Why Use Two Focal Dates?
A staged annuity problem may contain payments in several periods and an interest rate that changes at a specified date. A focal date is the date at which the cash flows are brought together for an equation of value. Choosing a convenient focal date can make the arithmetic simpler; choosing a second one can provide a useful check. The important qualification is that, under one consistent deterministic interest model, the two equations describe the same equivalence condition. They are not generally two independent equations for finding two unknown amounts.
The method here assumes annual cash flows at integer times and a known, deterministic effective rate for each year. A cash flow is positive if it is received and negative if it is paid. We assume the same accumulation basis applies to every cash flow. If different rates or other valuation rules apply to different streams, the relationship between focal dates must be established from that model rather than assumed.
For a rate that varies by year, define \(A(s,t)\) as the product of the annual accumulation factors from time \(s\) to time \(t\), for \(s<t\). For example, if the effective annual rate is 5% in the first year and 10% in the second, then \(A(0,2)=1.05(1.10)\). Rates and dates must be matched before any annuity shortcut is used.
Why the Two Focal-Date Equations Are Linked
Suppose the focal date moves from \(f\) to a later date \(g\). Let \(A(f,g)\) be the accumulation factor between those dates. A cash flow before \(f\) first accumulates to \(f\), then accumulates from \(f\) to \(g\). A cash flow between \(f\) and \(g\) is discounted back to \(f\) and then accumulated forward to \(g\). In either case, its value at \(g\) is its value at \(f\) multiplied by \(A(f,g)\). For a cash flow after \(g\), the same conclusion follows because accumulation factors multiply across adjacent intervals.
Therefore, if \(V_f\) is the net value of all cash flows at \(f\), the net value at \(g\) is \(V_g=A(f,g)V_f\). Since \(A(f,g)>0\), \(V_f=0\) if and only if \(V_g=0\). This is why a second focal date can confirm an equation but ordinarily cannot supply a new independent constraint.
Worked Valuation at Two Focal Dates
The following process is useful when solving a staged annuity problem. First write each dated cash flow explicitly. Then value the complete set at a convenient focal date, solve for the unknown, and translate the equation to a second date as a check.
Distinguish a payment at a stage boundary from one at the end of the preceding period.
Build each accumulation factor from the rates applying between the cash-flow date and the focal date.
Use the focal date that makes the number of discounting or accumulation steps manageable.
Multiply the entire net-value equation by the accumulation factor between the focal dates, then check the result.
Worked Example: Solve a Level Payment and Check It at a Later Date
An amount of $1,000 is invested at time 0. The effective annual rate is 5% in years 1 and 2, then 10% in years 3 and 4. The investment is exchanged for four equal payments \(R\) at times 1, 2, 3, and 4. Find \(R\). Use focal dates 2 and 4, and explain why the second equation is a check rather than a new condition.
Method and assumptions. Treat the initial investment as a receipt and the four payments as outflows. Use annual effective rates and end-of-year payments. At time 2, the investment has accumulated to \(1000(1.05)^2=1102.50\). The first two payments accumulate to time 2, while the third and fourth payments are discounted back using the 10% rate that applies in years 3 and 4.
Thus \(R=1102.50/3.78553719\), or, equivalently, we can use the focal-date-4 equation below to obtain the same result. At time 4, the investment has accumulated through both rate stages. The payments at times 1 and 2 accumulate across the remaining periods, and the payment at time 3 earns 10% for one year.
The left side is \(1000(1.1025)(1.21)=1334.025\). The payment coefficient is \(1.2705+1.21+1.10+1=4.5805\). Therefore \(R=1334.025/4.5805=291.240039\ldots\). To the nearest cent, \(R=\$291.24\).
Check. The time-2 payment coefficient multiplied by the accumulation factor from time 2 to time 4 is \(3.78553719(1.21)=4.5805\). The time-2 investment value likewise becomes \(1102.50(1.21)=1334.025\). Thus the two equations are the same equation scaled by 1.21. The unrounded payment satisfies both; using $291.24 introduces only the expected small rounding residual.
Common mistake avoided: Discounting the time-3 and time-4 payments at 5% because the first two years use 5%. Those payments cross into the 10% stage, so their values at time 2 use 10%.
Using One Date to Solve and Another to Verify
A second focal date is especially effective as an audit when the unknown payment occurs between the two focal dates. At the earlier date, that payment is discounted; at the later date, it is accumulated. These are not different unknowns or different financial conditions: the two coefficients are linked by the accumulation factor between the dates.
Worked Example: Find an Intermediate Lump Sum
A fund receives $1,000 at time 0 and makes payments of $200 at time 1, $300 at time 2, an unknown amount \(K\) at time 3, and $400 at time 4. The effective annual rate is 5% in years 1 and 2 and 10% in years 3 and 4. Find \(K\) using time 2, then check the equation at time 4.
Method. At time 2, the initial receipt is worth \(1000(1.05)^2=1102.50\). The time-1 payment is worth \(200(1.05)=210\), and the time-2 payment is worth $300. The later payments are discounted at 10% per year.
