Solve for present or future value with regular cash flows

Solving a TVM value

A time-value-of-money model combines interest with a regular cash flow. Here V_n is the balance after n periods, R is the interest multiplier per period, D is the regular cash-flow amount, N is the number of periods, and s is +1 for an addition or −1 for a repayment. Map the known endpoint, use consistent signs and keep the exact balance until the requested rounding boundary

1. Map the known endpoint Calculate forward when the starting value is known, or rearrange the model when the ending value is known \(\displaystyle V_N=R^NV_0+sD\frac{R^N-1}{R-1}\)
2. Apply interest before the cash flow The model first multiplies the previous balance and then adds or subtracts the end-of-period cash flow \(\displaystyle V_1=1.05\times\$800-\$60=\$780\)
3. Carry regular cash flows Each regular cash flow is carried through the periods that follow it \(\displaystyle V_2=1.05^2\times\$800-\$60\left(1.05+1\right)=\$759\)
4. Round only the marked balance Keep the exact balance between periods and round once at the requested boundary \(\displaystyle \$1{,}234.567\mathrel{\to}\$1{,}234.57\)

Worked example

A loan starts with a balance of $5,000. Interest is charged at 6% p.a. and a $400 repayment is made at the end of each year for 3 years. Find the balance after 3 years

1. \(\displaystyle R=1+\frac{6}{100}=1.06\)Convert the annual interest rate to the multiplier for one year
2. \(\displaystyle V_1=1.06\times\$5{,}000-\$400=\$4{,}900\)Apply interest to the opening balance before the first repayment
3. \(\displaystyle V_2=1.06\times\$4{,}900-\$400=\$4{,}794\)Apply the same transition to the balance after the first year
4. \(\displaystyle V_3=1.06\times\$4{,}794-\$400=\$4{,}681.64\)Use the second-year balance for the third transition and retain the cents

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