Sinking Funds With Changing Fund Yields

Why Changing Fund Yields Need Careful Timing

A sinking fund may receive regular deposits while earning different rates in different periods. The central task is to apply each period’s growth factor to the balance that is already in the fund, rather than applying one average rate to every deposit. A deposit made late has fewer periods to grow than an earlier deposit.

This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The focus here is the accumulation of scheduled sinking-fund deposits when the fund yield varies over time. The loan’s interest rate and the fund’s investment yield are separate rates: the accumulation calculations below use the fund yield.

Definition: Suppose the fund is divided into periods $1,2,\ldots,n$. Let $j_k$ be the effective fund rate earned during period $k$, and let $D_k$ be the deposit made at the end of period $k$. The fund balance immediately after that deposit is $B_k$. Assume the rates are known, apply to the stated periods, and there are no fees, taxes, withdrawals, or other cash flows.

The end-of-period timing convention matters. The deposit $D_k$ is made after the fund has earned the return for period $k$, so it does not earn $j_k$. It can earn the fund rates in periods $k+1$ through $n$. If the problem instead specifies beginning-of-period deposits, their accumulation periods differ; do not apply the end-of-period formula without changing the timing.

Balance Recurrence and Accumulation Formula

During period $k$, the existing balance $B_{k-1}$ earns the rate $j_k$. The deposit $D_k$ is then added at the end of that period. Thus, with no initial fund balance, $B_0=0$, and

B_k=B_{k-1}(1+j_k)+D_k.

This recurrence is often the safest way to keep the timeline straight. It also gives a direct derivation of the maturity value. Starting with $B_1=D_1$, each subsequent period multiplies the accumulated balance by its growth factor and adds the new end-of-period deposit. Repeating this process gives

B_n = D_1\prod_{r=2}^{n}(1+j_r) +D_2\prod_{r=3}^{n}(1+j_r) +\cdots +D_{n-1}(1+j_n)+D_n.

To see why each term has this form, consider a particular deposit $D_k$. It enters the balance at the end of period $k$. In period $k+1$, it is multiplied by $1+j_{k+1}$; in period $k+2$, the resulting amount is multiplied by $1+j_{k+2}$, and so on. By the end of period $n$, its accumulated value is $D_k\prod_{r=k+1}^{n}(1+j_r)$. Adding the accumulated values of all deposits gives $B_n$. For the last deposit, the product has no factors, so its value at the end of period $n$ is simply $D_n$.

For equal end-of-period deposits $D$, define the total accumulation factor

A_n=\sum_{k=1}^{n}\prod_{r=k+1}^{n}(1+j_r).

Then $B_n=DA_n$. If the required maturity fund is $F$, the level deposit is $D=F/A_n$. This uses the fact that each deposit has the same amount but a different number, and sequence, of growth factors.

Set the period timeline.
Record the rate earned in each period and the date each deposit is made.

Assign growth factors to deposits.
An end-of-period deposit in period $k$ receives only the factors for periods $k+1$ through $n$.

Accumulate or solve for the deposit.
Add the accumulated values for a known deposit schedule. For equal deposits, divide the target fund by the sum of the accumulation factors.

Check the result.
Use the recurrence $B_k=B_{k-1}(1+j_k)+D_k$ and verify the ending balance independently.

Worked Calculations

Worked Example: Equal Deposits Through Three Different Yields

A hypothetical fund receives a deposit of $1,000 at the end of each year for three years. The effective fund rates are 2% in year 1, 4% in year 2, and 1% in year 3. Find the fund balance immediately after the third deposit.

Method. Accumulate each deposit through only the rates earned after it is deposited. This avoids the common mistake of making every deposit earn the rate for its own deposit period.

The year 1 deposit earns the year 2 and year 3 rates, so its accumulated value is $ \$1{,}000(1.04)(1.01)=\$1{,}050.40 $. The year 2 deposit earns only the year 3 rate: $ \$1{,}000(1.01)=\$1{,}010.00 $. The year 3 deposit is made at the valuation date, so it remains $ \$1{,}000.00 $.

B_3=1{,}000(1.04)(1.01)+1{,}000(1.01)+1{,}000 =1{,}050.40+1{,}010.00+1{,}000.00 =\$3{,}060.40.

Check by recurrence. After year 1, $B_1=\$1{,}000.00$. After year 2, $B_2=1{,}000(1.04)+1{,}000=\$2{,}040.00$. After year 3, $B_3=2{,}040(1.01)+1{,}000=\$3{,}060.40$, agreeing with the deposit-by-deposit calculation.

Worked Example: Finding a Level Deposit for a Target Fund

A hypothetical sinking fund must contain $10,666.90 immediately after the fourth annual deposit. Equal deposits are made at the end of each year. The effective fund rates for years 1 through 4 are 2%, 3%, 4%, and 5%, respectively. Find the required annual deposit.

Method. Calculate the maturity accumulation factor for one dollar deposited at the end of each year, then divide the target by the sum of those factors. The deposit at the end of year 1 earns years 2, 3, and 4 rates, not the year 1 rate.