Since \(400/(1.10)^2=330.578512\ldots\), the amount left for the discounted value of \(K\) is \(1102.50-210-300-330.578512=261.921488\ldots\). Hence \(K=1.10(261.921488\ldots)=288.113636\ldots\). The exact model amount is approximately $288.1136; to the nearest cent it is $288.11.
Check at time 4 using the unrounded amount. The initial receipt accumulates to \(1000(1.05)^2(1.10)^2=1334.025\). The payments accumulate to \(200(1.05)(1.10)^2=254.10\), \(300(1.10)^2=363\), \(1.10K=316.925\), and $400. Their sum is \(254.10+363+316.925+400=1334.025\), as required.
Common mistake avoided: Rounding \(K\) to cents before checking it. The unrounded amount verifies the exact equation; a cent-rounded contractual payment can leave a small residual.
What Two Dates Can and Cannot Determine
If two amounts are unknown, writing the same equivalence equation at two focal dates does not generally determine both. Multiplying the first equation by the accumulation factor gives the second, so eliminating an unknown by subtracting the scaled equations also eliminates the remaining information. This dependency is an important diagnostic: a second focal date can expose an inconsistent calculation, but it cannot manufacture another independent condition.
Worked Example: Eliminate an Unknown and Identify the Missing Information
An investment of $1,000 at time 0 is exchanged for payments of $100 at time 1, $200 at time 2, \(X\) at time 3, and \(Y\) at time 4. The effective annual rate is 5% in years 1 and 2 and 10% in years 3 and 4. Write the equations at times 2 and 4, eliminate \(X\), and determine whether \(X\) and \(Y\) can both be found. Then find \(X\) if the contract specifies \(Y=\$304.975\).
At time 2, the investment is worth $1,102.50. The earlier payments are worth $105 and $200; the later ones are discounted to time 2. Therefore
Multiplying by 1.21 gives \(964.975=1.10X+Y\). At time 4, the investment is worth $1,334.025. The two known payments are worth \(100(1.2705)=127.05\) and \(200(1.21)=242\). Thus
Subtracting the known amounts on the right gives \(1.10X+Y=964.975\), exactly the same relation. In particular, subtracting the time-2 equation after scaling it by 1.21 from the time-4 equation produces \(0=0\): the \(X\) terms cancel, but so does all independent information. Without another condition, \(X\) and \(Y\) cannot both be uniquely determined.
If \(Y=304.975\), then \(1.10X=964.975-304.975=660\), so \(X=600\). Substitution gives \(1.10(600)+304.975=964.975\), confirming the relation.
Common mistake avoided: Treating two focal-date equations as two independent equations just because they look different before simplification. Always check whether one is a scalar multiple of the other.
Common Focal-Date Errors
- Using the wrong stage rate. Assign rates by the periods each cash flow actually crosses, not by the rate applying at the focal date alone.
- Misplacing a boundary payment. A payment at time 2 is not a payment one full year after time 2. Mark its date before accumulating or discounting.
- Changing the sign convention midway. Either equate receipts and payments as positive totals or use signed cash flows throughout. Do not mix the two approaches within one equation.
- Calling a second valuation an independent condition. Under a common deterministic rate schedule, it is a scaled version of the first. Use it as a check or as an alternative computational route.
- Comparing rounded values too early. Keep full precision through the verification, then round the final payment as instructed.
Check Your Understanding
For each problem, use the stated annual effective rates and payment dates. Keep full precision until the final answer.
- An investment of $1,000 at time 0 earns 5% in year 1 and 10% in year 2. It is exchanged for equal payments at times 1 and 2. Find the payment using time 2 as focal date. (Answer: $550)
- A fund earns 5% in years 1 and 2 and 10% in years 3 and 4. It receives $100 at time 1, $100 at time 2, $200 at time 3, and $200 at time 4. Find the accumulated value at time 4. (Answer: $668.05)
- An initial $500 at time 0 is exchanged for payments of $100 at times 1 and 2, an unknown \(K\) at time 3, and $200 at time 4. The rate is 4% in years 1 and 2 and 8% in years 3 and 4. Find \(K\), using time 2 as focal date; round to cents. (Answer: $178.56)
- Under 5% in years 1 and 2 and 10% in years 3 and 4, an investment of $800 at time 0 is exchanged for payments of $100 at time 1, $200 at time 2, \(X\) at time 3, and \(Y\) at time 4. Can the two focal dates determine both unknowns? State the relation they must satisfy. (Answer: Not uniquely; \(1.10X+Y=698.17\).)
- For the first example in this tutorial, suppose each payment is $300 instead of the amount found there. At time 2, by how much do the payment values exceed the investment value? At time 4, by how much do they exceed it, and how should the two excesses compare? (Answer: At time 2, $33.16; at time 4, $40.13. Before rounding, the time-4 excess is 1.21 times the time-2 excess.)