CFP Class 2 - Module 7 - Valuation and Analysis of Fixed-Income Investments

Make the following assumptions unless the problem specifically states otherwise: Face Value, N & END Mode

$1,000. Positive FV as money paid to client when bond matures. Since payments received twice a year, number of compounding periods (N) is twice number of years to maturity. Assume six months until next semiannual payment so each payment occurs at the end END mode.

New bond issued with a 6% coupon; it pays $60 of interest per year, in semiannual payments of $30. Assume market rates have declined, and the bond now sells for $1,100. The current yield is:

$60 divided by $1,100, or 5.45%. In bond market terminology, the bond yield is now 55 basis points lower.

X Itemizes deductions In 32% marginal tax bracket. 10% state marginal tax bracket. Municipal bond issued by his state of residence with yield of 4.2%. Also considering corporate bonds with same maturity with yield of 5.5%. Which should he purchase based solely on yield?

(1 - .32)(1 - .10) 1 - .1 = .9 .1 - .32 = .68 .68 x .9 = .612 .042 divided by .612 = Tax Equivalent Yield 6.86%, so buy the Muni. To find Tax Free Equivalent Yield TFEY: Taxable yield = 6.86%. 6.86 multiplied by (1 - SMTB) (1 - FMTB 1- .1 = .9 x 1 - .31 = .68. .68 x .9 = .612 x .686 = TFEY 4.2%

On what four factors does the calculation of a bond's price depend?

(1) the interest paid by the bond, (2) the interest rate available on comparable bonds of the same maturity and grade or market interest rate, (3) the maturity date of the bond, and (4) the bond's principal or call amount.

Bond with a 10% coupon. Current market price of $1,030. Duration of 3.5 using annual compounding, and interest rates currently 8%. What is the approximate price change in this bond if interest rates rise 1%? Note: this would be 100 basis points, which is .0100.

- D = 3.5, triangleY = .01, y = .08. .01 dived by 1.08 = .009259 x - 3.5 = -.03241 = 3.24% decrease in price. Multiply this by the current price of the bond $1,030 obtain the approximate price movement: $33.38. 1,030 - 33.38 = $996.62. Bond currently at $1,030 would decline to approximately $997 if interest rates were to increase by 1%.

Bond with a 10% coupon. Current market price of $1,030. Duration of 3.5 using annual compounding, and interest rates currently 8%. What is the approximate price change in this bond if interest rates fall 0.5%? Note that this would be 50 basis points, which is .0050.

-.005 divided by 1.08 = .00463 x - 3.5 = .0162 x 1030 = $16.69 + 1030 = $1,046.69 new bond price. 1.62% increase in the bond price. If have semiannual compounding just divided 8% by 2 = .04 for y. Everything else the same.

Annual compounding. What is the duration of a bond that has 20 years to maturity and a coupon of 8% when the current market interest rate is 6%? Assume annual compounding.

1 + .06 = 1.06 divided by .06 = 17.67 1 + .06 = 1.06 .08 - .06 = .02 x 20 = .40 1.06 + .40 = 1.46 1 + .06 = 1.06 SHIFT yx (on the "x" key) 20 = 3.2071 (3.21 rounded) - 1 = 2.21 x .08 = .18 + .06 = .24 1.46 divided by .24 = 6.08 17.67 - 6.08 = 11.59 periods or years

Putting It All Together—Yield Calculations: Natasha purchases a 6% coupon bond for $985. The bond matures in 20 years, and is callable in 10 years at $1,010. What is the current yield, yield-to maturity, and yield-to-call for her bond?

1,000 x .06 = $60. Current yield: $60 ÷ $985 = .0609, or 6.09% 1,000 x .06 = 60 divided by 2 = 30 YTC: END, 2 P/YR, 10 SHIFT N, 30 PMT, - 985 PV, 1,010 FV, I/YR = 6.28% YTM: END 2 P/YR, 20 SHIFT N, 30 PMT, - 985 PV, 1,000 FV, I/YR = 6.13%. Discount bond, so YTC should be highest, then YTM, then CY.

What is the YTC on an investment in a bond with a call price of $1,020, a current market price of $1,040, a 7% coupon, and eight years until call?

