Investment Yield and Asset Duration

Why Yield and Investment Horizon Matter

Investment income depends on two related assumptions: the return earned on invested funds and the length of time those funds remain invested. A higher assumed yield increases the income earned over a given period. A longer investment horizon generally also increases income, because funds earn returns for more time and, under compounding, earlier returns can earn returns of their own.

For a rate indication or projection, an investment assumption is meaningful only when its timing basis is clear. The annual yield might be an effective annual rate, and the horizon might be one year or several years. Applying an annual rate to a multi-year horizon as though it were a one-time percentage can materially understate projected income.

Definition: In this tutorial, investment yield is the effective annual rate $i$ earned on invested funds, with annual compounding. The investment horizon $t$ is the time, in years, that a specified amount remains invested. For an asset balance $A$ held for $t$ years at a constant yield, projected investment income is the accumulated value less the original balance.

The formulas below assume that the yield is constant over the stated horizon, the investment earns the full assumed yield, and there are no taxes, fees, defaults, withdrawals, or intermediate contributions. These assumptions make the relationship between yield, time, and income transparent. If rates vary by year, or funds enter or leave during the period, calculate income period by period or cash flow by cash flow instead.

Compound Income for One Investment Horizon

Under annual compounding, an initial balance $A$ grows to $A(1+i)^t$ after $t$ years. Therefore, the investment income is:

I(A,i,t)=A(1+i)^t-A=A\left[(1+i)^t-1\right].

When $t$ is a whole number, the accumulation can also be calculated one year at a time. The formula with a fractional $t$ assumes an effective annual yield and uses the corresponding fractional power. A different convention for partial years, such as simple interest within the year, must be stated and applied consistently.

Derivation: By definition of an effective annual yield, each year multiplies the invested balance by $1+i$. After one year the balance is $A(1+i)$; after two years it is $A(1+i)(1+i)=A(1+i)^2$. Repeating this multiplication $t$ times gives $A(1+i)^t$. Investment income excludes the principal originally invested, so subtracting $A$ gives $A[(1+i)^t-1]$.

For a short horizon or a rough approximation, the income may be estimated as $Ait$, which is simple interest. The approximation omits the effect of earning returns on accumulated interest. It is not the exact compound-income formula, and its accuracy generally decreases as the yield or horizon increases.

Worked Example: Comparing Yield and Horizon for One Asset

A hypothetical insurer expects to have $100,000 available to invest. Assume the entire balance earns a constant effective annual yield, compounded annually, with no withdrawals or expenses. Calculate investment income under three assumptions: 4% for two years, 6% for two years, and 4% for three years. Report dollar amounts to the nearest cent.

Method. Apply $I=A[(1+i)^t-1]$ to each yield-horizon pair. This isolates the effect of changing the yield or the horizon while holding the other assumption fixed.

4% for two years. After year one, the balance is $100{,}000(1.04)=104{,}000$, so year-one income is $4,000. After year two, the balance is $104{,}000(1.04)=108{,}160$, so year-two income is $4,160. Total income is $4{,}000+4{,}160=\$8{,}160.00$.

6% for two years. After year one, the balance is $100{,}000(1.06)=106{,}000$, and year-one income is $6,000. After year two, the balance is $106{,}000(1.06)=112{,}360$, and year-two income is $6,360. Total income is $6{,}000+6{,}360=\$12{,}360.00$.

4% for three years. The first two years produce the $8,160 of income already calculated. In year three, the $108,160 balance earns $108{,}160(0.04)=\$4{,}326.40$. Total income is $8{,}160+4{,}326.40=\$12{,}486.40$.

Check. At the same two-year horizon, increasing the yield from 4% to 6% raises income by $4,200. At a 4% yield, adding a third year raises income by $4,326.40. The third-year interest is greater than $4,000 because it is earned on the accumulated balance, not just the original principal.

Common mistake avoided: Multiplying $100,000 by 4% and then by two gives $8,000, not the exact compound income. That calculation applies simple interest to the original principal only. The correct compound result is $8,160 because the $4,000 earned in year one also earns 4% in year two.

Multiple Assets and an Average Investment Horizon

An insurer may invest different amounts for different lengths of time. If each amount has the same yield $i$, but amount $A_j$ is held for $t_j$ years, calculate each amount's income using its own horizon and then add the results. The total initial invested balance is $\sum_j A_j$.

