Assets, beginning of year | 10,000,000 |
Premium income | 1,000,000 |
Gross investment income | 530,000 |
Claims paid | 420,000 |
Investment expenses | 20,000 |
Other expenses | 180,000 |
5 Asset-Liability Management
5.1 Supplementary Concepts
- The following notation will be used in this unit:
| \(A\) | Fund balance at the beginning of the period |
| \(B\) | Fund balance at the end of the period |
| \(I\) | Amount of interest earned during the period |
| \(C_t\) | Net cash flow at time \(t\), where \(0 \leq t \leq 1\) |
| \(C=\sum_{t} \,C_t\) | Total amount contributed during the period |
| \(B_k^*\) | Fund amount immediately before each cash flow |
| \(C_k^*\) | Net cash flow to the fund at time \(t_k\) |
- The dollar-weighted formulas for calculating the yield rate earned by an investment fund are shown below: \[ \boxed{ \begin{array}{rl} \\ i & \approx \large{\frac{I}{A+\sum _t {C_t \, (1-t)}}} & \, \\ \\ i & \approx \large{\frac{2I}{A+B-I}} & \text{(if} \,\, t=1/2, \forall \, C_t)\\ \\ \end{array} } \]
- The time-weighted formulas for calculating the yield rate earned by an investment fund are shown below: \[ \boxed{ \begin{array}{rl} \\ 1 + i & \approx (1+j_1)\,(1+j_2) \cdots (1 + j_m) \\ \\ j_k &= \large{\frac{B_k^*}{B_{k-1}^* \, + \, C_{k-1}^*}}- 1\\ \\ \end{array} } \]
- Spot interest rates are the current market rates for immediate transactions, while forward rates are forecasted rates for transactions that will occur in the future. Let \(f_t\) be the forward rate at time \(t\), and let \(s_t\) be the spot rate at time \(t\). Then, \[ \boxed{ \begin{array}{rl} \\ 1 + f_t = \large{\frac{(1+s_{t+1})^{t+1}}{(1+s_t)^t}} \\ \\ \end{array} } \]
- The Macaulay duration is a measure of the sensitivity of a security’s price to changes in interest rates. It represents the weighted average time until a security’s cash flows are received or paid, with the weights based on the present value of each cash flow. \[ \boxed{ \begin{array}{rl} \\ \text{MacDur}=\large{\frac{\sum _t \, t \, v^t \, CF_t}{\sum _t \, v^t \, CF_t}} \\ \\ \end{array} } \]
- Let \(P(i)=\sum _t \, v^t\, CF_t\) be the present value of cash flows (i.e., price). Then, the modify duration, a measure of the volatility of the present value of cash flows is given by: \[ \boxed{ \begin{array}{rl} \\ \text{ModDur}&=-\large{\frac{P'(i)}{P(i)}} = -\large{\frac{\large{\frac{d}{di}\sum (1+i)^{-t}\,CF_t}}{\sum (1+i)^{-t}\,CF_t}} = \large{\frac{\sum t\,(1+i)^{-t-1}\,CF_t}{\sum (1+i)^{-t}\,CF_t}} \\ \\ & \therefore \, \text{ModDur} = \large{\frac{\text{MacDur}}{1+i}}\\ \\ \end{array} } \]
- The Macaulay convexity is a measure of how the duration of a security changes as interest rates change. It is given by: \[ \boxed{ \begin{array}{rl} \\ \text{MacCon}=\large{\frac{\sum _t \, t^2 \, v^t \, CF_t}{\sum _t \, v^t \, CF_t}} \\ \\ \end{array} } \]
- The modified convexity is a refinement of the Macaulay convexity, taking into account the impact of changes in yield on the modified duration. It is given by: \[ \boxed{ \begin{array}{rl} \\ \text{ModCon}=\large{\frac{P''(i)}{P(i)}}=\large{\frac{\text{MacDur}+\text{MacCon}}{(1+i)^2}} \\ \\ \end{array} } \]
- First- and second-order approximations to price changes can be derived from a Taylor series. The resulting formulae are: \[ \boxed{ \begin{array}{rl} \\ P(i+\Delta i) &\approx P(i)\,[\,1 - \text{ModDur} \cdot \Delta i\,]\\ \\ P(i+\Delta i) &\approx P(i)\,[\,1 - \text{ModDur} \cdot \Delta i\,+\frac{1}{2}\,\text{ModCon} \cdot (\Delta \dot i)^2]\\ \\ \end{array} } \]
