Deriving Realized Yield From Terminal Value

From Terminal Value to Realized Yield

An investment can produce cash receipts before its final date. If those receipts are reinvested, the amount available at the horizon depends both on the original investment and on how each receipt grows after it arrives. Once that terminal value is known, we can ask: what single annual effective rate would turn the initial investment into exactly that terminal value over the full term? That equivalent rate is the realized yield considered here.

The calculation has two distinct stages. First, accumulate each interim receipt to the end of the investment using the reinvestment rates that apply during the remaining periods. Second, compare the resulting terminal value with the initial outlay and solve for the annual effective rate. The realized yield is therefore determined by the terminal value; it need not equal any of the reinvestment rates.

Definition: Suppose an initial investment of $P>0$ is made at time 0 and held for $n$ annual periods. Let $W_n>0$ be its terminal value at time $n$, including the accumulated value of interim cash receipts and any final payment. The annual effective realized yield, denoted $j_R$, is the rate satisfying $$ P(1+j_R)^n=W_n. $$ When $W_n>0$, the solution is $$ j_R=\left(\frac{W_n}{P}\right)^{1/n}-1. $$

Building the Terminal Value

Assume cash receipts occur at the ends of annual periods. Let $C_k$ be a positive receipt at time $k$, for $k=1,\ldots,n$. Let $r_j$ be the effective rate earned during the year from time $j-1$ to time $j$. Thus, a receipt at time $k$ is reinvested only during the periods after it is received: periods $k+1$ through $n$. A receipt at time $n$ has no remaining period in which to earn interest.

If $M_n$ is a separate final amount, such as the return of the invested principal, the terminal value is

W_n=M_n+\sum_{k=1}^{n}C_k\prod_{j=k+1}^{n}(1+r_j).

An empty product, as occurs for a receipt at time $n$, is defined to equal 1. The formula assumes that each reinvestment rate applies to the stated annual period, interest is compounded annually, receipts are reinvested in full, and there are no interim withdrawals or additional contributions. If a cash flow is already part of the final amount $M_n$, do not count it again in the sum.

Set the horizon and identify each cash flow.
Record the initial outlay, the time of each interim receipt, and any final payment. Keep the initial outlay separate from receipts.

Accumulate receipts to the horizon.
For each receipt, multiply by the factors for the periods after that receipt arrives. Add those accumulated amounts and the final payment to obtain $W_n$.

Solve for the equivalent annual rate.
Use $P(1+j_R)^n=W_n$, so $j_R=(W_n/P)^{1/n}-1$.

Check the result.
Substitute the calculated rate into $P(1+j_R)^n$ and verify that it reproduces the terminal value, subject to rounding.

Why the Yield Formula Holds

The definition treats the entire investment as though its initial outlay had grown at one constant annual effective rate for $n$ years. At an annual effective rate $j_R$, one dollar at time 0 becomes $1+j_R$ dollars after one year, $(1+j_R)^2$ after two years, and $(1+j_R)^n$ after $n$ years. Therefore the accumulated value of $P$ is $P(1+j_R)^n$. Requiring this equivalent accumulated value to equal the actual terminal value gives the defining equation. Since $P>0$ and $W_n>0$, taking the positive $n$th root and isolating $j_R$ gives

P(1+j_R)^n=W_n \quad\Longrightarrow\quad 1+j_R=\left(\frac{W_n}{P}\right)^{1/n} \quad\Longrightarrow\quad j_R=\left(\frac{W_n}{P}\right)^{1/n}-1.

This is an equivalence based on the total terminal value. It does not claim that the original investment literally earned $j_R$ in every period, nor that each receipt was reinvested at $j_R$.

Worked Example: Different Rates for Successive Receipts

A hypothetical investment costs $1,000 and lasts three years. It returns $1,000 of principal at the end of year 3. It also pays $100 at the end of year 1 and $200 at the end of year 2. The year-2 and year-3 reinvestment rates for the year-1 receipt are both 10%; the year-3 reinvestment rate for the year-2 receipt is 5%. Find the annual effective realized yield.

Method. Accumulate each interim receipt separately to year 3, add the principal repayment, and then solve the terminal-value equation.

The $100 received at time 1 earns interest in years 2 and 3:

$$ 100(1.10)(1.10)=100(1.21)=121. $$

The $200 received at time 2 earns interest in year 3:

$$ 200(1.05)=210. $$

Thus the terminal value is

W_3=1{,}000+121+210=1{,}331.

Equate this with the accumulation of the initial outlay over three years:

1{,}000(1+j_R)^3=1{,}331, \qquad (1+j_R)^3=1.331, \qquad 1+j_R=1.10.

Therefore $j_R=0.10$, or 10% per year. As a check, $1{,}000(1.10)^3=1{,}331$, matching the terminal value. The reinvestment rates were 10% and 5%, but the realized yield comes from the combined terminal value, not from selecting one of those rates.

