different interest rates, and use the array of cash flows above.
The following example operates under the precept that the bond
is issued at par ($1,000) in year 0. From this point, the example
sets three conditions for interest rates to follow: interest rates
stay constant at 10%, interest rates fall to 5%, or interest rates
rise to 15%. Then the price of the bond over the fifteen years of
its life is determined for each of the scenarios.Suppose interest
rates rose to 15% or fell to 5% immediately after the bond was
issued, and they remained at the new level for the next 15 years.
What would happen to the price of the bond over time?We could
set up data tables to get the data for this problem, but instead
we simply inserted the PV formula into the following matrix to
calculate the value of the bond over time. Note that the formula
takes the interest rate from the column heads, and the value of
5. N from the left column. Note that the N = 0 values for the 5%
and 15% rates are consistent with the results in the data table
above. We can also plot the data, as shown in the graph
below.Value of Bond in Given
Year:N5%10%15%0$1,519$1,000$7081$1,495$1,000$7142$1,4
70$1,000$7213$1,443$1,000$7294$1,415$1,000$7385$1,386$1,
000$7496$1,355$1,000$7617$1,323$1,000$7768$1,289$1,000$
7929$1,254$1,000$81110$1,216$1,000$83211$1,177$1,000$85
712$1,136$1,000$88613$1,093$1,000$91914$1,048$1,000$957
15$1,000$1,000$1,000If rates fall, the bond goes to a premium,
but it moves toward par as maturity approaches. The reverse
hold if rates rise and the bond sells at a discount. If the going
rate remains equal to the coupon rate, the bond will continue to
sell at par. Note that the above graph assumes that interest
rates stay constant after the initial change. That is most
unlikely--interest rates fluctuate, and so do the prices of
outstanding bonds.Market rate =5%NBond PriceReturn Due to
Coupon PaymentReturn Due to Price ChangeTotal
Return0$1,518.981$1,494.936.58%-
1.58%5.00%2$1,469.686.69%-1.69%5.00%3$1,443.166.80%-
1.80%5.00%4$1,415.326.93%-1.93%5.00%5$1,386.097.07%-
2.07%5.00%6$1,355.397.21%-2.21%5.00%7$1,323.167.38%-
2.38%5.00%8$1,289.327.56%-2.56%5.00%9$1,253.787.76%-
2.76%5.00%10$1,216.477.98%-2.98%5.00%11$1,177.308.22%-
3.22%5.00%12$1,136.168.49%-3.49%5.00%13$1,092.978.80%-
3.80%5.00%14$1,047.629.15%-4.15%5.00%15$1,000.009.55%-
4.55%5.00%Market rate =10%NBond PriceReturn Due to
Coupon PaymentReturn Due to Price ChangeTotal
Return0$1,0001$1,00010.00%-0.00%10.00%2$1,00010.00%-
0.00%10.00%3$1,00010.00%0.00%10.00%4$1,00010.00%-
0.00%10.00%5$1,00010.00%0.00%10.00%6$1,00010.00%-
0.00%10.00%7$1,00010.00%0.00%10.00%8$1,00010.00%0.00
%10.00%9$1,00010.00%0.00%10.00%10$1,00010.00%0.00%10
.00%11$1,00010.00%-
0.00%10.00%12$1,00010.00%0.00%10.00%13$1,00010.00%0.0
0%10.00%14$1,00010.00%-
6. 0.00%10.00%15$1,00010.00%0.00%10.00%Market rate
=15%NBond PriceReturn Due to Coupon PaymentReturn Due to
Price ChangeTotal
Return0$707.631$713.7814.13%0.87%15.00%2$720.8414.01%0
.99%15.00%3$728.9713.87%1.13%15.00%4$738.3113.72%1.28
%15.00%5$749.0613.54%1.46%15.00%6$761.4213.35%1.65%1
5.00%7$775.6313.13%1.87%15.00%8$791.9812.89%2.11%15.0
0%9$810.7812.63%2.37%15.00%10$832.3912.33%2.67%15.00
%11$857.2512.01%2.99%15.00%12$885.8411.67%3.33%15.00
%13$918.7111.29%3.71%15.00%14$956.5210.88%4.12%15.00
%15$1,000.0010.45%4.55%15.00%BONDS WITH
SEMIANNUAL COUPONS (Section 5.5)Since most bonds pay
interest semiannually, we now look at the valuation of
semiannual bonds. We must make three modifications to our
original valuation model: (1) divide the coupon payment by 2,
(2) multiply the years to maturity by 2, and (3) divide the
nominal interest rate by 2.Problem: What is the price of a 15-
year, 10% semi-annual coupon, $1,000 par value bond if the