1,000 x .07 = 70 divided by 2 = 35 PMT. END. 2 P/YR, 8 SHIFT N, PMT 35, - 1,040 PV, 1,020 FV, I/YR = 6.54%

Paid $1,035. Has an 8% coupon and matures in 25 years. There is a call provision in 12 years at $1,005. What is the current yield, yield-to-maturity, and yield-to-call?

1,000 x .08 = 80. CY: 80 ÷ 1,035 = 7.73% YTC: END, 2 P/YR, 12 SHIFT N, 40 PMT, -1,035 PV, 1,005 FV, I/YR = 7.58% YTM: END, 2 P/YR, 25 SHIFT N, 40 PMT, -1,035 PV, 1,000 FV, I/YR = 7.68%. Premium bond. YTC should be the lowest, then YTM, and current yield the highest, and this is the case.

Which has the highest duration: 1.a. 5% coupon with 10-year maturity 1.b. 5% coupon with 15-year maturity. 2.a. 6% coupon with 8-year maturity 2.b. 7% coupon with 8-year maturity. 3.a. 7% coupon with 15-year maturity 3.b. 0% coupon with 15-year maturity

1.b. The longer the maturity, the higher longer, if expressed as years, the duration. 2.a. Lower the coupon rate, the higher or longer the duration. As coupon rate declines, duration increases. 3.b. Lower the coupon rate, the higher or longer the duration.

What is the price or intrinsic value of a bond with a $1,000 face value, a 10% coupon, and three years to maturity, if comparable bonds of the same maturity and grade are yielding 11.5%?

10% coupon $100 converted into a $50 payment $100 ÷ 2 to reflect the semiannual payment. END mode. P/YR 2, 3 SHIFT N, I/YR 11.5, PMT 50, FV 1000, PV = $962.83

Calculating the Yield-to-Maturity for a Bond Investment: What is the YTM or IRR on an investment in a bond with a $1,000 face value, a current market price of $966, a 10% coupon, and three years to maturity?

10% of $1,000 is $100, divided by two for semiannual payments = $50 PMT. END. P/YR 2, 3 SHIFT N, PV - 966, PMT 50, FV = 11.37%. Discount bond so YTM should come out higher than coupon rate and that was the case: 11.37% vs. 10.0%.

Market price of a bond paying 12% coupon interest semiannually, matures in seven years and pays a face value of $1,000. Comparable bonds with similar maturities and of the same investment grade are yielding 14.9%. Price of this bond?

1000 x .12 = 120 divided by 2 = 60 END. 2 SHIFT P/YR. 7 SHIFT N. 14.9 I/YR, 60 PMT, 1,000 FV, = - 876.54 PV

What is the intrinsic value (price) of a newly issued bond with a 12% coupon rate, 30 years to maturity, and a $1,000 maturity value when current market rates for comparable bonds are at 12%?

1000 x .12 = 120 divided by 2 = 60. END 2 P/YR. 30 SHIFT N, 12 I/YR, 60 PMT, 1000 FV = - 1,000 PV

2-year, 3.0% bond with semiannual interest payments, and an YTM of 4.0%. What is the duration?

2 × 2 (semiannual) = 4 for "t" .03 ÷ 2 = .015 for "c" .04 ÷ 2 = .02 for "y" 1 + .02 = 1.02 divided by .02 = 51 .015 - .02 = - .005 x 4 = - .02 + 1.02 = 1 1.02 SHIFT yx (on the "x" key) 4 = 1.08 - 1 = .082 x = .0012 + .02 = .02124 1 divided by .02124 = 47.08 51 - 47.08 = 3.92 divided by 2 = 1.96

What is the duration of a bond that has 20 years to maturity and a coupon of 8% when the current market interest rate is 6%? Assume semiannual compounding. Semiannual results in a lower duration because compounding more frequently than annual compounding.

20 × 2 (semiannual) = 40 for "t" .08 ÷ 2 = .04 for "c" .06 ÷ 2 = .03 for "y" 1 + .03 = 1.03 divided by .03 = 34.33 .04 - .03 = .01 x 40 = .40 + 1.03 = 1.43 1.03 SHIFT yx (on the "x" key) 40 = 3.26 - 1 = 2.26 x .04 = .09 + .03 = .12 1.43 divided by .12 = 11.92 34.33 - 11.92 = 22.41 divided by 2 = 11.21

Assume market rates have risen and the bond now sells for $900. Bond has 20 years until its maturity date 40 semiannual periods. What is Yield to Maturity?

