Bonds, Duration, Convexity, and
Immunization
Julian Mauricio Fajardo
Introduction
A bond converts a contractual sequence of future payments into a
value today. For a conventional fixed-coupon bond, the main inputs are
the coupon payments, the redemption value, the payment dates, and the
yield used to discount the cash flows.
The coupon rate and the yield play different roles. The coupon rate
is contractual and determines the periodic payment. The yield is the
rate used to value those payments. A change in market yield does not
modify the contractual coupon; it modifies the present value of the
remaining cash flows and therefore the bond price.
This vignette starts from the usual actuarial equations and then uses
tidyactuarial to reproduce and audit the calculations. The
mathematical structure is established first; the software is used
afterward as a computational tool.
Bond cash flows and the value equation
Consider a conventional bond with face value \(F\), coupon rate \(r\) per coupon period, redemption value
\(C\), effective yield \(i\) per coupon period, and \(n\) remaining coupon periods. The periodic
coupon is \(Fr\).
Taking the valuation date as time \(0\), the future cash flows are
\[
Fr,\; Fr,\; \ldots,\; Fr+C.
\]
Therefore,
\[
P
=
Fr(v+v^2+\cdots+v^n)+Cv^n,
\]
where \(v=(1+i)^{-1}\). Since the
coupon payments form an annuity-immediate,
\[
P
=
Fr\,a_{\overline{n}|i}
+
Cv^n.
\]
Equivalently,
\[
P
=
Fr\frac{1-(1+i)^{-n}}{i}
+
C(1+i)^{-n}.
\]
This is the ordinary equation of value applied to a level stream of
coupons plus a single redemption payment. The coupon rate, the yield,
and the number of periods must refer to the same time unit.
Valuing a coupon bond
Consider a bond with
\[
F=C=1{,}000{,}000,
\]
a nominal annual coupon rate of \(9\%\) payable semiannually, and six years
to maturity. Suppose the investor requires a nominal annual yield of
\(10\%\) convertible semiannually.
The coupon rate per half-year is
\[
r=\frac{0.09}{2}=0.045,
\]
so the semiannual coupon is \(45{,}000\). The effective yield per
half-year is \(i=0.05\), and there are
\(12\) remaining coupon periods.
Hence,
\[
P
=
45{,}000a_{\overline{12}|0.05}
+
1{,}000{,}000(1.05)^{-12}
=
955{,}683.74.
\]
Only after constructing the equation of value do we move to the
computational representation.
face_value <- 1000000
redemption_value <- 1000000
annual_coupon_rate <- 0.09
years_to_maturity <- 6
coupons_per_year <- 2
bond_flows <-
bond_cash_flows(
face = face_value,
c = annual_coupon_rate,
n = years_to_maturity,
k = coupons_per_year,
R = redemption_value
)
bond_flows
#> # A tibble: 13 x 5
#> cashflow_id period t cf type
#> <int> <int> <dbl> <dbl> <chr>
#> 1 1 1 0.5 45000 coupon
#> 2 2 2 1 45000 coupon
#> 3 3 3 1.5 45000 coupon
#> 4 4 4 2 45000 coupon
#> 5 5 5 2.5 45000 coupon
#> 6 6 6 3 45000 coupon
#> 7 7 7 3.5 45000 coupon
#> 8 8 8 4 45000 coupon
#> 9 9 9 4.5 45000 coupon
#> 10 10 10 5 45000 coupon
#> 11 11 11 5.5 45000 coupon
#> 12 12 12 6 45000 coupon
#> 13 13 12 6 1000000 redemption
bond_cash_flows() represents the contractual payments.
It does not determine the bond price. The valuation basis is introduced
separately.
semiannual_yield <- 0.05
bond_value <-
bond_price(
face = face_value,
c = annual_coupon_rate,
n = years_to_maturity,
k = coupons_per_year,
y_effective_per_period = semiannual_yield,
R = redemption_value
)
bond_value
#> [1] 955683.7
The package result reproduces the analytical value. The comparison
verifies that the computational implementation follows the actuarial
equation established independently.
Yield to maturity as an inverse problem
Suppose the market price of the same bond is now
\[
P=935{,}000.
