# Annuity Payment

An annuity is a series of equal cash flows that occur after equal interval of time. If we know the interest rate and number of time periods, we can work out the annuity cash flow that corresponds to a specific present value and/or future value.

Let us assume you want to enroll in an 8-semester MBA program. If you have $100,000 in savings currently invested at 10% per annum, you need to find out the maximum amount that you can draw down at the start of each month. Now let us assume that the$100,000 that you have in savings are not enough, hence you must postpone your admission by 3 years and have at least $150,000 by then. You can find out the amount that you must save at the end of each month using the time value of money relationships. ## Formula In order to calculate the annuity cash flow (annuity receipt or payment), referred to as PMT, you need to essentially rearrange the equations for present value or future value of an (ordinary) annuity: $$\text{PMT}=\text{PV}\div\frac{\text{1}-{(\text{1}+\text{i})}^{-\text{n}}}{\text{i}}=\frac{\text{PV}}{\text{1}-{(\text{1}+\text{i})}^{-\text{n}}}\times \text{i}$$ $$\text{PMT}=\text{FV}\div\frac{{(\text{1}+\text{i})}^\text{n}-\text{1}}{\text{i}}=\frac{\text{FV}}{{(\text{1}+\text{i})}^\text{n}-\text{1}}\times \text{i}$$ Where, PV is the present value, FV is the future value, i is the periodic interest rate and n is the number of periods. PMT in case of an annuity due can be worked out by dividing the PMT obtained for an ordinary annuity by (1 + i): $$\text{PMT}\ (\text{Annuity Due})=\frac{\text{PMT}\ (\text{Ordinary Annuity})}{\text{1}+\text{i}}$$ The above equations are for a scenario when either PV or FV is zero. If we have to work with both PV and FV, the following expression can be used to find PMT: $$\text{FV}=\text{PV}\times{(\text{1}+\text{i})}^\text{n}+\text{PMT}\times\frac{{(\text{1}+\text{i})}^\text{n}-\text{1}}{\text{i}}$$ ## Example Let us use the equations to solve your time value of money problems. In the first scenario, you have$100,000 today (hence, it is the present value PV), your periodic interest rate i is 0.833% (10% divided by number of months) and number of time periods n is 24 (i.e. total number of months in the 2-year program). You can use the PV of annuity equation to find annuity payment:

$$\text{PMT}=\text{\100,000}\div\frac{\text{1}-{(\text{1}+\text{0.833%})}^{- \text{24}}}{\text{0.833%}}=\text{\4,614.49}$$

Since you need to make the draw down at the start of each month (i.e. most of your expenses are pre-paid), your series of cash flows is an annuity due and your annuity due payment would effectively be $4,576.36: $$\text{PMT}\ (\text{Annuity Due})=\frac{\text{\4,614.49}}{\text{1}+\text{0.833%}}=\text{\4,576.36}$$ This tells us that with$100,000 in savings you can draw down at the rate of $4,576.36 per month. In the second scenario, you want to have$150,000 (i.e. an additional $50,000) in 3 years (i.e. 36 months). If the monthly interest rate is 0.833%, the additional savings that you must make (annuity deposit) can be obtained using the FV of annuity equation: $$\text{PMT}=\text{\50,000}\div\frac{{(\text{1}+\text{0.833%})}^{\text{36}}-\text{1}}{\text{0.833%}}=\text{\1,196.69}$$ In this analysis, we have ignored the$100,000 that you already have. Since it will also earn interest each month for 3 years, it will reduce your saving requirement.

$$\text{PMT}=\left(\text{FV}-\text{PV}\times{(\text{1}+\text{i})}^\text{n}\right)\div\frac{{(\text{1}+\text{i})}^\text{n}-\text{1}}{\text{i}}$$

$$\text{PMT}=\frac{\text{FV}-\text{PV}\times{(\text{1}+\text{i})}^\text{n}}{{(\text{1}+\text{i})}^\text{n}-\text{1}}\times \text{i}$$

$$\text{PMT}=\frac{\text{\150,000}-\text{\100,000}\times{(\text{1}+\text{0.833%})}^{\text{36}}}{{(\text{1}+\text{0.833%})}^{\text{36}}-\text{1}}\times\text{0.833%}=\text{\363.36}$$