Derivation for number of periods finance formula

[issacnewton]

Homework Statement

Hi, I am reading this book called, Introduction to Corporate Finance, by Berk, Demarzo, Harford (second edition). In it they try to explain how to calculate the number of periods in a loan payment formula. Authors give the following equation.
$$ 0 =PV + PMT \times \frac{1}{I/Y}\left( 1 - \frac{1}{(1+I/Y)^N} \right) + \frac{FV}{(1+I/Y)^N} $$

where $PV$ is the present value of annuity, $PMT$ is cash flow or payment per period, and $I/Y$ is the discount rate. I don't understand how they arrive at this formula. Authors are math phobic, I think, as they don't derive difficult formulae from first principles.

Homework Equations

Formula for the annuity.
$$ PV = C \times \frac{1}{r} \left( 1 - \frac{1}{(1+r)^N} \right) $$

where $C$ is annuity for $N$ periods with interest rate $r$.

The Attempt at a Solution

I am stuck. Any hints would be helpful.

 

[issacnewton]

I think the authors are trying to explain how to get the rate of return. In the title of this post, I said number of periods. We can just use logarithms to find it. But to calculate the rate of return , we need to use numerical analysis.

 

[issacnewton]

I think we don't this complicated expression. We can just run the numerical analysis on the annuity formula.

 

[haruspex]

What is FV? I'll guess "future value".

In the first equation, something on the right must be negative. Is PV negative here, maybe measured from the opposite point of view from the PV in the second equation?

If that is right, and the second equation assumes future value is zero (all paid out) then the first equation is just a generalisation of the second to the case where FV is nonzero.

 

[Chestermiller]

Why don't you try deriving it yourself? Let $P_0$ be the initial principal of the loan, r the yearly interest rate, and n the number of payments per year. Let P be the remaining principal after N payments (say, P is a final balloon payment). Once you have derived the equation for P, you will have established all the cash flows, and you can derive an expression for the NPV.

 

[issacnewton]

I think I resolved this. All I can do is to use the formula for annuity.

 

[scottdave]

What is FV? I'll guess "future value".

In the first equation, something on the right must be negative. Is PV negative here, maybe measured from the opposite point of view from the PV in the second equation?

If that is right, and the second equation assumes future value is zero (all paid out) then the first equation is just a generalisation of the second to the case where FV is nonzero.

Most financial calculators (and the Payment function in MS Excel) will return a negative number for the payment if the present value is positive, and future value is less than the present value, so the Payment is likely the one which will be negative. Think about it - if you owe some money, and positive is the convention for how much you owe, then each payment will contribute to reducing how much you owe, so the payment should be opposite sign of the present value.

 

[scottdave]

Typically the interest rate, is per period. Example, if the payments were monthly, and you had a rate of 12% per year, then the rate (per period) is 1%, since there are 12 periods in a year. If it is quarterly, then the rate per period is 3%, etc.

 

[Chestermiller]

I've done what I suggested the OP do in post #5. After n payments, the remaining principal is given by:
$$P_n=P_0r^n-m\frac{(r^n-1)}{r-1}$$where $P_0$ is the original principal, n is the number of payments, m is the payment at the end of each interest interval, and $r=1+i$, with i representing the interest rate applied to each payment. If we divide this equation by $r^n$ (the discount factor), we obtain: $$\frac{P_n}{r^n}=P_0-m\frac{(1-r^{-n})}{i}$$ The loan is repaid when $P_n=0$. So, $$0=P_0-m\frac{(1-r^{-n})}{i}$$. This is the present value of the stream of cash flows received by the loanee until he has paid back the loan in full. The second term on the right hand side is the present value of the annuity received by the loanor reckoned at time zero, and is equal to the amount originally loaned.