Put a deferred annuity on the right date
A deferred annuity is an ordinary series shifted later in time. Its P/A factor still anchors one period before the first payment.
Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
The annuity's local present date is year 3, not year 4.
Payments occur at years 4 through 7: 4 payments.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
What the model means
Suppose N equal payments begin at year s. Their local present worth sits at s−1. First compute that value, then discount it to year 0. If the last payment is at T, the payment count is T−s+1, not T.
Build the calculation
For a savings-then-withdrawals problem, first find the present worth of future withdrawals, then solve the required deposit series. Deposits that overlap withdrawals can still be handled by superposition, but their dates must be explicit.
Set up. Calculate. Explain.
Rebuild the starting scenario
Annual benefit: 5000 $/year; Interest rate: 10 % per year; Number of years: 4 years; First payment date: 4 year. Calculate the present worth at time 0.
Setup hint
Payments occur at years 4 through 7: 4 payments.
Open the visual solution
- Payments occur at years 4 through 7: 4 payments.
- At year 3, P = A(P/A,i,N) = $15,849.33.
- Discount that single local P through 3 periods to year 0.
Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
The annuity's local present date is year 3, not year 4.
Payments occur at years 4 through 7: 4 payments.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Transfer the idea to a new scenario
Six benefits of $7,000 occur at years 5 through 10. At 8%, calculate their present worth at year 0.
Setup hint
Payments occur at years 5 through 10: 6 payments.
Open the visual solution
- Payments occur at years 5 through 10: 6 payments.
- At year 4, P = A(P/A,i,N) = $32,360.16.
- Discount that single local P through 4 periods to year 0.
Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
The annuity's local present date is year 4, not year 5.
Payments occur at years 5 through 10: 6 payments.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Payments begin at year 4. Where is their local P/A equivalent?
Can you defend your setup?
Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?
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