Horizon Yield of a Fixed-Rate Bond
An investor buys an eight-year maturity bond and holds it for three years. The bond pays a 4.5% annual coupon on its 100 face value. Immediately after the purchase, the yield-to-maturity increases from 6% to 6.25%. The investor's horizon yield on this bond is _closest_ to:
Correct!
The purchase price of the bond can be obtained by taking the present value of all the coupons and the face value:
$$\displaystyle PV_{0}=\frac{PMT}{(1+r)^{1}}+\frac{PMT}{(1+r)^{2}}+\frac{PMT}{(1+r)^{3}}+ ... +\frac{PMT+FV}{(1+r)^{N}}$$
$$\displaystyle PV_{0}=\frac{4.5}{(1+0.06)^{1}}+\frac{4.5}{(1+0.06)^{2}}+\frac{4.5}{(1+0.06)^{3}}+ ... +\frac{4.5+100}{(1+0.06)^{8}}=90.6853$$
[[calc: 8 n 6 iy 4.5 pmt 100 fv cpt pv , 8 n 6 i 4.5 pmt 100 fv pv]]
Next, you need to find the value of the bond in Year 3 with the increased yield-to-maturity. This will be the selling price:
$$\displaystyle PV_{3}=\frac{4.5}{(1+0.0625)^{1}}+\frac{4.5}{(1+0.0625)^{2}}+\frac{4.5}{(1+0.0625)^{3}}+ ... +\frac{4.5+100}{(1+0.0625)^{5}}$$
>$$ \approx 92.6782$$
[[calc: 5 n 6.25 iy 4.5 pmt 100 fv cpt pv , 5 n 6.25 i 4.5 pmt 100 fv pv]]
Next, calculate the future value of the three coupons reinvested at 6.25%, which is:
$$\displaystyle 4.5(1+0.0625)^{2} + 4.5(1+0.0625)^{1} + 4.5 \approx 14.3613$$
[[calc: 3 n 6.25 iy 0 pv 4.5 pmt cpt fv , 3 n 6.25 i 0 pv 4.5 pmt fv]]
Next, calculate the total gain for the investor in Year 3, which is:
$$\displaystyle 92.6782+14.3613=107.0396$$
Finally, calculate the horizon yield $$r$$ by solving:
$$\displaystyle 90.6853=\frac{107.0396}{(1+r)^{3}}$$
$$\displaystyle r = \left( \frac{107.0396}{90.6853} \right) ^{1/3} - 1 \approx 0.056823 \approx 5.7 \%$$
[[calc: 3 n 90.6853 sign pv 0 pmt 107.0396 fv cpt iy , 3 n 90.6853 chs pv 0 pmt 107.0396 fv i]]
Incorrect.
The horizon yield is equal to the yield-to-maturity if the yield-to-maturity of the bond at the point of purchase does not change and the investor holds the bond until it matures.
Incorrect.
This answer appears close to the annualized yield on the coupons of the bond. Horizon yield includes both capital gain or loss and coupon yield.
5.0%.
5.7%.
6.0%.
