Question

A child’s grandparents purchase a $10,000 bond fund that matures in 18 years to be used for her college education. The bond fund pays 4% interest compounded semiannually. How much will the bond fund be worth at maturity? What is the effective rate of interest? How long will it take the bond to double in value under these terms?

Short answer

• Worth of bond fund at maturity : $\$20398.87$
• Effective rate of interest : $4.04\%$
• Time it takes the bond to double in value : $17.5years$

Step-by-step solution

Step 1. Given information  

A child’s grandparents purchase a $10,000 bond fund that matures in 18 years to be used for her college education. The bond fund pays 4% interest compounded semiannually.

Step 2. Required to find  

• Worth of bond fund at maturity
• Effective rate of interest
• Time it takes the bond to double in value

Step 3. Finding the worth of bond fund at maturity 

Using the compound interest formula, $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

Put $P=10000,n=2,r=0.04,t=18$

$A=10000{\left(1+\frac{0.04}{2}\right)}^{2\times 18}$

$\Rightarrow A=10000{\left(1+0.02\right)}^{36}$

$\Rightarrow A=10000{\left(1.02\right)}^{36}$

$\Rightarrow A=20398.87$

Step 4. Finding effective rate of interest 

Effective rate of interest,$r_e={\left(1+\frac{r}{n}\right)}^n-1$

Put $r=0.04,n=2$

$r_e={\left(1+\frac{0.04}{2}\right)}^2-1$

$\Rightarrow r_e={\left(1+0.02\right)}^2-1$

role="math" localid="1646621717617" $\Rightarrow r_e={\left(1.02\right)}^2-1$

role="math" localid="1646621727568" $\Rightarrow r_e=1.0404-1$

$\Rightarrow r_e=0.0404$

$\Rightarrow r_e=4.04\%$

Step 5. Finding time it takes the bond to double in value 

Using the compound interest formula, $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

Put $A=2P,r=0.04,n=2$

$2P=P{\left(1+\frac{0.04}{2}\right)}^{2t}$

$\Rightarrow 2={(1+0.02)}^{2t}$

$\Rightarrow 2={(1.02)}^{2t}$

Taking log on both sides,

$\ln 2=\ln {(1.02)}^{2t}$

$\Rightarrow \ln 2=2t\ln (1.02)$

$\Rightarrow t=\frac{\ln 2}{2\ln 1.02}$

$\Rightarrow t=\frac{0.6931}{2\times 0.0198}$

$\Rightarrow t=\frac{0.6931}{0.0396}$

$\Rightarrow t=17.5$