- Let \(A\) denote assets and \(L\) denote liabilities. The conditions for the Redington immunization are:
- \(P_A(i)=P_L(i)\)
- \(\text{ModDur}_A=\text{ModDur}_L\)
- \(\text{ModCon}_A>\text{ModCon}_L\)
- The condition for full immunization are:
- \(P_A(i)=P_L(i)\)
- \(\text{MacDur}_A=\text{MacDur}_L\)
- Each liability cash flow must be surrounded by two asset cash flows
- If \(X\) and \(Y\) are assets, and the liability cash flow \(L_k\) takes place at time \(t=k\), and if exactly two out of \(X\), \(x\), \(Y\), and \(y\) are known. then a full immunization strategy can be derived by solving the following equations (which may not have a solution, or a unique solution): \[ \boxed{ \begin{array}{rl} \\ P(i) &= X(1+i)^{(k-x)}+Y(1+i)^{-(k+y)}-L_k =0 \\ \\ P'(i) &= -(k-x)X(1+i)^{(k-x)-1}-(k+y)Y(1+i)^{-(k+y)-1} =0 \\ \\ \end{array} } \]
5.2 Exercises
1
A fund of \(10\,000\) is established at \(t=0\).
A deposit of \(5\,000\) is made at the end of six months. The amount at year-end is \(18\,000\).
Find the absolute difference in basis points (\(\mathrm{bps}\)) between the approximate effective rate of interest using the dollar-weighted approach, and the exact calculation using compound interest (\(100\, \mathrm{bps}=1\%\))
2
A fund of \(1\,000\) is established at \(t=0\).
A deposit of \(500\) is made at the end of four months.
Withdrawals of \(200\) and \(100\) were made at the end of six and eight months, respectively.
The amount at year-end is \(1\,272\).
Find the approximate effective rate of interest using the dollar-weighted approach.
3
Find the effective rate of interest earned in a calendar year by an insurance company using the dollar-weighted method and the following data:
4
Find the effective rate of interest earned in a calendar year by an insurance company using the dollar-weighted method and the following data:
Date | Fund value | Deposits | Withdrawals |
|---|---|---|---|
January 1, 2023 | 100,000 | - | - |
May 1, 2023 | 112,000 | 30,000 | - |
November 1, 2023 | 125,000 | - | 42,000 |
January 1, 2024 | 100,000 | - | - |
5
Find the yield rate of a fund using the time-weighted method and the following data:
\(\\\)
Date | Fund value | Deposits | Withdrawals |
|---|---|---|---|
January 1, 2023 | 100,000 | - | - |
May 1, 2023 | 112,000 | 30,000 | - |
November 1, 2023 | 125,000 | - | 42,000 |
January 1, 2024 | 100,000 | - | - |
6
A fund has \(500\,000\) at the beginning of the year, and \(680\,000\) at the end of the year.
Gross interest earned is \(60\,000\), and investment expenses are \(5\,000\).
Find the yield rate.
7
A fund earning \(4\%\) effective has a balance of \(10\,000\) at the beginning of the year.
If \(2\,000\) is added at the end of three months, and \(3\,000\) is withdrawn at the end of nine months, find the ending balance using the dollar-weighted method.
8
Deposits of \(10\,000\) are made into a fund at times \(t=0\) and \(t=1\).
The fund balance is \(12\,000\) at \(t=1\), and \(22\,000\) at \(t=2\).
What is the annual effective yield rate which is equivalent to that produced by the time-weighted method?
9
Given the spot rates below, calculate \(f_{4}\).
Lenght of investment (years) | Spot Rate |
|---|---|
1 | 7.00% |
2 | 8.00% |
3 | 8.75% |
4 | 9.25% |
5 | 9.50% |
10
Consider a five-year annuity with payments of \(1\,000\) whose present value is calculated using the spot rates below.