6.93%, consisting of a current yield of 6.67% and a compound semiannual return over the 20 years of 0.26% $100 of appreciation compounded over 40 periods. If YTM includes depreciation of asset's value from $1,100 to $1,000, the YTM will be less than the current yield.

Bond at Par

A bond at par $1,000 when 7% coupon rate equal to 7% current market interest rate or yield to maturity (YTM) for comparable bonds.

Negative Convexity

A bond declining more in value than duration alone would explain in a rising interest rate environment. Bond will not rise as much in value as duration alone would indicate with declining interest rates. Callable bonds and mortgage backed bonds typical examples.

Bond's Current Yield

Annual coupon payment divided by current price of the bond. When originally issued at par, current yield and coupon rate the same. If bond price rises, then coupon divided by a higher price and the current yield is less than the coupon yield.

Make the following assumptions unless the problem specifically states otherwise: Coupon Interest

Annual percentage rate based on face value of $1,000 unless stated otherwise. Paid twice a year, so payment received every six months. Positive input. Payment found by dividing annual coupon interest earned by two.

Bond Selling at a Premium or a Discount

As discount rate rises, present value of a bond decreases. Discount rate falls, present value increases. Bond sells above its par value which is generally $1,000, said to be selling at a premium. Sells below par value, selling at a discount.

Premium Bonds have Highest Likelihood of Being Called

Because bonds trade at a premium when interest rates go down issuer may call and then borrow money at the lower current rate. If 7% bond outstanding and current rates 6%, call the 7% bond and then borrow at 6%, thereby saving 1% in interest charges.

Explain why bond prices and interest rates are inversely related.

Because dollar amount of the interest paid by a bond is constant i.e., there is a fixed flow of income its price or intrinsic value changes in the opposite direction of a change in interest rates, which would encourage investors to make a purchase.

0 Coupon Bond Duration

Because the entire payment is received at maturity for zero-coupon bonds, a zero-coupon bond's maturity and duration are the same. So if 15 year maturity, duration is 15.

Duration Example

Bond has duration of 4. Interest rates rise 1%. Bond declines in price about 4%. If bond has a duration of 9, and interest rates were to fall 1%, the bond would rise in price about 9%.

Duration and Coupon Rates

Bonds with higher coupon rates have lower durations and are less volatile to interest rate changes than bonds with lower coupon rates. Duration of a zero-coupon bond is equal to its maturity, as only cash flow from a zero-coupon bond $1,000 principal payment at maturity.

Compare the price volatility of the following types of bonds. a. bonds with long maturities compared to bonds with short maturities, assuming both have the same coupon rate

Bonds with long maturities are more volatile than bonds with short maturities. The principal payment and coupon payments for longer-term bonds occur further into the future, which raises the duration.

Duration for a Bond Similar to Beta for a Stock

Both measures multiplied by expected change in interest rates for bonds or the expected market risk premium for stocks to arrive at expected change in market value of the bond or the expected risk premium of the stock.

Bond Trading at a Premium Current Yield vs. Yield to Maturity vs. Yield to Call

CY highest, YTM next, YTC lowest as YTM and YTC take into account decrease of the premium paid for the bond back to the par value at maturity or to the call value at call date. Yield-to-call lowest yield. If the bond called before maturity, decrease in value back to par happens faster.

Bond Trading at a Discount Current Yield vs. Yield to Maturity vs. Yield to Call

CY lowest, YTM next, YTC highest. as YTM and YTC take into account the accretion from the discount price back up to par. Would be level if the bond is trading at par. Assuming there is no call premium, the CY, YTC, and YTM would all be the same.

Calculating the Yield-to-Call for a Bond Investment Differences from Calculating Yield to Maturity

Call date before maturity date so fewer compounding periods than YTM. Often call premium paid if called before maturity of $10 or $20 on a $1,000 bond, 1% to 2% premium. Provides extra return to help compensate for the bond being called before maturity.

Yield-to-Maturity (YTM)

Compound yield earned from purchase until maturity date. Includes periodic cash income and capital gains or losses if principal amount greater/smaller than current market price. Market rate of return. Interest rate equating interest payments and par value at maturity to current price.

Conventional Bonds Present Value

Coupon and par value fixed at bond issuance and not changed during bond's life. A bond's present value, or market value, changes as current market interest rates change. Current market interest rates are the discount rates used to calculate the present value of a bond.

