Solve a time-value-of-money parameter

Solving a TVM parameter

A time-value-of-money parameter is found by leaving one field unknown in the same cash-flow model. Apply the required cent, whole-period or rate boundary and verify the result in the original model

1. Name the unknown field Leave the requested payment, period count or nominal rate as the quantity to solve \(\displaystyle B_n=R^nP+sD\frac{R^n-1}{R-1}\)
2. Use the stated boundary Apply the boundary in the question after solving the exact model \(\displaystyle D^*=\$99.604\Rightarrow D=\$99.61\)
3. Check adjacent periods A first qualifying period is proved by checking the period before and the period at the crossing \(\displaystyle B_4<T\le B_5\Rightarrow n=5\)
4. Convert a solved period rate Multiply a rate for one period by the number of periods per year before reporting the nominal annual rate \(\displaystyle r_{\mathrm{nominal}}=4\times r_{\mathrm{period}}\)

Worked example

An account starts with $1,000 and earns 10% p.a. An addition of D is made at the end of each year for 3 years. The balance after 3 years is $1,827.50. Find D.

1. \(\displaystyle R=1+\frac{10}{100}=1.10\)Form the annual multiplier from the nominal annual rate
2. \(\displaystyle R^3=1.10^3=1.331,\quad S=\frac{1.331-1}{0.10}=3.31\)Calculate the principal multiplier and the accumulated effect of one end-of-year addition
3. \(\displaystyle D=\frac{\$1{,}827.50-1.331\times\$1{,}000}{3.31}=\frac{\$496.50}{3.31}=\$150\)Subtract the grown principal from the target and divide by the accumulation factor

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