Assume that when market yields were 4.254% an investor purchased a $25,000 strip bond 20 years before maturity. The purchase price was $10,772.52. The investor then sold the bond five years later, when current yields dropped to 3.195%. The selling price was $15,539.94. The investor wants to know the gain and the actual yield realized on her investment.

Solution:

Step 1: PV = $10,772.52. This is the purchase price paid for the strip bond.

Step 2: FV = $15,539.94. This is the selling price received for the strip bond.

Step 3: C/Y = 2; Years = 5 (between the purchase and sale price);

[latex]n=C/Y \times \text{(Number of Years)}=2 \times 5=10[/latex]

Step 4: Calculate the periodic interest rate, i.

[latex]\begin{align} FV&=PV(1+i)^n\\ \$15,\!539.94&=\$10,\!772.52(1+i)^{10}\\ 1.442554&=(1+i){10}\\ 1.442554^{\frac{1}{10}}&=1+i\\ 1.037321&=1+i\\ i&=0.037321 \end{align}[/latex]

Step 5: Calculate the nominal interest rate, I/Y.

[latex]i=\frac{I/Y}{C/Y}[/latex]
[latex]\begin{align} I/Y&=i \times C/Y\\ &=0.037321 \times 2\\ &=0.074642\%\; \text{or}\; 7.4642\% \end{align}[/latex]

Step 6: Calculate the actual gain (interest earned).

[latex]I=FV-PV=\$15,\!539.94-\$10,\!772.52=$4,\!767.42[/latex]

The investor gained $4,767.42 on the investment, representing an actual yield of 7.4642% compounded semiannually.

Cameron invested in a $97,450 face value Government of British Columbia strip bond with 25 years until maturity. Market yields at that time were 9.1162%. Ten and half years later Cameron needed the money, so he sold the bond when current yields were 10.0758%. Calculate the actual yield and gain that Cameron realized on his investment.

Solution:

The timeline below illustrates the situation.

Step 1: Calculate the purchase price (PV) based on the purchase date.

I/Y = 9.1162%; C/Y = 2; Years = 25

[latex]i=\frac{I/Y}{C/Y}=\frac{9.1162\%}{2}=4.5581\%[/latex]

[latex]n=C/Y \times \text{(Number of Years)}=2 \times 25=50[/latex]

[latex]\begin{align} PV&=\frac{\$97,\!450}{(1+0.045581)^{50}}\\ &=\$10,\!492.95 \end{align}[/latex]

Step 2: Calculate the selling price (PV) based on the selling date.

I/Y = 10.0758%; C/Y = 2; Years = 14.5

[latex]i=\frac{I/Y}{C/Y}=\frac{10.0758\%}{2}=5.0379\%[/latex]

[latex]n=C/Y \times \text{(Number of Years)}=2 \times 14.5=29[/latex]

[latex]\begin{align} PV&=\frac{\$97,\!450}{(1+0.050379)^{29}}\\ &=\$23,\!428.63 \end{align}[/latex]

This becomes the FV for subsequent calculations.

Step 3: C/Y = 2; Years = 14.5
[latex]n=C/Y \times \text{(Number of Years)}=2 \times 10.5=21[/latex]

Step 4: Calculate the periodic interest rate, i.

PV=$10,492.95; FV=$23,428.63

[latex]\begin{align} FV&=PV(1+i)^n\\ \$23,\!428.63&=\$10,\!492.95(1+i)^{21}\\ 2.232797&=(1+i){21}\\ 2.232797^{\frac{1}{21}}&=1+i\\ 1.038991&=1+i\\ i&=0.038991 \end{align}[/latex]

Step 5: Calculate the nominal interest rate, I/Y.

[latex]\begin{align} I/Y&=i \times C/Y\\ &=0.038991 \times 2\\ &=0.077982\%\; \text{or}\; 7.7982\% \end{align}[/latex]

Step 6: Calculate the actual gain (interest earned).

[latex]I=FV-PV=\$23,\!428.63-\$10,\!492.95=\$12,\!935.68[/latex]

Calculator instructions:

Cameron sold the strip bond when prevailing market rates were higher than the rate at the time of purchase. He has realized a lower yield of 7.7982% compounded semi-annually, which is a gain of $12,935.68.