Calculating Current Yield (CY): What is the current yield of a bond trading at $965, with a 6% coupon, and 20 years until maturity?

Current yield is the annual coupon payment divided by the current price. In this case we have a $60 annual coupon payment with a current price of $965. $60 ÷ $965 = .0622 = 6.22%

What do the terms "discount" and "premium" mean in relation to the pricing of a bond?

Discount amount by which bond sells below its maturity value to be competitive with bonds of comparable quality. Premium amount by which a bond's price exceeds its maturity value. Coupon rate less than the market yield, price below maturity value = discount bond.

Current Price of a Bond

Discounted present value of its future cash flow stream. Price or present value requires four inputs (1) semiannual payment, (2) par value or future value, (3) periods until maturity, and (4) current market interest rate for comparable bonds. All readily available.

Problem with Current Yield (CY) - Discount Bond Example

Does not take into account that bond will revert from current price to call price typically a slight premium over par, or the maturity price or par. Example above current yield understating total return. Does not take into account that bond will move from $965 back to $1,000 at maturity.

Duration Rule of Thumb

Duration is a useful tool to help investors determine the expected change in the price of a bond for a given change in interest rates. Multiply the duration by the expected change in rates.

Bank A offering 5.50% compounded monthly and Bank B offering 5.50% compounded quarterly. Which do you buy?

EAR = (1 + i divided by n) to the n power - 1. i = stated interest rate. n is number of compounding periods. Bank A EAR = [1 + (5.50 ÷ 12)]12 - 1 = 5.64% Bank B's EAR = [1 + (5.50 % ÷ 4)]4 - 1 = 5.61%

What will be the bond's price one year after issue if market rates drop to 9%? What will be the bond's price one year after issue if market rates rise to 15%?

END 2 P/YR, 29 SHIFT N, 9 I/YR, 60 PMT, 1000 FV = 1,307.38 END 2 P/YR, 29 SHIFT N 15 I/YR, 60 PMT, 1000 FV = 803.02

Your client purchased a bond for $950 with a coupon rate of 11%, matures in 17 years, and callable in five years at $1,110. What is the YTC for this bond?

END 2 P/YR. 5 SHIFT N, 55 PMT, - 950 PV, 1,110 FV = I/YR 14.02%

Recently purchased a zero-coupon bond for $630. $1,000 face value and matures in six years. What is the YTM for this bond?

END 2 P/YR. 6 SHIFT N, 0 PMT, - 630 PV, 1000 FV = I/YR 7.85%

Market price of $875. Pays 12% coupon interest semiannually. Will mature in seven years and pay a face value of $1,000. What is YTM (IRR)? What is the YTM if market price of $1,200?

END 2 P/YR. 7 SHIFT N, 60 PMT, - 875 PV, 1000 FV = I/YR 14.9% END 2 P/YR. 7 SHIFT N, 60 PMT, -1,200 PV, 1000 FV = I/YR 8.19%

Market price of a zero-coupon bond if matures in seven years and will pay a face value of $1,000. Comparable bonds with similar maturities and of the same investment grade yielding 14.9%. What should be the price of this bond?

END, 2 P/YR. 7 SHIFT N, 14.9% I/YR, 0 PMT, 1000 FV = PV - 365.69

Calculating the Price of a Zero-Coupon Bond: What is the intrinsic value or price of a zero-coupon bond with a $1,000 face value, a YTM of 8.20%, and nine years to maturity?

END. 2 P/YR, 9 SHIFT N, I/YR 8.2, PMT 0, FV 1,000, PV = $485.16. You do not need to enter any payment amount because there is no coupon but can enter "0" if you wish.

What is the yield-to-maturity or IRR of a zero-coupon bond with a current market price of $360, and 22 years until maturity?

END. P/YR 2, N SHIFT 22, PMT 0, PV - 360, FV 1000, I/YR = 4.7%

Alternate Effective Annual Rate Using Financial Calculater

Enter the nominal rate and press SHIFT, NOM% (I/YR key) 2. Enter the number of compounding periods and press SHIFT, P/YR (PMT key) 3. Calculate the effective rate by pressing SHIFT, EFF% (PV key). If you are given effective rate and need nominal rate reverse steps 1 & 3

Determining Price of a Perpetual Debt Instrument

Equal to the present value of an infinite stream of payments, which is determined as follows: annual interest payment divided by the current market interest rate.

