Financial Mathematics

Recapitulation and Sinking Fund

**Present Value:-**Present value describes how much a future sum of money is worth today. It accounts for the fact that money we receive today can be invested to earn a return.

If money is wroth

*i*per period, then present value

*P*of amount

*S*due

*n*periods hence is given by

*P = S*(1 +

*i*)

^{−n}

*.*

**Annuity:**An annuity is a series of payments made at equal intervals of time. Examples of annuities are regular deposits to a savings account, monthly insurance payments, and pension payments.

In an ordinary annuity, the first payment is made at the end of the first payment period.

**Amount of an Annuity:-**The future value of an annuity is the value of a group of recurring payments at a certain date in the future, assuming a particular rate of return, or discount rate.

The future value

*S*of an ordinary annuity of ₹

*R*per period for

*n*periods at the rate

*i*per period is given by

$S=R\left\{\frac{{\left(1+i\right)}^{n}-1}{i}\right\}$

**Present Value of an Annuity:-**The present value of an annuity is the current value of future payments from an annuity, given a specified rate of return, or discount rate.

The present value

*P*of an ordinary annuity of ₹

*R*per payment period for

*n*periods at the rate

*i*per period is given by

$P=R\left\{\frac{1-{\left(1+i\right)}^{-n}}{i}\right\}$

**Sinking Fund:-**

A sinking fund is a fund created by a corporation or business organisation by putting money aside over time to meet a future capital outlay or the repayment of long-term debt. It is a fund that is built up with the intention of paying off a financial obligation at a later date.

It is an annuity created for accumulating money that can be used for paying off a financial obligation at some future predecided date.

For example, sometimes an individual or a company accumulates money, probably by periodic deposits, either to repay the principal of a loan in one installment or for the expansion of a business, etc.

**Amount in a Sinking Fund:-**

The amount in a sinking fund at any time is the amount of the annuity formed by the payments. Thus, the amount S in a sinking fund at any time is given by

$S=R\left\{\frac{{\left(1+i\right)}^{n}-1}{i}\right\}\phantom{\rule{0ex}{0ex}}R=\mathrm{Periodic}\mathrm{deposit}\mathrm{or}\mathrm{payment}\phantom{\rule{0ex}{0ex}}n=\mathrm{Number}\mathrm{of}\mathrm{periodic}\mathrm{deposits}\phantom{\rule{0ex}{0ex}}i=\mathrm{Interest}\mathrm{per}\mathrm{period}$

**Sinking Fund Payment:-**

The periodic payment of ₹

*R*is required to accumulate a sum of ₹

*S*over n periods with interest charged at the rate

*i*per period is given by

$R=\frac{iS}{\left\{{\left(1+i\right)}^{n}-1\right\}}$

Let us understand these topics with examples.

**Example 1**: How much should a company set aside at the end of each year, if it has to buy a machine expected to cost ₹100,000 at the end of 3 years and the rate of interest is 10% per annum compounded annually? ( Given ${\left(1.1\right)}^{3}=1.331$)

**Solution**: Let ₹

*R*be set aside at the end of each year. Since the company wants ₹100,000 at the end of 3 years. Therefore,

$S=100,000,n=3,i=\frac{10}{100}=0.1\phantom{\rule{0ex}{0ex}}\therefore R=\frac{iS}{{\left(1+i\right)}^{n}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{0.1\times 100,000}{{\left(1.1\right)}^{3}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{10000}{{\left(1.1\right)}^{3}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{10000}{0.331}=30,211.48\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Thus, to accumulate ₹100,000 after 3 years the company should keep aside ₹30,211.48 every year at 10% per annum compounded annually.

**Example 2**: A sinking fund is created for the redemption of debentures of ₹200,000 at the end of 25 years. How much money should be provided out of profits each year for the sinking fund, if the investment can earn interest 6% per annum? ( Given ${\left(1.06\right)}^{25}=4.2918$)

**Solution**: Suppose ₹

*R*are provided out of profits each year for the sinking fund. Then,

$S=200,000,n=25\mathrm{and}i=\frac{6}{100}=0.06\phantom{\rule{0ex}{0ex}}R=\frac{iS}{{\left(1+i\right)}^{n}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{0.06\times 200,000}{{\left(1.06\right)}^{25}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{12000}{4.2918-1}\phantom{\rule{0ex}{0ex}}\Rightarrow 3645.42$

Thus, ₹3645.42 are provided out of profits each year for the sinking fund.