Lenght of investment (years) | Spot Rate |
|---|---|
1 | 7.00% |
2 | 8.00% |
3 | 8.75% |
4 | 9.25% |
5 | 9.50% |
Find the present value of the remaining payments immediately after two payments have been made.
The forward rates at that time are expected to be \(100\) basis points (bps) higher for all periods than the current spot rates.
11
Find the absolute difference in bps between the one-year deferred two-year forward rate and the two-year deferred one-year forward rate, if spot rates are \(5\%\), \(5.5\%\), \(6\%\), \(6.5\%\), for terms of one to four years, respectively.
12
If the effective rate of interest is \(8\%\), find the Macaulay duration of a \(10\)-year bond with \(8\%\) annual coupons.
13
If the effective rate of interest is \(8\%\), find the Macaulay duration of a \(10\)-year mortgage repaid with level payments of principal and interest.
14
If the effective rate of interest is \(8\%\), find the Macaulay duration of a preferred stock paying level dividends,
15
If the effective rate of interest is \(8\%\), find the Macaulay duration of a common stock which pays dividends at the end of each year, if it is assumed that each dividend is \(4\%\) greater than the prior dividend.
16
If the effective rate of interest is \(25\%\), find the Macaulay duration of a loan that is repaid with payments of \(1\,000\) at the end of two years, and \(3\,000\) at the end of three years.
17
A company purchased the following bonds to form a portfolio.
Bond | Price | MacDur |
|---|---|---|
X | 980 | 21.46 |
Y | 1 015 | 12.35 |
Z | 1 000 | 16.67 |
Calculate the Macaulay duration of the portfolio.
18
Find the modified convexity of a loan repaid with equal instalments overs four years, if \(i=0\).
19
If the effective rate of interest is \(8\%\), find the modified convexity of a \(3\)-year bond with \(8\%\) annual coupons.
20
If the effective rate of interest is \(8\%\), find the modified convexity of a \(3\)-year mortgage repaid with level payments of principal and interest.
21
If the effective rate of interest is \(8\%\), find the modified convexity of a preferred stock paying level dividends.
22
If the effective rate of interest is \(8\%\), find the modified convexity of a common stock which pays dividends at the end of each year, if it is assumed that each dividend is \(4\%\) greater than the prior dividend.
23
If the effective rate of interest is \(25\%\), find the absolute difference of the Macaulay convexity and the modified convexity of a loan that is repaid with payments of \(1\,000\) at the end of two years, and \(3\,000\) at the end of three years.
24
Find the modified convexity of a perpetuity that has payments at the end each year period.
The first payment is \(200\).
Each subsequent payment is \(1\,000\) more than the previous payment, and you are given \(i=25\%\).
25
A company must pay \(1\,000\,000\) at the end of \(10\) years.
It has a zero coupon bond that matures for \(413\,947.55\) in five years, and a zero coupon bond that matures for \(864\,580.82\) in \(20\) years.
The current annual effective yield is \(10\%\).
Find the absolute difference between the Macaulay duration of assets and liabilities.
26
Using a first-order linear approximation based on Macaulay duration, calculate the estimated change in price of a \(30\)-year mortgage repaid with level monthly payments of principal and interest, when the effective monthly interest rate increases from \(0.5\%\) to \(1\%\).
27
Using a first-order linear approximation based on modified duration, calculate the estimated change in price of a preferred stock paying level dividends, when the effective annual interest rate decreases from \(10\%\) to \(8\%\).
28
A company must make payments of \(95\,030\) and \(297\,330\) and the end of one and two years, respectively.
Bond \(A\) is a two-year \(1\,000\) par value with \(6\%\) annual coupons, and bond \(B\) is a one-year zero-coupon bond redeemable at \(1\,000\).
Determine the number of each type of bond the company should buy in order to exact match the liabilities.
29
An insurer must pay \(3\,000\) and \(4\,000\) at the ends of years one and two, respectively.
The only investments available to the company are a one-year zero-coupon bond (par value of \(1\,000\); effective annual yield of \(5\%\)), and a two-year \(8\%\) annual coupon bond (par value of \(1\,000\); effective annual yield of \(6\%\)).