nominal rate (the YTM) is 5%? The bond is not callable.Use
the Rate function with adjusted data to solve the
problem.Periods to maturity = 15*2 =30
Christopher Buzzard: N=30, because of semi-annual
compounding (15*2 = 30).Coupon rate:10%Semiannual pmt =
$100/2 =$50.00
Bart Kreps: PMT=$50, because of semiannual payments
(100 ÷ 2) = 50
PV =$1,523.26Current price:$1,000.00Periodic rate = 5%/2
=2.5%
Christopher Buzzard: I=2.5%, because of semi-annual
compounding (5%/2 = 2.5%).Note that the bond is now more
valuable, because interest payments come in faster.BOND
YIELDS (Section 5.6)Yield to MaturityThe YTM is defined as
the rate of return that will be earned if a bond makes all
7. scheduled payments and is held to maturity. The YTM is the
same as the total rate of return discussed in the chapter, and it
can also be interpreted as the "promised rate of return," or the
return to investors if all promised payments are made. The
YTM for a bond that sells at par consists entirely of an interest
yield. However, if the bond sells at any price other than its par
value, the YTM consists of the interest yield together with a
positive or negative capital gains yield. The YTM can be
determined by solving the bond value formula for I. However,
an easier method for finding it is to use Excel's Rate function.
Since the price of a bond is simply the sum of the present values
of its cash flows, so we can use the time value of money
techniques to solve these problems.Problem: Suppose that you
are offered a 14-year, 10% annual coupon, $1,000 par value
bond at a price of $1,494.93. What is the Yield to Maturity of
the bond?Use the Rate function to solve the problem.Years to
Mat:14Coupon rate:10%Annual Pmt:$100.00Going rate, r
=YTM:5.00%Current price:$1,494.93Par value =
FV:$1,000.00The yield-to-maturity is the same as the expected
rate of return only if (1) the probability of default is zero, and
(2) the bond can not be called. If there is any chance of default,
then there is a chance some payments may not be made. In this
case, the expected rate of return will be less than the promised
yield-to-maturity.Finding the Yield to Maturity on Actual
DatesThus far we have evaluated bonds assuming that we are at
the beginning of an interest payment period. This is correct for
new issues, but it is generally not correct for outstanding bonds.
However, Excel has a function that uses the actual calendar
when finding yields. Consider the bond above, with 14 years
until maturity. Suppose the actual current date is 1/5/2012, so
the bond matures on 1/5/2026.Here is the data for the
bond.Settlement date (day on which you find bond price)
=01/05/12Maturity date =01/05/26Coupon rate =10.00%Price =
bond price per $100 par value =$149.49Redemption (100 means
the bond pays 100% of its face value at maturity)
=100Frequency (# payments per year) =1Basis (1 is for actual
8. number of days in month and year)1Using the YIELD function
with inputs that are cell references:Yield to maturity
=5.0%Yield to CallThe yield to call is the rate of return
investors will receive if their bonds are called. If the issuer has
the right to call the bonds, and if interest rates fall, then it
would be logical for the issuer to call the bonds and replace
them with new bonds that carry a lower coupon. The yield to
call (YTC) is found similarly to the YTM. The same formula is
used, but years to maturity is replaced with years to call, and
the maturity value is replaced with the call price.Problem:
Suppose you purchase a 15-year, 10% annual coupon, $1,000
par value bond with a call provision after 10 years at a call
price of $1,100. One year later, interest rates have fallen from
10% to 5% causing the value of the bond to rise to $1,494.93.
