Pesky Bond Pricing Question involving Redemption Yield

Question

Consider a bond with face value £100 with semi-annual coupons at a rate 3% per annum and redeemable at par in ten years. If the bond is to produce a gross redemption yield of 3.5% per annum, what is its price at issue?

So I've attempted this question in two ways and have gone wrong in both methods.

First attempt: Using the fact that D = 0.03, F=100, i=0.035, I equated the price with the present value of the sequence of coupon payments + the present value of the redemption value of £100. This gave me an answer of £96.06 which was wrong.

Second attempt: I used the formula $YTM = (C + (F - P)/n)/(F+P)/2)$, where YTM = 0.035, F = 100, n = 10, C = 3. This gave me an answer of £95.75 which was wrong.

Supposedly the correct answer is £95.81. Where have I gone wrong? Thanks.

Answer 1

The formula relating price and yield-to-maturity with semiannual coupon payments is

$$P = F\sum_{j=1}^n\frac{C/2}{(1+y/2)^j} + \frac{F}{(1+y/2)^n}$$

Using the closed form expression for a geometric sum $\sum_{j=1}^n \alpha^j = \frac{\alpha - \alpha^{n+1}}{1-\alpha}$, where $\alpha = (1+y/2)^{-1}$, we get

$$\tag{*}P = \frac{FC}{y}\left[1 - (1+y/2)^{-n} \right]+ F (1+y/2)^{-n} $$

In this case we have $F = 100$, $C = 0.03$, $y = 0.035$, and the number of semiannual payment periods is $n = 20$ (not $10$). Substituting these values into (*) we obtain

$$P \approx 95.81$$

Answer 2

I think the best way to answer this question is to use the formula for bond pricing.

\begin{align} P = F \cdot r \cdot a_{\overline{n}|} \ + \ C \cdot v^n \end{align}

which has the same intuition as what you did in your first attempt.

I might think that you made a mistake by not dividing the coupon rate into two since the coupon rate pays semi-annualy.

The given $r$ or the coupon rate is a nominal coupon rate convertible semi-annualy, so the coupon paid semi-annualy is $ \frac {r}{2} = 1.5 \% $

And not to forget that you have to convert the effective yield of

\begin{align} i = 3.5\% \end{align}

into the nominal yield rate convertible semi-annually

\begin{align} i^{(2)} = 2 \times [(1+i)^{\frac{1}{2}} - 1] \approx 3.4698995 \% \end{align}

and use

\begin{align} j = \frac{i^{(2)}}{2} \approx 1.7349497 \% \end{align} as your yield to maturity, and then you should be able to find the correct answer.