ECN226 Capital Markets 1 – 2011 Past Paper Questions and Answers

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Discounted prices compared to all other websites

ECN226 Capital Markets 1 – 2011
Questions and Answers
Question 1




a) Note that there is an error in the question. The value of a bond is equal to the present discounted
value of all of its coupon payment (CP) plus the discounted value of its Face value (F) payed at the
end of the bonds life. We assume that the bond pays 1 coupon per year for T years.

The present value of the bond is then:
𝐶𝑃 𝐶𝑃 𝐶𝑃 𝐹
𝑃𝑉 = + + ...+ +
1+𝑟 (1 + 𝑟) (1 + 𝑟) (1 + 𝑟)
1 1 1 𝐹
𝑃𝑉 = 𝐶𝑃[ + + ...+ ] +
1+𝑟 (1 + 𝑟) (1 + 𝑟) (1 + 𝑟)

To further simplify the present value formula, we first add ( )
+ ( )
+ . .. and take away
the same numbers. Then, the present value can be expressed as follows:
1 1 1 1 1 1
𝑃𝑉 = 𝐶𝑃[ + + ...+ + + +. . . −
1+𝑟 (1 + 𝑟) (1 + 𝑟) (1 + 𝑟) (1 + 𝑟) (1 + 𝑟)
1 𝐹
− − ...] +
(1 + 𝑟) (1 + 𝑟)

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1 1 1 1 1 1
𝑃𝑉 = 𝐶𝑃[ [1 + + + ...] − [1 + + + . . . ]]
1+𝑟 1+𝑟 (1 + 𝑟) (1 + 𝑟) 1+𝑟 (1 + 𝑟)
𝐹
+
(1 + 𝑟)

The terms [1 + + ( )
+ . . . ] are infinite sums. We can express this term using the infinite
geometric summation formula:
1 1 1
1 + + + . .. =
1+𝑟 (1 + 𝑟) 1
1 − 1+𝑟

Then,
1 1 1 1 1 1 1
+ + ...+ = [ ] − [ ]
1+𝑟 (1 + 𝑟) (1 + 𝑟) 1+𝑟 1 − 1 (1 + 𝑟) 1
1 − 1+𝑟
1+𝑟
Simplifying this, we arrive at:
1 1 1 1 1
+ + ...+ = −
1+𝑟 (1 + 𝑟) (1 + 𝑟) 𝑟 𝑟(1 + 𝑟)
Substituting this into the formula defined previously:
1 1 𝐹
𝑃𝑉 = 𝐶𝑃[ − ] +
𝑟 𝑟(1 + 𝑟) (1 + 𝑟)


b) Here, bond A would have a lower volatility as compared to bond B. The relationship between
bond price volatility and the coupon rate is an inverse one – the higher the coupon rate, the less
volatile the bond price is to interest rate change, and vice versa. Bonds with higher coupon rates pay
higher coupon payments, allowing investors to be paid back their initial investment costs sooner in
terms of time value of money, and thus subjecting bond prices to interest rate change to a lesser
degree. This is because a higher coupon payment results in the bond having a shorter duration, and
in general, bonds with a long duration have a higher price fluctuation than bonds with a short
duration.

c) Yield to maturity (YTM) is the total return anticipated on a bond if the bond is held until it
matures. Yield to maturity is considered a long-term bond yield but it is expressed as an annual rate.

Bond D will have a higher yield to maturity as compared to Bond C. This is because there is an
inverse relationship between market price of the bond and its yield. The higher the market price, the
lower the return and the lower the market price the higher the return in bond. To see this
mathematically, consider the equation below, which relates the given price of a bond (P) with the
return expected by investors (y):
1 1 𝐹
𝑃 = 𝐶𝑃[ − ] +
𝑦 𝑦(1 + 𝑦) (1 + 𝑦)
It can be shown that as y increases, P decreases, which demonstrates the inverse relationship
between price and the yield to maturity.