Determining Price of A Bond with Maturity Date

Equal to the present value of the interest payments plus the present value of the principal to be received at maturity. The present value of a bond, also known as its intrinsic value, can be determined with a financial function calculator.

The YTC is higher than the YTM. The YTC is lower than the YTM.

For a discount bond, the YTC is higher than the YTM if the bond is called and the principal is redeemed early. If a bond is selling at a premium and it is called by the issuing firm at par, then the YTC would be lower than the YTM.

Positive Convexity Corrects for Duration Over/Under Statements

General rule is duration understates the price increase when rates fall and duration overstates the price decrease when rates rise. As the percentage change in interest rates increases, with convexity the price change in the bond is more than duration alone would explain.

Effective Annual Rate (EAR)

Helps compare the impact of different compounding periods.

Convexity

Helps to measure the impact of interest rate changes greater than 1%. Positive, zero, or Negative convexity. Zero convexity looks like a straight line. Positive like a simley face, negative frowny. Most straight bonds have positive convexity.

High Durations Similar to High Betas and Vice Versa

High durations, like high betas, indicate high risk and high volatility; low durations indicate low risk and low volatility. Treasury bills have low durations and 30-year zero coupon bonds have high durations.

The current yield is higher than the YTM. The current yield is lower than the YTM.

If a bond sells at a premium, the current yield is higher than the YTM. If a bond sells at a discount, the current yield is lower than the YTM.

New bond issued with a 6% coupon; it pays $60 of interest per year, in semiannual payments of $30. Assume market rates have risen and the bond now sells for $900. The current yield is:

If bond price declines because market interest rates have risen, coupon divided by a lower price so current yield greater than coupon yield. bond now sells for $900. Current yield is $60 divided by $900, or 6.67%. The bond yield is now 67 basis points higher.

Duration and Interest Rate Expectations

If expectations were for declining interest rates, then you would seek out higher duration investments. However, if expectations were for interest rates to rise, you would lower the overall duration of a fixed income portfolio.

- D. Negative sign in front of the duration number in Change in Bond Price Formula.

If interest rates go up negative duration number times a positive interest rate number, meaning a negative answer or bond down in price. Interest rates go down negative duration number times a negative change in interest rates, so a positive answer. Bond price up.

Vanguard Long-Term Bond Index has the highest yield, at 3.96%, and the Vanguard Short-Term Bond Index has the lowest yield, at 2.80%. How much more risk for additional 1.16% in yield?

If interest rates were to suddenly move 1% higher, then the Vanguard Short Term Bond Index fund would be expected to lose about 2.7% in value, whereas the Vanguard Long-Term Bond Index fund would lose about 14.7%, which is a large price swing. Not appropriate for risk averse.

Premium Bond

If market interest rates decrease the buyer of the bond will receive the 7% coupon payments, but by paying a premium, $1,050, over the par value, the buyer of the bond will realize a 6% YTM.

Discounted Bond

If market interest rates increase nobody will buy a 7% bond at par when interest rates have increased to 8%. Will pay less than par i.e., buy at discounted price of 975 in order to achieve the 8%YTM.

Bond Price and Interest Rates Inversely Related Example - Interest Rate Increase

If market interest rates of comparable bonds increase, value or price of the bond declines, which makes its flow of income attractive to investors who could otherwise receive a larger flow of income from other newly issued, higher-coupon bonds.

If a municipal bond is free from federal income tax only and has a yield of 5.5%, for an investor in the 24% tax bracket this bond would have a taxable-equivalent yield of 7.24%. Should they buy it?

If the investor can find a taxable bond with an equivalent credit rating and characteristics but with a yield greater than 7.24%, then the taxable bond will yield more, after tax, than the tax free bond. The taxable bond should probably be purchased in that case.

Which contribute more to the present value of a bond, an interest payment received in the near future or an interest payment received in the distant future?

Interest payments received in the near future contribute more to the present value of a bond because dollars received in the distant future have less value today. The present value of $100 received in 3 years is greater than the present value of $100 received in 20 years.

Inverse Relationship Between Duration and Coupon Rates and Current Market Interest Rate

Interest rates increases duration decreases. Interest rates decrease duration increases. But with maturity tandem relationship. Maturity increases so does duration. Maturity decreases so does duration.

