Formulate and prove the Fisher rule.

The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates under inflation. It is named after Irving Fisher, who was famous for his works on the theory of interest. In finance, the Fisher equation is primarily used in YTMcalculations of bonds or IRR calculations of investments. In economics, this equation is used to predict nominal and real interest rate behavior. Economists generally use the Greek letter as the inflation rate, not the constant 3.14159....

Letting denote the real interest rate, denote the nominal interest rate, and let denote the inflation rate, the Fisher equation is:

This is a linear approximation, but as here, it is often written as an equality:

The Fisher equation can be used in either ex-ante (before) or ex-post (after) analysis. Ex-post, it can be used to describe the real purchasing power of a loan:

Rearranged into an expectations augmented Fisher equation and given a desired real rate of return and an expected rate of inflation over the period of a loan, , it can be used ex-ante version to decide upon the nominal rate that should be charged for the loan:

This equation existed before Fisher, but Fisher proposed a better approximation which is given below. The approximation can be derived from the exact equation:

Derivation

Although time subscripts are sometimes omitted, the intuition behind the Fisher equation is the relationship between nominal and real interest rates, through inflation, and the percentage change in the price level between two time periods. So assume someone buys a $1 bond in period t while the interest rate is . If redeemed in period, t+1, the buyer will receive dollars. But if the price level has changed between period t and t+1, then the real value of the proceeds from the bond is therefore

From here the nominal interest rate can be solved for.

(1)

In expanded form, (1) becomes:

Assuming that both real interest rates and the inflation rate are fairly small, (perhaps on the order of several percent, although this depends on the application) is much larger than and so can be dropped, giving the final approximation:

.

More formally, this linear approximation is given by using two 1st order Taylor expansions, namely:

Combining these yields the approximation:

and hence

24. Annuity. Ordinary annuity. Formulae for the accumulated value and present value of final ordinary

yearly annuities.

Annuity

In finance theory, an annuity is a terminating "stream" of fixed payments, i.e., a collection of payments to be periodically received over a specified period of time.^[1] The valuation of such a stream of payments entails concepts such as the time value of money, interest rate, and future value.^[2]

Examples of annuities are regular deposits to a savings account, monthly home mortgage payments and monthly insurance payments. Annuities are classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other interval of time.

Annuity-immediate

An annuity is a series of payments made at fixed intervals of time. If the number of payments is known in advance, the annuity is an annuity-certain. If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuity-immediate, or ordinary annuity. Mortgage payments are annuity-immediate, interest is earned before being paid.

	↓	↓	...	↓	payments
———	———	———	———	—
			...	n	periods

The present value of an annuity is the value of a stream of payments, discounted by the interest rate to account for the payments are being made at various moments in the future. The present value is given in actuarial notation by:

where is the number of terms and is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or rent is:

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest is stated as a nominal interest rate, and .

The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by:

where is the number of terms and is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or rent is:

Example: The present value of a 5 year annuity with annual interest rate 12% and monthly payments of $100 is:

The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the principal of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal.

Future and present values are related as:

and

Proof of Annuity Formula

To calculate present value, the k-th payment must be discounted to the present by dividing by the interest, compounded by k terms. Hence the contribution of the k-th payment R would be R/(1+i)^k. Just considering R to be one, then:

We notice that the second factor is an infinite geometric progression of the form,

therefore,

Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (n −1) years. Therefore,

Example calculations

Formula for Finding the Periodic payment(R), Given A:

R = A/(1+〖(1-(1+(j/m))〗^(-(n-1))/(j/m))

Examples:

1. Find the periodic payment of an annuity due of $70000, payable annually for 3 years at 15% compounded annually.

· R = 70000/(1+〖(1-(1+((.15)/1))〗^(-(3-1))/((.15)/1))

· R = 70000/2.625708885

· R = $26659.46724

2. Find the periodic payment of an annuity due of $250700, payable quarterly for 8 years at 5% compounded quarterly.

· R= 250700/(1+〖(1-(1+((.05)/4))〗^(-(32-1))/((.05)/4))

· R = 250700/26.5692901

· R = $9435.71

Finding the Periodic Payment(R), Given S:

R = S\,/((〖((1+(j/m))〗^(n+1)-1)/(j/m)-1)

Ordinary Annuity

A series of equal payments made at the end of each period over a fixed amount of time. While the payments in an annuity can be made as frequently as every week, in practice, ordinary annuity payments are made monthly, quarterly, semi-annually or annually. The opposite of an ordinary annuity is an annuity due, where payments are made at the beginning of each period.

