Integrated Workshop on Mixed Level and Varying Annuities

One Timeline, One Focal Date

A mixed annuity problem may combine a lump sum, a deferred level annuity, and payments that increase or decrease over time. Its main challenge is not a new annuity formula. It is keeping the dates, payment amounts, and applicable interest rates consistent. A reliable solution begins by recording every cash flow on one timeline and choosing a focal date at which to compare values.

The focal date can be the present, the last payment date, or an intermediate date. Any choice works if every cash flow is accumulated or discounted to that same date using the rates that apply over the relevant intervals. The calculations below use effective rates over the periods between dated cash flows. When the quoted rate is nominal, convert it to the effective rate for the period before valuing the cash flows.

Definition: A focal date is the date to which each cash flow in a value equation is moved. Under a specified interest model, two sets of cash flows are financially equivalent at that date when their accumulated values there are equal.

A Common-Date Valuation Rule

Suppose the effective rate in period $k$ is $i_k$, and a cash flow $C_j$ occurs at time $j$. To move it forward from $j$ to a later focal time $N$, multiply it by the accumulation factors for all intervening periods. To move it backward, divide by those factors. If a constant effective rate $i$ applies throughout, the familiar factors are $(1+i)^{N-j}$ and $v^j=(1+i)^{-j}$, respectively.

\text{Value at time }N = \sum_{j} C_j\prod_{k=j+1}^{N}(1+i_k)

This expression assumes that the listed cash flows occur at the stated times, that each $i_k$ applies over its specified period, and that there are no unlisted cash flows. A payment at the focal date needs no accumulation or discounting. If rates change, do not apply one rate to periods that belong to another rate regime.

The rule follows from repeated accumulation. A cash flow $C_j$ becomes $C_j(1+i_{j+1})$ one period later, then $C_j(1+i_{j+1})(1+i_{j+2})$ after a second period. Continuing this step until time $N$ gives the product of all intervening factors. Summing the resulting values is valid because accumulation is linear: each cash flow earns interest independently of the others.

C_j \longrightarrow C_j(1+i_{j+1}) \longrightarrow C_j(1+i_{j+1})(1+i_{j+2}) \longrightarrow \cdots \longrightarrow C_j\prod_{k=j+1}^{N}(1+i_k)

For a level annuity-immediate of $n$ payments $R$, with the first payment one period after the valuation date and a constant effective rate $i$, its present value is $R a_{\overline{n}|}$. A deferred annuity is valued by first locating its actual payment dates: deferral does not change the payment pattern, but it does change how far each payment is from the focal date. For a varying annuity, the safest general method is to write the actual payment amount and date for every payment. The standard arithmetic and geometric annuity formulas can then shorten the calculation when their timing and pattern fit.

Worked Examples

Worked Example: Nominal Rate, Deferral, and an Arithmetic Stage

A fund earns a nominal annual rate of $8\%$, convertible semiannually. At time 0, an investor deposits \$500. The investor receives \$200 at the ends of half-year periods 2, 3, and 4, followed by payments of \$100, \$150, and \$200 at the ends of periods 5, 6, and 7. Find the value of all these cash flows at time 0.

Method. Convert the nominal rate to the rate per half-year, then discount every dated cash flow to time 0. Time is measured in half-year periods, so period 2 is one year after time 0.

The nominal rate is convertible twice per year, so the effective rate per half-year is $0.08/2=0.04$. The corresponding effective annual rate is $(1.04)^2-1=0.0816$, or $8.16\%$. Since all payment dates are separated by half-year periods, use $i=0.04$ per period and $v=1/1.04$.

The time-0 deposit is already at the focal date and has value \$500. The deferred level payments occur at periods 2, 3, and 4, so their present value is:

200(v^2+v^3+v^4) = 200(0.9245562+0.8889964+0.8548042) = \$533.67

The final three payments are an arithmetic-increasing sequence. Discount them at their actual dates rather than treating them as if the sequence began at period 1:

100v^5+150v^6+200v^7 = 100(0.8219271)+150(0.7903145)+200(0.7599178) = \$352.72

Adding the three components gives $500+533.67+352.72=\$1,386.39$, to the nearest cent. As a check, the same cash flows accumulated to period 7 have value approximately \$1,824.40; dividing by $(1.04)^7$ gives approximately \$1,386.39. The calculation avoids the common mistake of using the annual effective rate as though it were the rate per half-year.

Worked Example: A Payment Pattern Crossing a Rate Change

A hypothetical account earns $5\%$ effective per year during years 1 through 3 and $7\%$ effective per year during years 4 through 6. It receives \$300 at the end of each of years 1, 2, and 3. It then receives \$330, \$363, and \$399.30 at the ends of years 4, 5, and 6, respectively. Find the present value at time 0.

Method. Use time 3 as an intermediate focal date. Accumulate the first stage to time 3 using $5\%$, value the later stage at time 3 using $7\%$, add the two values, and discount the total back using the first three years’ rate.

At time 3, the first-stage payments have accumulated as follows: the year-1 payment earns interest for two years, the year-2 payment for one year, and the year-3 payment for no time.