What is the bond's YTC? Note that this is the same bond as in
the previous question, but now we assume it can be called.Use
the Rate function to solve the problem.Years to call:9
Christopher Buzzard: N is equal to 9, because the bond can be
called 10 years after issuance and one year has already gone
by.Coupon rate:10%Annual Pmt:$100.00Rate = I = YTC
=4.21%Current price:$1,494.93Call price = FV$1,100.00Par
value$1,000.00This bond's YTM is 5%, but its YTC is only
4.21%. Which would an investor be more likely to actually
earn?This company could call the old bonds, which pay $100
per year, and replace them with bonds that pay somewhere in
the vicinity of $50 (or maybe even only $42.10) per year. It
would want to save that money, so it would in all likelihood call
the bonds. In that case, investors would earn the YTC, so the
YTC is the expected return on the bonds.Current YieldThe
current yield is the annual interest payment divided by the
bond's current price. The current yield provides information
regarding the amount of cash income that a bond will generate
in a given year. However, it does not account for any capital
gains or losses that will be realized fi the bond is held to
maturity or call. Problem: What is the current yield on a
9. $1,000 par value, 10% annual coupon bond that is currently
selling for$985?Simply divide the annual interest payment by
the price of the bond. Even if the bond made semiannual
payments, we would still use the annual interest.Par
value$1,000.00Coupon rate:10%Current Yield =10.15%Annual
Pmt:$100.00Current price:$985.00The current yield provides
information on a bond's cash return, but it gives no indication of
the bond's total return. To see this, consider a zero coupon
bond. Since zeros pay no coupon, the current yield is zero
because there is no interest income. However, the zero
appreciates through time, and its total return clearly exceeds
zero.THE DETERMINANTS OF MARKET INTEREST RATES
(Section 5.7)Quoted market interest rate = rd = r* + IP + DRP +
LP + MRPr* =Real risk-free rate of interestIP =Inflation
premiumDRP =Default risk premiumLP =Liquidity
premiumMRP =Maturity risk premiumTHE REAL RISK-FREE
RATE OF INTEREST, r* (Section 5.8)r* = Real risk-free rate
of interestr* = Yield on short-term (1-year) U.S. Treasury
Inflation-Protected Security (TIPS)r* = 1.54%(March 2009)THE
INFLATION PREMIUM (IP) (Section 5.9)Maturity5 Years20
YearsNon-indexed U.S. Treasury
Bond1.91%3.93%TIPS1.41%2.44%Inflation
premium0.50%1.49%THE NOMINAL, OR QUOTED, RISK-
FREE RATE OF INTEREST, rRF (Section 5.10)Nominal, or
quoted, rate = rd = rRF + DRP + LP + MRPTHE DEFAULT
RISK PREMIUM (DRP) (Section 5.11)Table 5-1Rating
AgencyaPercent defaulting within:bMedian RatioscPercent
upgraded or downgraded in 2008:bS&P and FitchMoody’s1
year5 yearsReturn on capitalTotal debt/Total
capitalDownUpYieldd(1)(2)(3)(4)(5)(6)(7)(8)(9)Investment
grade
bonds:AAAAaa0.000.0027.6012.4013.60NA5.50AAAa00.12728
.321.805.62AA0.10.617.537.581.85.79BBBBaa0.32.913.442.56.
42.67.53Junk
bonds:BBBa1.48.211.353.715.16.811.62BB1.89.28.775.910.85.6
13.7CCCCaa22.336.93.2113.526.18.726.3Notes: aThe ratings
10. agencies also use “modifiers” for bonds rated below triple-A.