**Present Value:-**Present value describes how much a future sum of money is worth today. It accounts for the fact that money we receive today can be invested to earn a return.

If money is wroth

*i*per period, then present value

*P*of amount

*S*due

*n*periods hence is given by

*P = S*(1 +

*i*)

^{−n}

*.*

**Annuity:**An annuity is a series of payments made at equal intervals of time. Examples of annuities are regular deposits to a savings account, monthly insurance payments, and pension payments.

In an ordinary annuity, the first payment is made at the end of the first payment period.

**Amount of an Annuity:-**The future value of an annuity is the value of a group of recurring payments at a certain date in the future, assuming a particular rate of return, or discount rate.

The future value

*S*of an ordinary annuity of ₹

*R*per period for

*n*periods at the rate

*i*per period is given by

$S=R\left\{\frac{{\left(1+i\right)}^{n}-1}{i}\right\}$

**Present Value of an Annuity:-**The present value of an annuity is the current value of future payments from an annuity, given a specified rate of return, or discount rate.

The present value

*P*of an ordinary annuity of ₹

*R*per payment period for

*n*periods at the rate

*i*per period is given by

$P=R\left\{\frac{1-{\left(1+i\right)}^{-n}}{i}\right\}$

**Sinking Fund:-**

A sinking fund is a fund created by a corporation or business organisation by putting money aside over time to meet a future capital outlay or the repayment of long-term debt. It is a fund that is built up with the intention of paying off a financial obligation at a later date.

It is an annuity created for accumulating money that can be used for paying off a financial obligation at some future predecided date.

For example, sometimes an individual or a company accumulates money, probably by periodic deposits, either to repay the principal of a loan in one installment or for the expansion of a business, etc.

**Amount in a Sinking Fund:-**

The amount in a sinking fund at any time is the amount of the annuity formed by the payments. Thus, the amount S in a sinking fund at any time is given by

$S=R\left\{\frac{{\left(1+i\right)}^{n}-1}{i}\right\}\phantom{\rule{0ex}{0ex}}R=\mathrm{Periodic}\mathrm{deposit}\mathrm{or}\mathrm{payment}\phantom{\rule{0ex}{0ex}}n=\mathrm{Number}\mathrm{of}\mathrm{periodic}\mathrm{deposits}\phantom{\rule{0ex}{0ex}}i=\mathrm{Interest}\mathrm{per}\mathrm{period}$

**Sinking Fund Payment:-**

The periodic payment of ₹

*R*is required to accumulate a sum of ₹

*S*over n periods with interest charged at the rate

*i*per period is given by

$R=\frac{iS}{\left\{{\left(1+i\right)}^{n}-1\right\}}$

Let us understand these topics with examples.

**Example 1**: How much should a company set aside at the end of each year, if it has to buy a machine expected to cost ₹100,000 at the end of 3 years and the rate of interest is 10% per annum compounded annually? ( Given ${\left(1.1\right)}^{3}=1.331$)

**Solution**: Let ₹

*R*be set aside at the end of each year. Since the company wants ₹100,000 at the end of 3 years. Therefore,

$S=100,000,n=3,i=\frac{10}{100}=0.1\phantom{\rule{0ex}{0ex}}\therefore R=\frac{iS}{{\left(1+i\right)}^{n}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{0.1\times 100,000}{{\left(1.1\right)}^{3}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{10000}{{\left(1.1\right)}^{3}-1}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{10000}{0.331}=30,211.48\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Thus, to accumulate ₹100,000 after 3 years the company should keep aside ₹30,211.48 every year at 10% per annum compounded annually.

**Example 2**: A sinking fund is created for the redemption of debentures of ₹200,000 at the end of 25 years. How much money should be provided out of profits each year for the sinking fund, if the investment can earn interest 6% per annum? ( Given ${\left(1.06\right)}^{25}=4.2918$)

**Solution**: Suppose…

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