# How long will it take?

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Assuming that I invest a certain amount of money into a single corporate bond, with a rate of 7.5%, then, assuming the corporation does not default, and that the bond is automatically renewed each term unless I explicitly ask for it, how long will it take before the value of the bond doubles?

For example, if I invest $10,000 into a 7.5% corporate bond, how long will it take me to rack up$20,000? If I invest $25k into that same bond, how long will it take me to rack up$50k?

Thanks.

EDIT: By the way: Interest would compound quarterly, because I feel that is the most common compounding period when an individual is the one investing, rather than a finance institution.

Edited by dstebbins

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For a given return on investment, call it p, the amount of money after 1 compounding is given by

$N_1 = N_0(1+p)$

For example, let the return be 2%, p = 0.02, and given an initial seed of $100.$100*(1.02) = $102. The above can be generalized for any number of compoundings: $N_k = N_0(1+p)^k$ N_k is the money after being compounded k times. To answer your question the above, the last thing to figure out is the effective interest rate. Most interest rates are quoted on an annual basis, even if compounded on a different period. What you need to figure out is whether the quarterly interest rate would be 0.075/4 or if the quarterly rate is the rate where the effective annual compounded interest ends up being 7.5%. That is the p_q (q for quarterly) is the solution to: $(1+p_q)^4 = (1 + p_y)$ where p_y is the yearly rate, 7.5% in this example. Or, in other words, if you put in$100 and let it be compounded quarterly at that p_q 4 times, you would then have \$107.50.

In this case, p_q = 1.8245%. Which is different than 0.075/4 = 1.875%. You'll have to read the fine print in the contract to figure out which one it is. But, as with a lot of things in life, whichever way it benefits the bank is probably the way it will be. I.e. compounded at 1.875% for a loan, and compounded at 1.8245% for a savings account.

Edited by Bignose

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The standard formula for compounding interest is:

$A\left(t\right)=P\left(1+r\right)^{t}$

Where:

$A\left(t\right)=$ amount in account

$t=$ time periods passed[

$P=$ the principle deposit

$r=$ the growth rate

So in your case of 7.5% interest:

$A\left(t\right)=P\left(1+.075\right)^{\frac{t}{4}}$

The $\frac{t}{4}$ comes from the fact that I am letting $t$ represent the number of months since investment, but since your interest compounds quarterly you must divide the number of months by four.

As for your questions about how long it will take a certain deposit to reach a certain amount, it seems like you are interested in doubling your money. In that case it does not matter what your principle is. To prove this:

$P=p$

$A\left(t\right)=2p$ ie: twice your starting amount

$A\left(t\right)=2p=p\left(1+.075\right)^{\frac{t}{4}}$

$\frac{2p}{p}=\frac{p}{p}\left(1+.075\right)^{\frac{t}{4}}=2$

$log_{1.075}\left(2\right)=log_{1.075}\left(\left(1+.075\right)^{\frac{t}{4}}\right)$

$log_{1.075}\left(2\right)=\frac{t}{4}$

$4log_{1.075}\left(2\right)=\frac{4t}{4}=t$

$t=4log_{1.075}\left(2\right)=38.337$

So at 7.5% interest it will take you about 39 months to double your principle investment.

Edited by DJBruce

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Hi

Thanks. It solves some of my problems..

regards

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