I_{\text{total}}=\sum_j A_j\left[(1+i)^{t_j}-1\right].

An amount-weighted average investment horizon, used here as a simple duration summary, is:

\bar{t}=\frac{\sum_j A_jt_j}{\sum_j A_j}.

This summary gives more weight to the holding periods associated with larger invested amounts. It can be useful for comparing portfolios or making a first-order simple-interest estimate. However, the average horizon alone does not generally determine exact compound income. In general, $\sum_j A_j[(1+i)^{t_j}-1]$ is not equal to $(\sum_j A_j)[(1+i)^{\bar{t}}-1]$. The exact calculation depends on the separate horizons.

Worked Example: Different Asset Amounts and Holding Periods

A hypothetical portfolio has $60,000 invested for one year and $40,000 invested for three years. Both amounts earn a constant 5% effective annual yield, compounded annually, with no other cash flows. Calculate the amount-weighted average horizon, exact total income, and the simple-interest estimate based on that average horizon. Report amounts to the nearest cent.

Average horizon. The total invested balance is $60{,}000+40{,}000=\$100{,}000$. The amount-weighted average horizon is $(60{,}000(1)+40{,}000(3))/100{,}000=(60{,}000+120{,}000)/100{,}000=1.8$ years.

Exact income from the one-year amount. The $60,000 grows to $60{,}000(1.05)=\$63{,}000$, so its income is $3,000.

Exact income from the three-year amount. The yearly balances are $40,000, then $40{,}000(1.05)=\$42{,}000$, then $42{,}000(1.05)=\$44{,}100$, then $44{,}100(1.05)=\$46{,}305$. Its total income is $46{,}305-40{,}000=\$6{,}305$. Equivalently, $(1.05)^3=1.157625$, so the income is $40{,}000(0.157625)=\$6{,}305$.

Total exact income. Add the two incomes: $3{,}000+6{,}305=\$9{,}305.00$.

Simple-interest estimate using the average horizon. The estimate is $100{,}000(0.05)(1.8)=\$9{,}000.00$. It is $305.00 below exact compound income.

Check and interpretation. The exact total is also the sum of the two independently accumulated asset amounts, less the $100,000 initially invested: $63{,}000+46{,}305-100{,}000=\$9{,}305$. The 1.8-year average horizon is a useful summary, but it does not replace the two separate accumulation calculations.

Common mistake avoided: Compounding the full $100,000 for 1.8 years treats every dollar as if it had the same horizon. That is not the portfolio described. The correct exact method respects the one-year and three-year holding periods separately.

How to Compare Assumptions

When both yield and horizon assumptions change, calculate the income under each complete set of assumptions rather than combining percentages informally. A sequential comparison can help explain the result: first change the yield while holding the horizon fixed, then change the horizon while holding the new yield fixed. The allocation of the total change between those steps depends on the order, because yield and time interact through $(1+i)^t$.

Fix the invested amount and timing convention.
Identify which funds are invested, when each amount enters, and how the yield applies over the period.

Calculate the baseline.
Use the current yield and horizon assumptions to determine baseline income.

Change one assumption at a time for explanation.
Hold yield fixed when measuring a horizon change, or hold horizon fixed when measuring a yield change.

Calculate the final scenario directly.
Apply the new yield and horizon together, then compare final income with baseline income.

Worked Example: Separating Yield and Duration Effects

A hypothetical $200,000 asset balance is invested as a single amount. The baseline assumption is a 4% effective annual yield for two years. An alternative uses a 5% yield and a three-year horizon. Assume annual compounding, no cash flows during the period, and no taxes or expenses. Calculate income at the baseline, after changing only the yield, and under the full alternative. Report amounts to the nearest cent.

Baseline: 4% for two years. Year-one income is $200{,}000(0.04)=\$8{,}000$, leaving a balance of $208,000. Year-two income is $208{,}000(0.04)=\$8{,}320$. Baseline total income is $8{,}000+8{,}320=\$16{,}320.00$.