High vs Low Durations & Interest Rates

Inverse relationship between bond prices and direction of an interest rate move. In a declining interest rate environment, the higher the duration, the greater the total return that an investor will realize. Rising interest rate environment, high durations greatest losses.

Compare the price volatility of the following types of bonds: bonds with low coupon rates compared to bonds with high coupon rates, assuming both have the same maturity

Low coupon rates more volatile than high. Low-coupon have higher durations than high-coupons as present value of time-weighted cash flows is lower. If interest rates to decline, then choose low coupon higher duration.

Duration

Measure of approximate bond price volatility given a change in interest rates. A measure of interest rate risk. Longer a bond's duration, the more sensitive investment is to changes in interest rates. Effective in immunizing bond portfolios against interest rate risk.

Current Yield

Measure of bond's return based on the stated cash interest per year and the bond's current market price. Divide annual interest payment by market price. Does not take into account the difference between a bond's purchase price and its redemption value.

Yield-to-Call (YTC)

Measure of the yield for bonds that are likely to be called. In calculating, the number of periods until the call date is used instead of the number of periods until maturity, and the call price is used instead of the face value.

if instead you knew the taxable yield was 6.18%, and wanted to know what the tax-free equivalent was:

Multiply the 6.18% by 1 minus the marginal tax bracket: 6.18% × (1 - .32) = 4.2% 1 - .32 = .68. .0618 x .68 = 4.2%

Make the following assumptions unless the problem specifically states otherwise: Present value of the bond PV

Negative number as considered a cash outflow. Any time an investor spends money, or purchases an investment, the amount is entered as a negative number. Assume rate of return assumes any interest payments are being reinvested at the same rate.

Assume Semi-Annual Bond Interest Payments

Or accrue interest semiannually in the case of zero-coupon bonds unless told otherwise.

Calculating the Price of a Bond

Or intrinsic value primarily a function of interest rates. As they change so will intrinsic value. "I/YR" function reserved for current interest rate or current YTM. The coupon rate is converted into a semiannual coupon and entered as a payment PMT.

Problem with Current Yield (CY) - Premium Bond Example

Premium bonds selling for over $1,000. Current yield overstates total return investor will achieve because it does not take into account the fact that the bond will decline from the premium price, let's say $1,030, back to $1,000 at maturity. Need to also consider YTM & YTC.

What factors determine the amount of price fluctuation in a bond?

Price fluctuations are affected by a bond's grade (credit/default risk), its coupon rate, its length of time to maturity, its duration, and any changes in market interest rates.

Bond Duration Formula

Provided. Three inputs. DUR = 1 + y divided by y minus (1 + y) + t(c - y) divided by c[(1 + y)t shift X key minus 1] + y. y = Yield-to-maturity per period c = Coupon rate per period t = Number of periods until maturity. Think "t" for time.

Minimize Uncertainty of Realized Compound Yield of a Bond?

Purchase only noncallable bonds. Noncallable bonds tend to sell for lower yields. The uncertainty associated with changes in interest rates remains.

Semi-Annual Coupon Payment Formula

Semiannual coupon payment (PMT) = $1,000 x Annual Coupon Rate divided by 2.

Yield-to-Call (YTC)

Similar to YTM, except number of periods until call date less than periods until maturity date. YTC on bond selling at a discount always higher than YTM as appreciation will be returned faster. YTC on a bond selling at a premium lower than its YTM as depreciation incurred faster.

Bond Yield and Valuation Calculations: Keystrokes for computing price, yield-to-maturity, and yield-to-call for bonds same as those used for single sums combined with annuities.

Single sums are the bond's present value or purchase or current market price, and its future value generally $1,000. Annuities are semiannual coupon payments. Use top row of keys Five variables N, I/YR, PV, PMT, and FV. Input four variables and solve for unknown fifth.

Yield-to-Maturity (YTM)

Sum of the current yield and the appreciation or depreciation the bond will experience between the current date and its maturity date.

In the 32% marginal tax bracket. Considering investing in a municipal bond with a yield of 4.2%. Also considering Treasury bonds with the same maturity that have a yield of 5.5%. Which should she purchase?

TEY = 1 - .32 = .68. .042 divided by .68 = 6.18%. Tax-free bond has a TEY of 6.18%, which is higher than the 5.5% yield of the Treasury bonds. Based just on yield, Brad would choose the municipal bond.

