Reinvestment Rates and Realized Yield
A quoted yield secretly assumes every coupon gets reinvested at that same rate. Redeploy the cash at a different reinvestment rate and the yield you actually earn shifts too — you have to recompute it from the true accumulated value, not just read it off the price.
By the end you'll be able to accumulate a reinvested payment stream at its own rate, add back any returned principal, and solve the resulting equation of value for the realized yield.
Predict: this bond's stated yield is 10%, but that number only holds if coupons reinvest at 10% too. If coupons can only be reinvested BELOW 10%, will the realized yield end up above or below 10%? Drag the reinvestment-rate slider to check.
Buy this bond for X = 1000. It pays 100 at the end of years 1 and 2, then 1100 (last coupon + principal) at the end of year 3. The two early coupons get reinvested at rate j until year 3; the final payment arrives right at year 3 and needs no reinvestment.
AV = 100(1.04)² + 100(1.04) + 1100 = 1312.16 · realized yield i′ = 9.48% (below the 10% stated yield)
When payments or coupons are reinvested at a rate different from the valuation rate, find the overall accumulated value by accumulating the reinvested income separately, then solve the equation of value for the realized yield.
The reliable procedure has two stages. First, accumulate the reinvested income at rate j: for level payments of P at the end of each of n periods, the side account they build grows to \(P\cdot s_{\overline{n}|j} = P\cdot\dfrac{(1+j)^n-1}{j}\). Add any return of principal that is not reinvested — like a bond's redemption value — at its own maturity date to get the total accumulated value AV. Second, solve the equation of value for the realized yield i′: with price X paid at t = 0, \(X(1+i')^n = AV \Rightarrow i' = (AV/X)^{1/n} - 1\). The coupons never earn the pricing rate i once received — they earn the market reinvestment rate j, and only this two-step calculation recovers the true yield.
This is exactly the risk a bondholder faces: the coupon rate printed on the bond and the yield quoted at purchase both assume coupons get redeployed at that same rate. If market rates drop after you buy, every coupon you receive can only be reinvested at the new, lower rate — so even though nothing about the bond itself changed, the yield you actually walk away with at maturity is lower than what was quoted on day one.
Buy a 3-year investment for X = 1000 paying 100 at the end of years 1 and 2, then 1100 at year 3. Coupons reinvest at j = 4% (below the 10% coupon rate). Accumulated coupons: \(100\cdot s_{\overline{3}|4\%} = 100\cdot\dfrac{(1.04)^3-1}{0.04} = 100(3.1216) = 312.16\). Add the 1000 principal returned at t = 3: \(AV = 312.16 + 1000 = 1312.16\). Realized yield: \(i' = (1312.16/1000)^{1/3} - 1 \approx\) 9.48%.
Same idea, different numbers: buy a 3-year investment for X = 1000 that pays 60 at the end of years 1 and 2, then 1060 at year 3, with coupons reinvested at j = 3%. Accumulated coupons: \(60\cdot s_{\overline{3}|3\%} = 60\cdot\dfrac{(1.03)^3-1}{0.03} = 60(3.0909) = 185.45\). Add the 1000 principal returned at t = 3: \(AV = 185.45 + 1000 = 1185.45\). Now finish it: \(i' = (1185.45/1000)^{1/3} - 1 = \) ____
Reveal the answer
\(i' = (1.18545)^{1/3} - 1 \approx 1.0583 - 1 = \) 5.83% — well below the bond's original 6% coupon rate, because the reinvestment rate (3%) is below it. Set the slider above to j = 3% on the demo's own numbers and compare the shape of the drop.
More info — why the realized yield isn't just an average of i and j
It's tempting to guess that mixing a 10% coupon rate with a 4% reinvestment rate should land you somewhere around 7%. It doesn't, because most of the money at stake — the 1000 principal — is returned at maturity untouched by the reinvestment rate at all; only the relatively small coupon payments are exposed to j. So the realized yield sits much closer to the stated yield than a naive average would suggest (9.48% in the worked example, not 7%). This is exactly why the Internal rate of return / NPV approach can mislead when comparing two projects that pay out cash at different times — see Dive deeper below for more on reinvestment risk.
Check your understanding
A 1000 investment pays 50 at the end of each of years 1 and 2, then 1050 at the end of year 3. Coupons reinvest at j = 6%. What is the 3-year realized yield i′?
A bond's stated (quoted) yield implicitly assumes every coupon reinvests at that same rate. If the actual reinvestment rate available in the market is BELOW the stated yield, what happens to the yield you actually realize over the holding period?
Now flip it: if coupons could instead be reinvested at a rate ABOVE the bond's stated yield, would the realized yield end up above or below the stated yield?
You've already learned that a level annuity-immediate's accumulated value is \(s_{\overline{n}|} = \dfrac{(1+i)^n - 1}{i}\). A different 3-year bond pays 100 at the end of each of 3 years (its principal is folded into that last 100 for this step), with coupons reinvested at j = 5%. What is \(s_{\overline{3}|5\%}\), the factor you'd multiply by 100 to get the reinvested stream's accumulated value?
You've also learned that the internal rate of return (IRR) of a cash-flow stream is the rate that sets its net present value to zero. Once you've accumulated the reinvested income into AV at time n, how does solving \(X(1+i')^n = AV\) for i′ relate to that IRR idea?
Recap
- A quoted yield assumes coupons reinvest at that same rate; reinvesting at a different rate j changes the yield you actually realize.
- Accumulate the reinvested stream separately at j — \(P\cdot s_{\overline{n}|j} = P\cdot\dfrac{(1+j)^n-1}{j}\) — then add back any non-reinvested principal to get AV.
- Solve the equation of value \(X(1+i')^n = AV\) for the realized yield: \(i' = (AV/X)^{1/n} - 1\).
- Reinvesting below the stated yield pulls the realized yield below it; reinvesting above pushes the realized yield above it — it is never simply the average of the two rates.
Dive deeper
- Reinvestment risk — Wikipedia Explain why coupons reinvested at a different rate change the realized return.
- Marcel Finan — Exam FM study guide (PDF) Work reinvestment problems that accumulate income at a separate rate and solve for realized yield.
Sources
- Reinvestment Rates and Realized Yield