Calculating the Future Value of an Ordinary Annuity
If you know how much you can invest per period for a certain time period, the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate. If you are making payments on a loan, the future value is useful for determining the total cost of the loan.

Let's now run through Example 1. Consider the following annuity cash flow schedule:

In order to calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let's assume that you are receiving $1,000 every year for the next five years, and you invested each payment at 5%. The following diagram shows how much you would have at the end of the five-year period:

Since we have to add the future value of each payment, you may have noticed that, if you have an ordinary annuity with many cash flows, it would take a long time to calculate all the future values and then add them together. Fortunately, mathematics provides a formula that serves as a short cut for finding the accumulated value of all cash flows received from an ordinary annuity:

C = Cash flow per period i = interest rate n = number of payments

If we were to use the above formula for Example 1 above, this is the result:

= $1000*[5.53] = $5525.63

Note that the one cent difference between $5,525.64 and $5,525.63 is due to a rounding error in the first calculation. Each of the values of the first calculation must be rounded to the nearest penny - the more you have to round numbers in a calculation the more likely rounding errors will occur. So, the above formula not only provides a short-cut to finding FV of an ordinary annuity but also gives a more accurate result.

Calculating the Present Value of an Ordinary Annuity
If you would like to determine today's value of a series of future payments, you need to use the formula that calculates the present value of an ordinary annuity. This is the formula you would use as part of a bond pricing calculation. The PV of ordinary annuity calculates the present value of the coupon payments that you will receive in the future.

For Example 2, we'll use the same annuity cash flow schedule as we did in Example 1. To obtain the total discounted value, we need to take the present value of each future payment and, as we did in Example 1, add the cash flows together.

Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many future payments. As such, there is a mathematical shortcut we can use for PV of ordinary annuity.

C = Cash flow per period i = interest rate n = number of payments

The formula provides us with the PV in a few easy steps. Here is the calculation of the annuity represented in the diagram for Example 2:

= $1000*[4.33] = $4329.48

Not that you'd want to use it now that you know the long way to get present value of an annuity

25. Infinite annuities (perpetuities). Formula for present value.

Perpetuity

A perpetuity is an annuity for which the payments continue forever. Since:

even a perpetuity has a finite present value when there is a non-zero discount rate. The formula for a perpetuity are:

where is the interest rate and is the effective discount rate.

A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely. It is sometimes referred to as a perpetual annuity. Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. Scholarships paid perpetually from an endowment fit the definition of perpetuity.

The value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low present value (present value of the future cash flows). Unlike a typical bond, because the principal is never repaid, there is no present value for the principal. Assuming that payments begin at the end of the current period, the price of a perpetuity is simply the coupon amount over the appropriate discount rate or yield, that is

Where PV = Present Value of the Perpetuity, A = the Amount of the periodic payment, and r = yield, discount rate or interest rate.

26. Internal rate of return of a cash flow (define and justify the existence).

Definition

Showing the position of the IRR on the graph of NPV(r) (r is labelled 'i' in the graph)

The internal rate of return on an investment or project is the "annualized effective compounded return rate" or "rate of return" that makes the net present value (NPV as NET*1/(1+IRR)^year) of all cash flows (both positive and negative) from a particular investment equal to zero.

In more specific terms, the IRR of an investment is the discount rate at which the net present value of costs (negative cash flows) of the investment equals the net present value of the benefits (positive cash flows) of the investment.

IRR calculations are commonly used to evaluate the desirability of investments or projects. The higher a project's IRR, the more desirable it is to undertake the project. Assuming all projects require the same amount of up-front investment, the project with the highest IRR would be considered the best and undertaken first.

A firm (or individual) should, in theory, undertake all projects or investments available with IRRs that exceed the cost of capital. Investment may be limited by availability of funds to the firm and/or by the firm's capacity or ability to manage numerous projects.

Calculation

Given a collection of pairs (time, cash flow) involved in a project, the internal rate of return follows from the net present value as a function of the rate of return. A rate of return for which this function is zero is an internal rate of return.

Given the (period, cash flow) pairs (, ) where is a positive integer, the total number of periods , and the net present value , the internal rate of return is given by in:

The period is usually given in years, but the calculation may be made simpler if is calculated using the period in which the majority of the problem is defined (e.g., using months if most of the cash flows occur at monthly intervals) and converted to a yearly period thereafter.

Any fixed time can be used in place of the present (e.g., the end of one interval of an annuity); the value obtained is zero if and only if the NPV is zero.

In the case that the cash flows are random variables, such as in the case of a life annuity, the expected values are put into the above formula.

Often, the value of cannot be found analytically. In this case, numerical methods or graphical methods must be used.

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