The four accumulation factors are as follows:

End of year 1 deposit: $(1.03)(1.04)(1.05)=1.12476$.
End of year 2 deposit: $(1.04)(1.05)=1.092$.
End of year 3 deposit: $1.05$.
End of year 4 deposit: $1$.

Their sum is $1.12476+1.092+1.05+1=4.26676$. If the level deposit is $D$, then $4.26676D=\$10{,}666.90$, giving

D=\frac{\$10{,}666.90}{4.26676}=\$2{,}500.00.

Check. The four deposits accumulate to $2{,}500(1.12476)+2{,}500(1.092)+2{,}500(1.05)+2{,}500 =\$2{,}811.90+\$2{,}730.00+\$2{,}625.00+\$2{,}500.00 =\$10{,}666.90$. The target is met exactly.

Worked Example: Why the Order of Fund Rates Matters

A hypothetical fund receives $500 at the end of each of three years. Compare its year 3 balance under two yield sequences: (A) 2%, 6%, 4%; and (B) 6%, 2%, 4%. All rates are effective annual rates.

Method. Apply the actual chronological rate sequence to the recurrence. The common mistake is to treat the same set of annual rates as interchangeable or replace them with an average rate.

Sequence A. The first deposit earns 6% and then 4%, the second earns 4%, and the third earns no interest before valuation: $500(1.06)(1.04)=\$551.20$, $500(1.04)=\$520.00$, and $\$500.00$. The ending balance is $\$551.20+\$520.00+\$500.00=\$1{,}571.20$.

Sequence B. The first deposit earns 2% and then 4%, the second earns 4%, and the third earns no interest: $500(1.02)(1.04)=\$530.40$, $500(1.04)=\$520.00$, and $\$500.00$. The ending balance is $\$530.40+\$520.00+\$500.00=\$1{,}550.40$.

Check. For sequence A, the recurrence gives $B_1=500$, $B_2=500(1.06)+500=1{,}030$, and $B_3=1{,}030(1.04)+500=\$1{,}571.20$. For sequence B, it gives $B_1=500$, $B_2=500(1.02)+500=1{,}010$, and $B_3=1{,}010(1.04)+500=\$1{,}550.40$. Sequence A finishes with $\$20.80$ more. Although the same rates occur in both cases, deposits are exposed to them for different periods, so the chronological order can change the ending fund.

Common Traps and Checks

Changing yields do not require a new principle for each period; they require careful matching of cash-flow dates and growth factors. A rate $j_k$ applies to the balance held during period $k$. If an interest rate is quoted on a different basis from the fund period, first express it as the effective rate for the period being modeled. A nominal annual rate is not automatically the effective annual rate.

A tempting shortcut is to average the rates and then use that average for every deposit. In general, this does not reproduce the correct maturity value: each deposit starts earning returns on a different date, and the period-specific growth factors compound in sequence. Another frequent error is to let a deposit made at the end of period $k$ earn $j_k$ as well as the later rates. The recurrence makes the timing explicit: first grow $B_{k-1}$ for period $k$, then add $D_k$.

Question to check	Correct interpretation
Which balance earns the period $k$ rate?	The balance already in the fund at the start of period $k$.
Does the end-of-period $k$ deposit earn the period $k$ rate?	No. It is added after that period’s growth.
Which rates does deposit $D_k$ earn by time $n$?	The rates $j_{k+1},\ldots,j_n$, each applied in its own period.
Can the fund rate be replaced with the loan rate?	No. Use the fund’s actual yield sequence for fund accumulation.

Key takeaway: For a changing-yield sinking fund, accumulate each deposit through the growth factors for the periods after it is made. The recurrence $B_k=B_{k-1}(1+j_k)+D_k$ is the reliable timing check, and for equal end-of-period deposits the required level amount is the target fund divided by the sum of the deposits’ accumulation factors.

Check Your Understanding

For each problem, assume the stated rates are effective annual fund rates and deposits are made at the end of each year. Round dollar answers to the nearest cent.

A fund receives $400 at the end of each of two years. The year 1 and year 2 yields are 5% and 2%. Find the balance immediately after the second deposit. (Answer: $808.00)
A fund receives $600 at the end of each of three years. The yields are 1%, 3%, and 4%. Find the balance after the third deposit. (Answer: $1,866.72)
A three-year fund must reach $3,060.40 with equal end-of-year deposits. The yields are 2%, 4%, and 1%. Find the deposit. (Answer: $1,000.00)
A fund receives $1,000 at the end of each year for three years. Its yields are 4%, 2%, and 3%. Find the balance after the third deposit. (Answer: $3,080.60)
A four-year fund must reach $5,000 with equal end-of-year deposits. The yields are 0%, 2%, 0%, and 3%. Find the required deposit. (Answer: $1,216.37)

Question to check	Correct interpretation
Which balance earns the period \(k\) rate?	The balance already in the fund at the start of period \(k\).
Does the end-of-period \(k\) deposit earn the period \(k\) rate?	No. It is added after that period’s growth.
Which rates does deposit \(D_k\) earn by time \(n\)?	The rates \(j_{k+1},\ldots,j_n\), each applied in its own period.
Can the fund rate be replaced with the loan rate?	No. Use the fund’s actual yield sequence for fund accumulation.

What You'll Learn