\]
The YTM is the rate \(i\)
satisfying
\[
935{,}000
=
45{,}000\frac{1-(1+i)^{-12}}{i}
+
1{,}000{,}000(1+i)^{-12}.
\]
For this bond,
\[
f(0.05)>0
\qquad\text{and}\qquad
f(0.06)<0,
\]
so the root lies in
\[
5\%<i<6\%.
\]
Since the price of a conventional bond with positive future cash
flows is strictly decreasing in the yield, the solution is unique in
this interval. Numerically,
\[
i\approx5.2435\%
\]
per half-year.
market_price <- 935000
ytm_result <-
bond_ytm(
P = market_price,
face = face_value,
c = annual_coupon_rate,
n = years_to_maturity,
k = coupons_per_year,
R = redemption_value,
interval = c(0.05, 0.06)
)
semiannual_ytm <-
ytm_result |>
dplyr::pull(i_period) |>
dplyr::first()
ytm_result
#> # A tibble: 1 x 5
#> price i_period j_nominal i_effective_annual k
#> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 935000 0.0524 0.105 0.108 2
The interval does not come from the software. It is identified from
the equation of value before the numerical solver is used.
Book value
Let \(BV_t\) denote the book value
at coupon date \(t\).
Prospectively,
\[
BV_t
=
Fr\,a_{\overline{n-t}|i}
+
Cv^{n-t}.
\]
Retrospectively,
\[
BV_t
=
P(1+i)^t
-
Fr\,s_{\overline{t}|i}.
\]
These are two representations of the same financial equation.
For the bond purchased at \(935{,}000\), after three years, or six
semiannual coupon periods,
\[
BV_6
\approx
962{,}555.82.
\]
valuation_time <- 3
book_value <-
bond_book_value(
face = face_value,
c = annual_coupon_rate,
n = years_to_maturity,
t = valuation_time,
k = coupons_per_year,
y_effective_per_period = semiannual_ytm,
R = redemption_value
)
book_value
#> [1] 962555.8
The recurrence
\[
BV_t=BV_{t-1}(1+i)-Fr
\]
shows how premiums and discounts are recognized through time. Book
value should not be confused with market value: the former follows the
yield basis adopted for amortization, whereas the latter uses the yield
currently required by the market.
Callable bonds
A callable bond gives the issuer the right to redeem the bond before
maturity under specified contractual conditions. The investor does not
control the final redemption date.
Suppose scenario \(s\) redeems the
bond after \(n_s\) coupon periods for
\(C_s\). At target yield \(i\),
\[
P_s(i)
=
Fr\,a_{\overline{n_s}|i}
+
C_sv^{n_s}.
\]
To guarantee at least the target yield under every admissible
scenario, the purchase price must satisfy \(P\le P_s(i)\) for every \(s\). Hence,
\[
P_{\max}(i)
=
\min_s P_s(i).
\]
The minimum follows from the contractual structure: the investor must
be protected against the least favorable redemption scenario the issuer
is allowed to impose.
Consider a bond with face value \(1{,}000{,}000\), a nominal annual coupon
rate of \(10\%\) payable semiannually,
and final maturity in eight years. It may be redeemed at year 4 for
\(1{,}040{,}000\), at year 6 for \(1{,}020{,}000\), or at year 8 for \(1{,}000{,}000\).
At a target semiannual yield of \(4\%\), the scenario prices are
approximately \(1{,}096{,}555.06\),
\(1{,}106{,}342.68\), and \(1{,}116{,}522.96\). Therefore,
\[
P_{\max}=1{,}096{,}555.06.
\]
call_dates <- c(4, 6)
call_values <- c(1040000, 1020000)
target_yield <- 0.04
callable_price <-
bond_callable_price(
face = 1000000,
c = 0.10,
n = 8,
k = 2,
call_t = call_dates,
call_R = call_values,
y_effective_per_period = target_yield,
R = 1000000
)
callable_price
#> [1] 1096555
The same contractual logic applies when price is known and yields are
compared: the yield to worst is the minimum among the relevant
yields.