Change yield only: 5% for two years. Year-one income is $200{,}000(0.05)=\$10{,}000$, leaving $210,000. Year-two income is $210{,}000(0.05)=\$10{,}500$. Total income is $10{,}000+10{,}500=\$20{,}500.00$, an increase of $20{,}500-16{,}320=\$4{,}180.00$ from baseline.

Full alternative: 5% for three years. The third-year income is $220{,}500(0.05)=\$11{,}025$, since the balance after two years is $220,500. Total income is $20{,}500+11{,}025=\$31{,}525.00$, an increase of $31{,}525-16{,}320=\$15{,}205.00$ from baseline.

Check and interpretation. The final accumulation is $200{,}000(1.05)^3=200{,}000(1.157625)=\$231{,}525$; subtracting the initial $200,000 gives $31,525 of income. In the sequence used here, $4,180 of the total increase arises from changing the yield at a two-year horizon, and $11,025 from extending the horizon at 5%. This decomposition is path-specific; changing the horizon first would assign the interaction differently.

Common mistake avoided: Adding a simple 5% times three-year amount to the principal gives income of $200{,}000(0.05)(3)=\$30{,}000$. The exact income is $31,525 because the annual return compounds. A second error would be comparing only the yields while ignoring that the alternative also extends the investment period.

Assumptions, Interpretation, and Common Traps

The calculations describe projected investment income under stated assumptions; they do not establish that an insurer will actually earn the assumed yield. Actual returns may vary, and assets may mature, be sold, or be reinvested at different yields. When the yield changes across years, apply each year's factor to the balance at the start of that year. For example, with yields $i_1$ and $i_2$ in successive years, the two-year accumulation is $A(1+i_1)(1+i_2)$, not $A(1+i_2)^2$.

The average horizon defined here is amount-weighted and intended to summarize how long invested amounts remain invested. It is not automatically the same as every technical duration measure used in asset-liability management. In particular, measures based on the timing and present values of future asset cash flows address cash-flow structure and sensitivity to changes in market yields. Do not infer that a single average holding period fully describes those features.

Do not mix yield conventions. An effective annual yield with annual compounding uses $(1+i)^t$. A simple-interest estimate uses $1+it$; these are different accumulation assumptions.
Match the horizon to each amount. If balances enter or leave at different times, calculate income for the corresponding invested periods rather than assigning every dollar the same duration.
Distinguish income from accumulated value. The accumulated value includes principal; investment income is accumulated value minus principal.
State the basis of the yield. A projected yield must be consistent with the stated period and with any assumed fees or other reductions. The formulas here do not include those reductions.

Key takeaway: For a constant effective annual yield and a single amount held for $t$ years, compound investment income is $A[(1+i)^t-1]$. For amounts with different horizons, calculate each amount's income separately and sum it. An average duration is a helpful summary, not a substitute for the exact cash-flow calculation.

Check Your Understanding

For each problem, use the stated annual compounding convention and round dollar answers to the nearest cent.

A hypothetical $80,000 balance earns a constant 3.5% effective annual yield for two years. Calculate compound investment income and compare it with the simple-interest estimate. (Answer: Compound income = $5,698.00; simple-interest estimate = $5,600.00.)
A hypothetical portfolio has $50,000 invested for one year and $50,000 for three years, both at a 4% effective annual yield. Calculate exact total compound income and the amount-weighted average horizon. (Answer: Exact income = $8,243.20; average horizon = 2 years.)
A hypothetical portfolio has $120,000 invested for two years and $80,000 for one year at a 6% effective annual yield. Find the amount-weighted average horizon, exact total income, and simple-interest estimate based on that average. (Answer: Average horizon = 1.6 years; exact income = $19,632.00; estimate = $19,200.00.)
A hypothetical $300,000 asset balance is invested for three years at 4%. An alternative assumes a 5% yield for two years. Calculate income under both assumptions and the difference between them. (Answer: At 4% for three years, $37,459.20; at 5% for two years, $30,750.00; difference = $6,709.20.)
A hypothetical portfolio has $100,000 invested for two years and $100,000 for four years, both at a 3% effective annual yield. Calculate exact total compound income, the amount-weighted average horizon, and simple interest using that average horizon. (Answer: Exact income = $18,640.88; average horizon = 3 years; simple-interest estimate = $18,000.00.)

What You'll Learn