What is the cost to the company today to match its liabilities?
30
A company has to pay \(72\,900\) after two years.
They wish to immunize this liability at an interest rate that corresponds to a factor of \(v=0.9\) by using \(M\), a one-year zero-coupon bond and \(N\), a three-year zero-coupon
Determine how much of \(M\) and \(N\) should be bought.
31
A company has the projected cash flows in the table below.
Year | 1 | 2 | 3 | 4 | 5 |
Liability cash flow | 179 | 679 | 144 | 3,144 | 824 |
Available investments are a \(100\) par value \(2\)-year bond with annual coupons of \(7\%\); a \(100\) par value \(4\)-year bond with annual coupons of \(4\%\); and a \(100\) par value \(5\)-year bond with annual coupons of \(3\%\).
The annual yield bonds is \(3\%\).
The company wishes to match the cash flows of assets and liabilities.
Determine the cost of establishing the investment portfolio.
32
A company must pay \(100\,000\) in five years.
The company funds this liability through the purchase of \(4\)-year zero-coupon bonds and \(10\)-year zero coupon bonds.
The annual effective yield for assets and liabilities is \(12\%\).
How much the company should invest in each investment to immunize its position using the Redington technique?
33
A company owes \(500\) and \(1\,000\) to be paid at \(t=1\) and \(t=4\), respectively.
The company will set up an investment program to match the Macaulay duration and the present value of the above liabilities using an annual effective interest rate of \(10\%\).
The investment program produces asset cash flows of \(X\) at \(t=0\), and \(Y\) at \(t=3\).
Calculate \(X\) and determine whether the program satisfies the conditions for Redington immunization.
34
A company has a liability of \(120\,000\) due in eight years.
This liability will be met with payments of \(50\,000\) in five years, and \(B\) in \(8+b\) years (\(b>0\)).
The company is employing a full immunization strategy using an annual effective rate of \(3\%\).
Calculate \(B/b\).
35
An insurance company has to pay \(1\,000\) after two years and \(2\,000\) after four years.
The current market interest rate is \(10\%\), and the yield curve is assumed to be flat at any time.
The company wishes to immunize the interest rate risk by purchasing zero-coupon bonds which mature after one, three, and five years.
The company will purchase the following bonds:
- One-year zero-coupon bond of \(44.74\)
- Three-year zero-coupon bond of \(2\,450.83\)
- Five-year zero-coupon bond of \(500.00\)
You are asked to
Show that the investment strategy satisfies the conditions of duration matching; and
Calculate the surplus when there is an immediate one-time change of interest rates from \(10\%\) to \(9\%\), \(11\%\), \(15\%\), and \(20\%\)
36
An insurance company has to pay \(1\,000\) afer two years and \(2\,000\) after four years.
The current market interest rate is \(10\%\), and the yield curve is assumed to be flat at any time.
The company wishes to immunize the interest rate risk by purchasing zero-coupon bonds which mature after one, three, and five years.
The company will purchase the following bonds:
- One-year zero-coupon bond of \(154.16\)
- Three-year zero-coupon bond of \(2\,186.04\)
- Five-year zero-coupon bond of \(660.18\)
You are asked to
Show that the investment strategy satisfies the conditions of the Redington immunization; and
Calculate the surplus when there is an immediate one-time change of interest rates from \(10\%\) to \(9\%\), \(11\%\), \(15\%\), and \(20\%\)
37
An insurance company has to pay \(1\,000\) afer two years and \(2\,000\) after four years.
The current market interest rate is \(10\%\), and the yield curve is assumed to be flat at any time.
The company wishes to immunize the interest rate risk by purchasing zero-coupon bonds which mature after one, three, and five years.
The company will purchase the following bonds:
- One-year zero-coupon bond of \(454.55\)
- Three-year zero-coupon bond of \(1\,459.09\)
- Five-year zero-coupon bond of \(1\,100.00\)
You are asked to
Calculate the number of units of each asset that the company must buy to be fully immunized against changes in interest rates; and
Calculate the surplus when there is an immediate one-time change of interest rates from \(10\%\) to \(9\%\), \(11\%\), \(15\%\), and \(20\%\)