From price to sensitivity
For a conventional bond with positive future cash flows,
\[
P(i)
=
\sum_{t=1}^{n}CF_t(1+i)^{-t}.
\]
Differentiating,
\[
\frac{dP}{di}
=
-\sum_{t=1}^{n}tCF_t(1+i)^{-(t+1)}<0,
\]
and
\[
\frac{d^2P}{di^2}
=
\sum_{t=1}^{n}t(t+1)CF_t(1+i)^{-(t+2)}>0.
\]
The price-yield relationship is therefore decreasing and convex.
These two derivatives lead naturally to duration and convexity.
Duration and convexity of a cash-flow stream
For payments \(C_j\) at times \(t_j\),
\[
P(i)
=
\sum_j C_j(1+i)^{-t_j}.
\]
Define present-value weights
\[
w_j(i)
=
\frac{C_j(1+i)^{-t_j}}{P(i)}.
\]
For positive cash flows, Macaulay duration is
\[
D_{\text{Mac}}(i)
=
\sum_j t_jw_j(i),
\]
while modified duration is
\[
D_{\text{mod}}(i)
=
-\frac{1}{P(i)}\frac{dP(i)}{di}
=
\frac{D_{\text{Mac}}(i)}{1+i}.
\]
Thus,
\[
\frac{\Delta P}{P}
\approx
-D_{\text{mod}}\Delta i.
\]
Relative convexity is
\[
C(i)
=
\frac{1}{P(i)}\frac{d^2P(i)}{di^2},
\]
so the second-order approximation is
\[
\frac{\Delta P}{P}
\approx
-D_{\text{mod}}\Delta i
+
\frac12 C(\Delta i)^2.
\]
Duration gives a first-order approximation; convexity adds the effect
of curvature.
A general cash-flow example
Consider payments of 120, 160, 220, and 300 million at years 1, 2, 4,
and 6, valued at an annual effective rate of \(7\%\). The analytical results are
approximately
\[
P_0=619.64,
\qquad
D_{\text{Mac}}=3.6512,
\qquad
D_{\text{mod}}=3.4123,
\qquad
C=18.0646.
\]
cash_flows <- c(120, 160, 220, 300)
payment_times <- c(1, 2, 4, 6)
valuation_rate <- 0.07
macaulay_duration <-
duration_cash_flow(
cf = cash_flows,
t = payment_times,
rate = valuation_rate,
duration = "macaulay",
output = "value"
)
modified_duration <-
duration_cash_flow(
cf = cash_flows,
t = payment_times,
rate = valuation_rate,
duration = "modified",
output = "value"
)
cash_flow_convexity <-
convexity_cash_flow(
cf = cash_flows,
t = payment_times,
rate = valuation_rate,
output = "value"
)
tibble::tibble(
measure = c("Macaulay duration", "Modified duration", "Convexity"),
value = c(macaulay_duration, modified_duration, cash_flow_convexity)
)
#> # A tibble: 3 x 2
#> measure value
#> <chr> <dbl>
#> 1 Macaulay duration 3.65
#> 2 Modified duration 3.41
#> 3 Convexity 18.1
The functions reproduce quantities already defined from the value
function and its derivatives.
Specialization to bonds
For a bond with face value \(F\),
annual coupon rate \(c\), \(k\) coupon payments per year, maturity
\(n\) years, and redemption value \(R\),
\[
C=\frac{Fc}{k},
\qquad
N=nk.
\]
Let \(j\) be the effective yield per
coupon period and \(y\) the annual
effective yield. Then
\[
1+y=(1+j)^k,
\qquad
j=(1+y)^{1/k}-1.
\]
If \(D_p\) denotes Macaulay duration
in coupon periods, duration in years is
\[
D=\frac{D_p}{k}.
\]
The rate variable used in the sensitivity measure must match the rate
change being studied. A duration defined with respect to \(j\) must be combined with \(\Delta j\); one defined with respect to
\(y\) must be combined with \(\Delta y\).
Consider
\[
F=R=1000,
\qquad
c=8\%,
\qquad
n=10,
\qquad
k=2,
\qquad
y=9\%.
