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Bonds

Formulas & Examples

Basic bond valuation

Excel functions

Examples

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Bond Val

Basic Bond Valuation

Bond price (PV annuity + PV principal)

PV=PMT*(1-1/(1+R)^n)/R PV=FV/(1+R)^n

Example Coupon rate 8 FV 100

Yield 8.00% Yield 8.00%

Periods 10 Periods 10

PV 53.7 PV 46.32

Answer Price $100.00

Bond price with semi-annual payments

PV=(PMT/m)*(1-1/(1+R/m)^n*m)/(R/m) PV=FV/(1+R/m)^n*m

Example Coupon rate 8 FV 100

Yield 8.00% Yield 8.00%

Frequency 2 2

Periods 10 Periods 10

PV 54.4 PV 45.64

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Excel functions

Bond calculations with Excel functions

Use PRICE function to find a bonds price

Settle date 1/1/1997 Mat date 1/1/2007 Rate 8% Yield 9% Frequency 2 Price 93.50

Use YIELD function to find a bond's yield to maturity

Settle date 1/1/1997 Mat date 1/1/2007 Rate 8% Price 95 Frequency 2 Yield 8.76%

Use MDURATION function to find a bond's modified duration

Settle date 1/1/1997 Mat date 1/1/2007 Rate 8% Price 8% Frequency 2 Modified duration 6.8

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Examples

Examples

1 Atlantic Bell issued 10-year bonds one year ago at a coupon rate of 9.25 percent. The bonds make semiannual payments. If the YTM on these bonds is 7.15 percent, what is the current bond price? Use the Excel PV function to calculate the current value of the bond.

Coupon rate 8.25%

Yield to maturity 12.00%

Years till maturity 7

Bond value $82.57

2 IBM issued 12-year bonds two years ago at a coupon rate of 8.4 percent. The bonds make semiannual payments. If these bonds currently sell for 97.5 percent of par value, what is the YTM?

Use Excel YIELD function

Bond value $95.88 41.16606

Par value (redemption value) $100

Coupon rate (annual) 4.00%

Years till maturity 5

Today 6/6/2014 Maturity 6/6/2019

Yield to maturity 4.95%

3 Stern Investments has 14 percent coupon bonds issued by Rotten Tree Inc with seven years to maturity. The bonds make semiannual payments and currently sell for 105 percent of par. What is the current yield on the bonds?

The YTM? The effective annual yield?

Use Excel YIELD and EFFECT functions

Bond value $105.00

Par value (redemption value) $100

Coupon rate (annual) 14.00%

Years till maturity 7

Today 6/6/2014 Maturity 6/6/2021

Current yield 13.33%

Yield to maturity 12.89%

Effective annual yield 13.31%

4 The YTM on a bond is the interest rate you earn on your investment if interest rates don't change. If you actually sell the bond before it matures, your realized return is known as the holding period yield (HPY).

a. Suppose that today you buy a GE 9 percent coupon bond making annual payments of $1,200. The bond has 10 years to maturity. What rate of return do you expect to earn on your investment?

b. Two years from now, the YTM on your bond has declined by 2.5 percent, and you decide to sell. What price will your bond sell for? What is the HPY on your investment? Compare this to the YTM when you first bought the bond. Why are they different?

Use Excel YIELD and PRICE functions

Bond value $120.00

Par value (redemption value) $100

Coupon rate (annual) 9.00%

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Examples

Today 6/6/2014 Maturity 6/6/2024

Yield to maturity 6.28%

2 years later

New yield 3.78%

Par value (redemption value) $100

Coupon rate (annual) 9.00%

Years till maturity 8

Today 6/6/2016 Maturity 6/6/2024

New Price 135.77

Holding period yield

Bond value $120.00

Par value (redemption value) 135.77

Coupon rate (annual) 9.00%

Years till maturity 2

Today 6/6/2014 Maturity 6/6/2016

Holding period yield 13.45%

5 An EDS Corp. bond carries an 8 percent coupon, paid semiannually. The par value is $1,000 and the bond matures in six years. If the bond currently sells for $911.37, what is its yield to maturity?

What is the effective annual yield?

