Question

Report bond prices to the nearest .01, dollar amounts to the nearest dollar, and returns or yields to the nearest basis point. You do not need to express bond prices in 1/32^nd format.

1. You are considering purchasing a newly issued annual coupon bond with 3 years to maturity on the settlement date. The bond pays a coupon of 8% and will be priced with a yield-to-maturity of 5%, and you will purchase $10,000,000 par value.  

1. What is the quoted price of the bond? How much will you need to pay at settlement?

2. What are the bond’s Macaulay duration and modified duration?

3. Assume you must sell the bond after 6 months have passed. The market yield-to-maturity is now 4.5%. Calculate the bond’s dirty price, clean price, and your proceeds from the sale.

Answer 1

Solution:-

Maturity of bond = 3Years

Coupon Rate = 8%

Coupon Amount = $ 80 (Assume Face Value to be $1000)

Yield to Maturity/Discounting rate = 5%

A. Quoted Price of Bond

Quoted price of a bond is a price at which bond is traded or sell in a market. Quoted price of a bond is calculated by discounting all the coupon payments and maturity value receive at the end of maturity period. Quoted Price is calculated as below:-

Price (B₀) ₌ Coupon Amount / (1+ytm)¹ + Coupon Amount / (1+ytm)² + Coupon Amount / (1+ytm)³ + Maturity Value / (1+ytm)³

= $80 / (1+ 0.05)¹ + $80 / (1+ 0.05)² + $80 / (1+ 0.05)³ + $1000 / (1+ 0.05)³

= $80 * 0.952 + $80 * 0.907 + $80 * 0.864 + $1000 * 0.864

= $76.16 + $75.56 + $69.12 + $864

= $1081.84 0r rounded off to $1081.80

Amount need to pay at settlement

Total amount of bond purchases = $10,000,000 at par value

No of Bond Purchased = $10,000,000 / $1000 (Assume Face Value of one bond to be $1000)

= 10,000 bonds to be purchased

Amount Paid for Bond at settlement date = 10,000 Bonds * $1081.80

= $10,818,000

B. Bond’s Macaulay duration and modified duration of Bond

The Macaulay duration of bond s the weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price. Macaulay duration is frequently used by portfolio managers who use an immunization strategy.

The Macaulay duration of bond is calculated as follow by using following formula

= ∑ⁿ_t=1 [(t×C)​ / (1+y)_^t + (n×M​)​ /1+y)ⁿ ] / ( Current Bond Price)

Where, t=respective time period

C=periodic coupon payment

y=yield to maturity

n=total no of period or maturity period

M=Maturity value

= [(1 * $80)/(1+0.05)¹ + (2 * $80)/(1+0.05)² + (3 * $80)/(1+0.05)³ + (3 * $1000)/(1+0.05)³] / $1081.80

= [($80* 0.952) + ($160 * 0.907) + ($240 * 0.864) + ($3000 * 0.864)] / $1081.80

= [$76.16 + $145.12 + $207.36 + $2592] / $1081.80

= 2.792 years or rounded off to 2.80 years

Modified duration of bond expresses the measurable change in the value of security in response to change in interest rates. Modified duration follows the concept that interest rates and bond prices move in opposite directions. This formula is used to determine the effect that a 100-basis-point (1 percent) change in interest rates will have on the price of a bond

Modified Duration = Macaulay duration / (1+ytm)

= 2.80 years / (1+.05)

= 2.66 or rounded off to 2.70%

It means that 1% change in interest rate leads to 2.70% change in bond price in opposite direction.

C. Calculation of  bond’s dirty price, clean price, and your proceeds from the sale

A dirty price is a bond pricing quote, which refers to the cost of a bond that includes accrued interest based on the coupon rate. Bond price quotes between coupon payment dates reflect the accrued interest up to the day of the quote

A Clean price is a bond pricing quote, which refers to the cost of a bond that Excludes accrued interest based on the coupon rate

Clean Price = Dirty Price - Accrued Interest

Now We have to first calculate dirty price of bond

In question it is given that after 6 months bond have been sale and ytm changes to 4.5% now for calculating price after 6 months we have to first calculate price at the end of first year

B₀ = Coupon Amount / (1+ytm)¹ + Coupon Amount / (1+ytm)² + Maturity Value / (1+ytm)²

= $80 / (1+ 0.045)¹ + $80 / (1+ 0.045)² + $1000 / (1+ 0.045)²

= $80 * 0.957 + $80 * 0.916 + $1000 * 0.916

= $76.56 + $73.28 + $916

= $1065.84

Now we have to discount back the price calculated above for 6 months to calculate the bond price after 6 monts  

= $1065.84 / (1+0.0225) [Take 6 months doscount rate for 4.5% i.e 2.25%]

= $1042.38 or rounded off to $1042.40

Dirty price = $1042.40

Now for calculating clean price we have to calculate Accrued interest

Accrued Interest = $80 * 6/12

= $40

Therrfore Clean Price = Dirty Price - Accrued Interest

= $1042.40 - $40

= $1002.4

Proceeds From Sale = Current Market price after 6 months * No. Of Bonds Purchased

= $1042.40 * 10,000 Bonds

= $10,424,000