\]
bond_duration_result <-
bond_duration(
face = 1000,
c = 0.08,
n = 10,
k = 2,
y = 0.09,
y_type = "effective"
)
bond_convexity_result <-
bond_convexity(
face = 1000,
c = 0.08,
n = 10,
k = 2,
y = 0.09,
y_type = "effective"
)
bond_duration_result
#> # A tibble: 1 x 9
#> price macaulay_duration_periods macaulay_duration_years modified_duration_pe~1
#> <dbl> <dbl> <dbl> <dbl>
#> 1 947. 14.0 6.98 13.4
#> # i abbreviated name: 1: modified_duration_periods_j
#> # i 5 more variables: modified_duration_years_i <dbl>, yield_per_period <dbl>,
#> # yield_effective_annual <dbl>, k <int>, n_periods <int>
bond_convexity_result
#> # A tibble: 1 x 7
#> price discrete_convexity_periods discrete_convexity_years yield_per_period
#> <dbl> <dbl> <dbl> <dbl>
#> 1 947. 235. 58.7 0.0440
#> # i 3 more variables: yield_effective_annual <dbl>, k <int>, n_periods <int>
For this bond, the Macaulay duration is approximately \(13.9529\) coupon periods, or \(6.9764\) years.
Portfolio duration and convexity
Portfolio sensitivities are value-weighted averages when the
individual measures refer to the same rate movement.
portfolio <-
tibble::tibble(
asset = c("A", "B", "C"),
units = c(450, 300, 180),
price = c(96.20, 102.75, 88.50),
duration = c(2.80, 5.60, 9.40),
convexity = c(11.40, 43.20, 104.60)
) |>
dplyr::mutate(
market_value = units * price
)
portfolio_duration_result <-
portfolio_duration(
portfolio,
col_P = "market_value",
col_D = "duration",
col_portfolio = NULL
)
portfolio_convexity_result <-
portfolio_convexity(
portfolio,
col_P = "market_value",
col_C = "convexity",
col_portfolio = NULL
)
portfolio_duration_result
#> # A tibble: 1 x 3
#> D_P P_total n_positions
#> <dbl> <dbl> <int>
#> 1 4.93 90045 3
portfolio_convexity_result
#> # A tibble: 1 x 3
#> C_P P_total n_positions
#> <dbl> <dbl> <int>
#> 1 38.8 90045 3
The weights are based on economic value, not simply on face amount or
number of securities.
From sensitivity to immunization
For a positive cash-flow stream and a horizon \(H\), suppose all cash flows are valued and
reinvested using the same flat effective rate \(i\). The accumulated value at the horizon
is
\[
W_H(i)
=
(1+i)^HP(i).
\]
Therefore,
\[
\frac{W_H'(i)}{W_H(i)}
=
\frac{H-D_{\text{Mac}}(i)}{1+i}.
\]
If
\[
H=D_{\text{Mac}}(i_0),
\]
then \(W_H'(i_0)=0\). At a
horizon equal to Macaulay duration, the local price effect and the local
reinvestment effect offset each other to first order.
The surplus function and Redington
Let
\[
S(i)=PV_A(i)-PV_L(i).
\]
If
\[
PV_A(i_0)=PV_L(i_0),
\]
then \(S(i_0)=0\). If, in
addition,
\[
D_A(i_0)=D_L(i_0),
\]
then \(S'(i_0)=0\).
Since \(PV''(i)=PV(i)C(i)\),
under present-value equality,
\[
S''(i_0)
=
PV_A(i_0)\left[C_A(i_0)-C_L(i_0)\right].
\]
Thus the classical Redington conditions are
\[
PV_A(i_0)=PV_L(i_0),
\]
\[
D_A(i_0)=D_L(i_0),
\]
and
\[
C_A(i_0)>C_L(i_0).
\]
Redington is therefore a local curvature condition. By contrast,
duration-convexity matching imposes \(C_A=C_L\), so \(S''(i_0)=0\). These are different
constructions.
An integrated immunization example
Consider liabilities
\[
L=(30{,}000,\ 40{,}000,\ 50{,}000)
\]
payable at years \(3\), \(5\), and \(8\), valued at an annual effective rate of
\(5\%\). Their present value is
approximately \(91{,}098.14\), with
Macaulay duration \(5.5455\) and
convexity \(36.7184\).