Worked Example: A Receipt Reinvested at a Lower Rate

A hypothetical two-year investment requires an initial outlay of $5,000. It returns the $5,000 principal at the end of year 2 and pays a $1,000 distribution at the end of year 1. The distribution earns 5% during year 2. Find the annual effective realized yield over the two-year term.

Method. Accumulate the year-1 distribution for its one remaining year, then include the principal repayment.

The distribution’s year-2 value is

$$ 1{,}000(1.05)=1{,}050. $$

Adding the returned principal gives

W_2=5{,}000+1{,}050=6{,}050.

The realized yield satisfies

5{,}000(1+j_R)^2=6{,}050, \qquad (1+j_R)^2=\frac{6{,}050}{5{,}000}=1.21.

Taking the positive square root gives $1+j_R=1.10$, so $j_R=10\%$. The check is $5{,}000(1.10)^2=6{,}050$. The distribution was reinvested at 5%, but that rate applies only to that distribution’s remaining year; it is not the realized yield on the full investment.

Worked Example: Calculating a Noninteger Realized Yield

A hypothetical three-year investment costs $4,000 and returns $4,000 of principal at the end of year 3. It also pays $400 at the end of year 1 and $500 at the end of year 2. The year-1 receipt is reinvested at 10% in each of years 2 and 3; the year-2 receipt is reinvested at 10% in year 3. Find the realized yield, rounded to three decimal places as a percentage.

Method. Find terminal wealth first, then take the cube root of its ratio to the initial outlay.

The $400 receipt accumulates for two years:

$$ 400(1.10)^2=400(1.21)=484. $$

The $500 receipt accumulates for one year:

$$ 500(1.10)=550. $$

Including the principal repayment, terminal wealth is

W_3=4{,}000+484+550=5{,}034.

Then

j_R=\left(\frac{5{,}034}{4{,}000}\right)^{1/3}-1 =(1.2585)^{1/3}-1 \approx 0.0796535.

Rounded to three decimal places as a percentage, the realized yield is 7.965% per year. As a check, $4{,}000(1.0796535)^3\approx5{,}034.00$. The small difference from exact equality is due only to rounding the displayed rate.

Assumptions, Interpretation, and Common Traps

The formula gives one annual effective rate over the full term when time is measured in whole annual periods and terminal wealth is positive. The rates used to reinvest receipts may vary by period. Each receipt must be accumulated only over the periods after it arrives; applying a rate for a period before receipt would overstate its terminal value.

The calculation also relies on having a well-defined terminal value. If there are interim withdrawals or other negative cash flows, simply adding positive accumulated receipts is not enough; the timing and accumulation of those outflows must also be represented, and a positive realized yield may not exist. The formula here is for an initial outlay accumulated to a positive terminal value under the stated receipt-and-reinvestment model.

A frequent exam trap is to report a reinvestment rate as the realized yield without using the terminal value equation. For the two-year example above, using 5% as the yield would give

5{,}000(1.05)^2=5{,}512.50,

which does not equal the actual terminal value of $6,050. The correction is to accumulate the interim distribution separately, obtain $W_2=6{,}050$, and solve $5{,}000(1+j_R)^2=6{,}050$, giving 10%. The reinvestment rate describes how the distribution grows; the realized yield summarizes the total terminal value relative to the original outlay.

Key takeaway: Accumulate every interim receipt to the common horizon using the rates that apply after receipt. Then solve $P(1+j_R)^n=W_n$. The resulting annual effective realized yield is an equivalent rate for the whole investment term, not necessarily a rate earned by each cash flow.

Check Your Understanding

For each problem, first find the terminal value and then solve for the annual effective realized yield. Round percentage answers to three decimal places where needed.

A two-year investment has an initial outlay of $2,000 and returns $2,000 of principal at the end of year 2. A $300 receipt at the end of year 1 is reinvested at 4% during year 2. Find the realized yield. (Answer: 7.517% per year)
A hypothetical two-year investment costs $10,000, returns its principal at the end of year 2, and pays $600 at the end of year 1. The receipt earns 8% in year 2. Find the realized yield. (Answer: 3.189% per year)
A three-year investment costs $3,000 and returns $3,000 of principal at the end of year 3. It pays $200 at time 1, reinvested at 5% in year 2 and 6% in year 3; $300 at time 2, reinvested at 6% in year 3; and $100 at time 3. Find the realized yield. (Answer: 6.664% per year)
A two-year investment costs $2,500 and returns $2,500 of principal at the end of year 2. It pays $250 at the end of year 1, reinvested at 12% in year 2, and a further $220 at the end of year 2. Find the realized yield. (Answer: 9.545% per year)
A two-year investment costs $4,000 and returns $4,000 of principal at the end of year 2. It pays one receipt $C$ at the end of year 1, which is reinvested at 5% in year 2. What value of $C$ gives an 8% annual effective realized yield? (Answer: $633.90)

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