Settle date 1/1/1997 Mat date 1/1/2003 Rate 8% Price 91.137 Frequency 2 Yield 10.00% EAY 10.25%

6 A callable bond pays annual interest of $60, has a par value of $1,000,

matures in 20 years but is callable in 10 years

at a price of $1,100, and has a value today of $911.90. The yield to call on this bond is:

Settle date 1/1/1997 Mat date 1/1/2007 Rate 6% Price 91.19 Frequency 1 Yield 7.27%

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Examples

7 A coupon bond which pays interest annually, has a par value of $1,000, matures in 5 years and has a yield to maturity of 12%.

If the coupon rate is 9%, the value of the bond today will be: Settle date 1/1/1997 Mat date 1/1/2002 Rate 9% Yield 12% Frequency 1 Price 89.19

8 As a portfolio manager at Putnam Management, you bought a 10-year, 8.25% semiannual-pay bond at issuance 2 1/2 years ago. At that time the yield to maturity was 8.23%; it's now fallen to 7.10%. How much has the bond's modified duration changed?

Original Present Settle date 1/1/1997 7/1/1999 Mat date 1/1/2007 1/1/2007 Rate 8.25% 8% Yield 8.23% 9% Frequency 2 2 Modified duration 6.7 5.5

9 You are the risk manager at a new savings bank, Lostyur Trust. Mr Edgar Lostyur, sole shareholder, has been told by the regulatory authorities to provide them with the institution's modified duration in order to evaluate its sensitivity to interest rate fluctuations. The Prime rate is 8.25%.

You find that the initial balance sheet of the savings bank looks like this: Assets:

Cash on deposit 2 million face value

3-month loans at Prime + 2% 13 million face value

4-year, 6% s.a. Treasury bonds yielding 6.35% 20 million face value 7-year, 7.90% s.a. IBM bonds yielding 8.23% 6 million face value

Liabilities:

6-month CDs at 4% 3 million face value

5-year 9% s.a. bonds 33 million face value

Equity: 100,000 shares

What can you tell the regulators the bank's modified duration of assets is? Of liabilities? And what is the net modified duration?

Based on the net duration, what would be the effect on net worth of a 10bp increase in rates?

Item Cash Loans Treas note IBM bond 6-mo CD 5-yr bond Equity

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Examples Settle date 1/1/1997 1/1/1997 1/1/1997 1/1/1997 1/1/1997 1/1/1997 Mat date 1/1/1997 4/1/1997 1/1/2001 1/1/2004 7/1/1997 1/1/2002 Rate - 10.25% 6.00% 7.90% 4.00% 9.00% Yield - 10.25% 6.35% 8.23% 4.00% 9.00% Frequency - 2 2 2 2 2 Price 100 100.00 98.78 98.27 100.00 100.00 46.52 Value 2 13.00 19.76 5.90 3.00 33.00 4.65 Modified duration 0 0.2 3.5 5.3 0.5 4.0

If rates rose by 0.10% value would change by 0.028661 million

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Examples

IBM issued 12-year bonds two years ago at a coupon rate of 8.4 percent. The bonds make semiannual

Stern Investments has 14 percent coupon bonds issued by Rotten Tree Inc with seven years to maturity. The bonds make semiannual payments and currently sell for 105 percent of par. What is the current yield on the bonds?

The YTM on a bond is the interest rate you earn on your investment if interest rates don't change. If you actually

a. Suppose that today you buy a GE 9 percent coupon bond making annual payments of $1,200. The bond has

b. Two years from now, the YTM on your bond has declined by 2.5 percent, and you decide to sell. What price will your bond sell for? What is the HPY on your investment? Compare this to the YTM when you

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Examples

As a portfolio manager at Putnam Management, you bought a 10-year, 8.25% semiannual-pay bond at issuance 2 1/2 years ago. At that time the yield to maturity was 8.23%; it's now fallen to 7.10%. How much has the bond's

You are the risk manager at a new savings bank, Lostyur Trust. Mr Edgar Lostyur, sole shareholder, has been told by the regulatory authorities to provide them with the institution's modified duration in order to evaluate its

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Examples

40.65 40.65 2.54 3.25 -0.71

References

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