Suppose zero-coupon assets with redemption value 1,000 are available
at maturities 2, 5, and 10 years.
liabilities <- c(30000, 40000, 50000)
liability_times <- c(3, 5, 8)
reference_rate <- 0.05
asset_maturities <- c(2, 5, 10)
asset_redemption <- 1000
asset_prices <-
asset_redemption *
(1 + reference_rate)^(-asset_maturities)
asset_durations <-
asset_maturities
asset_convexities <-
asset_maturities *
(asset_maturities + 1) /
(1 + reference_rate)^2
Using the 2-year and 10-year assets:
duration_match <-
immunize_duration(
L = liabilities,
t = liability_times,
P = asset_prices[c(1, 3)],
D = asset_durations[c(1, 3)],
i = reference_rate,
i_type = "effective"
)
duration_match
#> # A tibble: 2 x 6
#> w PV_L D_L PV_A D_A n_assets
#> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 55.9 91098. 5.55 91098. 5.55 2
#> 2 65.8 91098. 5.55 91098. 5.55 2
immunize_duration() matches present value and duration.
It does not by itself prove Redington; the convexity inequality must be
checked separately.
With all three assets, present value, duration, and convexity can be
matched simultaneously:
duration_convexity_match <-
immunize_duration_convexity(
L = liabilities,
t = liability_times,
P = asset_prices,
D = asset_durations,
C = asset_convexities,
i = reference_rate,
i_type = "effective"
)
duration_convexity_match
#> # A tibble: 3 x 8
#> w PV_L D_L C_L PV_A D_A C_A n_assets
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
#> 1 7.34 91098. 5.55 36.7 91098. 5.55 36.7 3
#> 2 90.0 91098. 5.55 36.7 91098. 5.55 36.7 3
#> 3 22.7 91098. 5.55 36.7 91098. 5.55 36.7 3
immunize_duration_convexity() should be interpreted
exactly this way: it matches present value, duration, and convexity. It
is not a direct implementation of Redington and it does not establish
full immunization.
The immunization gap
The surplus function has a direct interpretation:
\[
S(i)>0 \Rightarrow \text{asset surplus},
\]
\[
S(i)=0 \Rightarrow \text{exact value equality},
\]
\[
S(i)<0 \Rightarrow \text{asset insufficiency}.
\]
plot_immunization_gap() can be used to inspect this gap
over a finite range of rates.
asset_cash_flows <- list(
list(
cf = 444998.22,
t = 3
),
list(
cf = 561800.00,
t = 7
)
)
plot_immunization_gap(
L = 1000000,
t = 5,
asset_cashflows = asset_cash_flows,
w = c(1, 1),
i = 0.06,
i_type = "effective",
delta = 0.02
)
Immunization gap around the reference
rate.
The graph is a diagnostic tool. A finite grid of rates can illustrate
the shape of \(S(i)\), but it cannot
prove \(S(i)\ge0\) for every admissible
rate. Full immunization requires a stronger structural argument.
Final actuarial reading
The valuation of a fixed-income position begins with cash flows and a
rate basis. Duration and convexity then describe how that value responds
locally to changes in the rate.
The sequence is
\[
P(i)
\longrightarrow
P'(i)
\longrightarrow
D
\longrightarrow
P''(i)
\longrightarrow
C,
\]
followed by
\[
PV_A-PV_L
\longrightarrow
\text{duration matching}
\longrightarrow
\text{immunization}.
\]
Throughout the vignette, tidyactuarial enters after the
actuarial structure has been identified. It reproduces bond values,
yields, sensitivities, portfolio measures, and immunization systems
while the mathematical interpretation remains explicit.
The same precision applies to the immunization functions:
immunize_duration() matches present value and duration;
immunize_duration_convexity() matches present value,
duration, and convexity; Redington additionally requires favorable
curvature; and full immunization requires a stronger structural
argument.
The software verifies the equations being solved. It does not replace
the conditions that define the actuarial strategy.