300(1.05)^2+300(1.05)+300 = 330.75+315+300 = \$945.75

The later payments are discounted to time 3 at $7\%$, because that rate applies during years 4 through 6. Their amounts increase by $10\%$ each year, but their actual dates and amounts can be valued directly:

\frac{330}{1.07}+\frac{363}{(1.07)^2}+\frac{399.30}{(1.07)^3} = 308.4112+317.0583+325.9477 = \$951.42

The total value at time 3 is $\$945.75+\$951.42=\$1,897.17$. Discount this three years at $5\%$, since years 1 through 3 are the periods between time 0 and time 3:

\frac{1,897.1672}{(1.05)^3} = \frac{1,897.1672}{1.157625} = \$1,638.84

Thus the present value is \$1,638.84, to the nearest cent. A useful check is to verify that the year-4 through year-6 payments were discounted only to time 3 at $7\%$, and that the entire time-3 value was then discounted through years 1 to 3 at $5\%$. Applying $7\%$ to the whole time-0-to-6 span would use the wrong rate for the first stage.

Worked Example: Arithmetic Payments, a Deferred Level Annuity, and a Lump Sum

A cash-flow arrangement earns $6\%$ effective annually. It pays \$400 at time 1, \$500 at time 2, and \$600 at time 3. It also pays \$700 at each of times 4 and 5, plus a separate lump sum of \$250 at time 5. Find the equivalent single amount at time 3.

Method. Use time 3 as the focal date. Accumulate payments before time 3 and discount payments after time 3. The \$700 payments form a deferred two-payment level annuity, but their payment dates must remain 4 and 5.

The first three payments are moved forward to time 3:

400(1.06)^2+500(1.06)+600 = 449.44+530+600 = \$1,579.44

The level payments at times 4 and 5 are discounted to time 3. The \$250 lump sum at time 5 is discounted over the same two-year interval:

\frac{700}{1.06}+\frac{700}{(1.06)^2}+\frac{250}{(1.06)^2} = 660.3774+622.9975+222.4991 = \$1,505.87

Combining the values gives $1,579.44+1,505.87=\$3,085.31$. Therefore, the single amount at time 3 equivalent to the full arrangement is \$3,085.31, to the nearest cent. This check also confirms that the payment at time 3 is not discounted: it is already at the focal date. Treating the time-4 payment as if it occurred at time 3 would overstate its value there.

A Practical Workflow for Integrated Problems

When a problem contains several stages, a short ledger is often clearer than trying to select one large annuity formula at the outset. Record each payment date, amount, and the rate applying between that date and the focal date. Group payments only after their timing is clear.

Set the time unit.
Use the interval that matches the cash-flow dates, such as years or half-years. Convert a nominal rate to an effective rate for this interval before calculating.

Mark every payment date.
Distinguish the first payment date from the deferral period. A deferred annuity-immediate’s first payment is still one full payment period after the deferral ends.

Write the amount at each stage.
For an arithmetic pattern, record the fixed dollar change. For a geometric pattern, record the growth ratio. Note whether the amount resets at a stage boundary.

Choose one focal date.
Move each payment to that date using the rate for every intervening period. Keep a rate-change boundary visible rather than blending rates into one.

Check the equation.
Confirm that every cash flow appears once, at its correct date, and that a payment at the focal date has factor 1. If practical, revalue the same cash flows at another date.

Common Traps and Why the Method Matters

The common-date method is useful because it makes the assumptions visible. A compact annuity formula can be efficient, but only if its payment frequency, first payment date, and rate match the cash flows in the problem. For an irregular start, a rate change, or a payment pattern that changes midway, explicitly listing the cash flows is usually safer.

A frequent error is to count the deferral as an extra payment period twice. If the first payment is at time 4, its present-value factor at time 0 is $v^4$, not $v^3$. Another is to apply an annual effective rate directly to half-year payments. If a nominal rate $i^{(m)}$ is convertible $m$ times per year, the rate per conversion period is $i^{(m)}/m$; the effective annual rate is $(1+i^{(m)}/m)^m-1$. Use the rate that matches the time steps in the valuation.

A third trap occurs when a payment sequence crosses a rate boundary. Payment growth and interest accumulation are different operations: a geometric increase changes the payment amount, while the interest rate determines how that amount moves through time. Record both separately. In particular, do not assume a geometric payment increase is also the rate earned on the account.

Key takeaway: Write the cash-flow dates and amounts first. Convert rates to the correct period, then move every amount to one focal date using only the rates that apply along its path. Group into annuity formulas only when the payment pattern and timing genuinely match.

Check Your Understanding

For each problem, identify the time unit and focal date before calculating.

A nominal rate of $6\%$ is convertible quarterly. Find the effective annual rate and the accumulated value of \$1,000 invested for two years. (Answer: $6.1364\%$ effective annually; \$1,126.49)
At $5\%$ effective annually, payments of \$200 are made at times 3, 4, and 5. Find their present value at time 0. (Answer: \$494.01)
At $4\%$ effective annually, payments of \$100, \$120, and \$144 are made at times 1, 2, and 3. Find the present value at time 0. (Answer: \$335.12)
An account earns $4\%$ effective in years 1 and 2, then $6\%$ effective in year 3. A deposit of \$500 is made at time 0 and another \$200 at time 2. Find the value at time 3. (Answer: \$785.25)
At $5\%$ effective annually, payments of \$300 occur at times 1 and 2, followed by \$400 at time 3 and \$500 at time 4. Find their accumulated value at time 4. (Answer: \$1,598.04)

What You'll Learn