S&P and Fitch use a plus and minus system; thus, A+ designates
the strongest A-rated bonds and A– the weakest. Moody’s uses a
1, 2, or 3 designation, with 1 denoting the strongest and 3 the
weakest; thus, within the double-A category, Aa1 is the best,
Aa2 is average, and Aa3 is the weakest.bDefault data are from
Fitch Ratings Global Corporate Finance 2008 Transition and
Default Study, March 5, 2009: see
http://www.fitchratings.com/corporate/reports/report_frame.cfm
?rpt_id=428182.cMedian ratios are from Standard & Poor’s
2006 Corporate Ratings Criteria, April 23, 2007: see
http://www2.standardandpoors.com/spf/pdf/fixedincome/Corpor
ate_Ratings_2006.pdf.dComposite yields for AAA, AA, and A
bonds can be found at
http://finance.yahoo.com/bonds/composite_bond_rates.
Representative yields for BBB, BB, B, and CCC bonds can be
found using the bond screener at http://screen.yahoo.com/bonds.
.html.Bond spreads are the difference between the yield on a
bond and the yield on some other bond of the same maturity.For
a bond with good liquidity, its spread relative to a T-bond of
similar maturity is a good estmat of the default risk
premium.Figure 5-3: Bond SpreadsData for chart to
rightDATEAAA - T-bondBAA - T-bond2009-022.405.212009-
012.535.622008-122.636.012008-112.595.682008-
102.475.072008-091.963.622008-081.753.262008-
071.663.152008-061.582.972008-051.693.052008-
041.873.292008-032.003.382008-021.793.082008-
011.592.802007-121.392.552007-111.292.252007-
101.131.952007-091.222.072007-081.121.982007-
070.731.652007-060.691.602007-050.721.642007-
040.781.702007-030.741.712007-020.671.562007-
010.641.582006-120.761.662006-110.731.602006-
100.781.692006-090.791.712006-080.801.712006-
070.761.672006-060.781.67THE LIQUIDITY PREMIUM (LP)
(Section 5.12)2006-050.841.642006-040.851.69A differential of
at least 2 percentage points (and perhaps up to 4 or 5 percentage
11. points) exists between the least liquid and the most liquid
financial assets of similar default risk and maturity. 2006-
030.811.692006-020.781.702006-010.871.822005-
120.901.85THE MATURITY RISK PREMIUM (MRP) (Section
5.13)2005-110.881.852005-100.891.84Bonds are exposed to
interest rate risk and reinvestment rate risk. The net effect is
the maturity risk premium.2005-090.931.832005-
080.831.70Interest Rate Risk2005-070.881.772005-
060.961.86Interest Rate Risk is the risk of a decline in a bond's
price due to an increase in interest rates. Price sensitivity to
interest rates is greater (1) the longer the maturity and (2) the
smaller the coupon payment. Thus, if two bonds have the same
coupon, the bond with the longer maturity will have more
interest rate sensitivity, and if two bonds have the same
maturity, the one with the smaller coupon payment will have
more interest rate sensitivity.2005-051.011.872005-
040.991.712005-030.901.562005-021.031.652005-
011.141.80Compare the interest rate risk of two bonds, both of
which have a 10% annual coupon and a $1,000 face value. The
first bond matures in 1 year, the second in 25 years.2004-
121.241.922004-111.332.012004-101.372.11Use the PV
function, along with a two variable Data Table, to show the
bonds' price sensitivity.2004-091.332.14Coupon rate:10%2004-
081.372.18Payment$100.002004-071.322.12Par
value$1,000.002004-061.282.05Maturity12004-
051.322.03Going rate = r = YTM10%2004-041.382.112004-
031.502.28Value of bond:$1,000.002004-021.422.192004-
011.392.292003-121.352.33Value of the Bond Under Different
Conditions2003-111.352.36Going rate, rYears to Maturity2003-
101.412.44$1,000.001252003-
091.452.520%$1,100.00$3,500.002003-
081.432.565%$1,047.62$1,704.702003-
071.512.6410%$1,000.00$1,000.002003-
061.642.8615%$956.52$676.792003-
051.652.8120%$916.67$505.242003-
041.782.8925%$880.00$402.272003-032.083.142003-