If a municipal bond is free from both federal and state income taxation a "double tax-exempt bond", and the taxpayer itemizes deductions, then the formulas are as follows:

TEY = TEY divided by 1 - [FMTB + SMTB (1 - FMTB)]. TEY = Taxable equivalent yield, TFEY = Tax-free equivalent yield, FMTB = Federal marginal tax bracket, SMTB = State marginal tax bracket

Taxable Equivalent Yield (TEY) Formula

TEY = Tax-Free-Yield divided by 1 - Marginal Tax Bracket

Jane owns a public purpose municipal bond that pays 6%. Assuming in the 35% marginal tax bracket, what yield on corporate bonds would be comparable to the yield on Jane's current investment?

TEY = Tax-Free-Yield divided by 1 - Marginal Tax Bracket 1-.35 = .65 .06 divided by .65 = 9.23%

Marginal tax bracket is 35%. Considering either a corporate bond that pays 8% annually or a tax-exempt municipal bond. What yield on the municipal bond would be comparable to the yield on the taxable corporate bond?

TEY = Tax-Free-Yield divided by 1 - Marginal Tax Bracket 1=.35 = .65 .08 = X divided by .65. Move the .65 to other side divided by becomes times .08 x .65 = 5.2%

What are the taxable-equivalent yields of municipal bonds with the following tax-free yields for investors in the following marginal tax brackets? Tax Free Yield 4%, 4.5%, 5%. TEY Tax Brackets of 24, 32, 35 and 37%

TEY = Tax-Free-Yield divided by 1 - Marginal Tax Bracket TEY 24% Bracket 4% = 5.26%, 4.5% = 5.92%, 5%= 6.58% TEY 32% Bracket 4% = 5.88%, 4.5% = 6.62%, 5% = 7.35% TEY 35% Bracket 4% = 6.15%, 4.5% = 6.92%, 5% = 7.69% TEY 37% Bracket 4% = 6.35%, 4.5% = 7.14%, 5% = 7.94%

Itemizes deductions and in 8% state marginal income tax bracket. What yield on corporate bonds would then be comparable to Jane's current investment?

Tax Equivalent Yield = Tax Free Equivalent Yield divided by 1 - SMTB x 1- FMTB 1 - .08 = .92, 1 - .35 = .65. .92 x .65 = .598 .06 divided by .598 = 10.03%

Discount Bonds Normally Not Called

Trade at a discount when interest rates go up. Issuer won't call as would have to borrow at a higher rate. 7% bond. Current rates now 8%. No incentive to call as borrowing costs have increased.

Change in Bond Price Formula Four Components

TriangleP = Change in price P = Price, -D = Duration of the bond expressed as a negative Triangley = Expected change in interest rates y = Current yield-to-maturity or current interest rate "y" is the current yield-to-maturity current interest rate, not the coupon rate. Coupon rate has already been taken into account when calculating duration.

Taxable Equivalent Yield (TEY)

Two reasons to calculate: (1) when a municipal bond is free from federal income tax but subject to state income tax (2) when a municipal bond free from both federal income tax and state income tax. CFP Board's approved formula assumes only federal taxation.

Duration Example with Falling/Rising Interest

Vanguard Long-Term Bond Index fund has a duration of 14.7, so if interest rates were to rapidly fall 1%, the fund's value would be expected to increase approximately 14.7%. However, if interest rates were to rise 1%, the fund would decline approximately 14.7%.

Duration Expressed in Years or as Measure of Interest Rate Risk

Weighted-average amount of time measured in years that it takes to collect a bond's principal and interest payments. Vanguard Long-Term fund duration "14.7 years," but duration's value for the financial planner is as a measure of interest rate risk.

Bond Price and Interest Rates Inversely Related Example - Interest Rate Decrease

When market rates decrease, price of the bond increases because its flow of income is more valuable to investors who would otherwise have to accept a smaller flow of income from other newly issued, lower-coupon bonds.

Make the following assumptions unless the problem specifically states otherwise: If a return on "comparable bonds of the same maturity and grade" or I/YR is given as an input for a bond problem:

Will be given as an average annual yield-to-maturity. If you are calculating the price of a bond, this annual rate is a necessary input.

Change in Bond Price Formula

triangleP divided by P = -D x TringleY divided by 1 + Y