Topic 10 - Term structure of Interest Rates Flashcards
Give a brief overview for Topic 10 - Term structure of Interest Rates
- Evaluate discrete and continuous spot and forward rates
- Theories of the term structure of interest rates
- Evaluate duration, convexity and the Redington’s
conditions for immunisation
What is a Discrete Time Spot Rate?
- Yield on a unit zero coupon bond with term n years, y,n
- Known as n-year spot rate of interest
- Effectively average interest rate over term
- If P,n is price of a zero coupon bond the equation is:
P,n = 1/(1+y,n)^n = (1+y,n)^(-n) or
(1+y,n) = P,n^(-1/n)
Discrete Time Spot Rates
Example
Prices for a portfolio of zero coupon bonds are:
1yr = £93, 3yr = £80, 5yr = £68, all per £100 nominal
Calculate spot rates
1yr: 100(1+y,1)^-1 = 93 y1 = 7.5%
3yr: 100(1+y,3)^-3 = 80 y3 = 7.7%
5yr: 100(1+y,5)^-5 = 68 y5 = 8.0%
Variation of interest rates known as term structure of
interest rates
What is a Discrete Time Forward Rate?
Discrete time forward rate ft,r is annual rate agreed at
t=0, for investment made at t>0 for period of r years
So if investor agrees (at t=0) to invest £100 at time t for
r years, the accumulated value at t + r is:
100(1+ft,r)r
ft,r is the average interest rate between times t and t + r
Show how forward rates, spot rates and zero coupon bond prices are all connected using formulae and a timeline
(1+y,t)^t(1+ft,r)^r = (1+y,t+r)^(t+r) = 1/P,t+r
(1+ft,r)^r = (1+y,t+r)^(t+r)/(1+y,t)^t = P,t/P,t+r
Timeline available page 7 week 10 lecture notes
Show how a one year forward rate is written
(1+f,t) = (1+y,t+1)^(t+1)/(1+y,t)^t = P,t/P,t+1
Discrete Time Forward Rates Example The 3,5 and 7 year spot rates are 6%, 5.7% and 5% pa respectively and the 3 yr forward rate from t=4 is 5.2%pa. Calculate: y4 f5,2 f3 f3,4
So y,3 = 6%, y,5 = 5.7%, y,7 = 5% and f4,3 = 5.2%
y,4
(1+y,4)^4 = (1+y,7)^7/(1+f4,3)^3= (1.05)^7/(1.052)^3
Therefore y,4 = 4.85%
f5,2
(1+f5,2)^2 = (1+y,7)^7/(1+y,5)^5= (1.05)^7/(1.057)^5
Therefore f5,2 = 3.27%
f,3
1+f,3 = (1+y,4)^4/(1+y,3)^3 = (1.0485)^4/(1.06)^3
Therefore f,3 = 1.47%
f3,4
(1+f3,4)^4 = (1+y,7)^7/(1+y,3)^3 = (1.05)^7/(1.06)^3
Therefore f3,4 = 4.26%
What is a Continuous Time Spot Rate?
This is the force of interest equivalent to an effective
rate of interest equal to the spot rate
If P,t is price of a zero coupon bond the equation is
P,t =e^(−txY,t)or
Y,t = −1/t ln(P,t)
Relationship of y,t & Y,t is same as i and δ
So y,t =e^(Y,t) -1 (analogous to i = e^(δ) -1)
What is a Continuous Time Forward Rate?
This (Ft,r) is the force of interest equivalent to the annual forward rate of interest ft,r Similar to the discrete forward rates - e^(tY,t)e^(rFt,r) = e^[(t+r)Y,t+r] Taking logs of both sides tY,t + rFt,r = (t+r)Y,t+r Ft,r = (t+r)Y,t+r−tY,t /r Using Y,t = −1/t ln(P,)t Ft,r = 1/r ln(P,t/P,t+r)
Continuous Time Rates
Example
Prices for a portfolio of zero coupon bonds are
5yr = £70, 10yr = £47, 15yr = £30 all per £100 nominal
Calculate Y,10 and F5,10
100e^(−10Y,10) = 47 Y,10 = -0.1xln(0.47) = 7.55%
F5,10 = 1/10 ln(P,5/P,15) = 0.1 ln (0.7/0.3) = 8.47%
Why do interest rates vary?
Due to expectations of lenders/borrowers
Rates vary due to expectations of lenders/ borrowers.
What factors affect expectations?
- Supply and demand
- Base rates
- Interest rates in other countries
- Expected future inflation
- Tax rates
- Risks associated with changes in interest rates
What is Expectations theory in regard to the term structure of interest rates?
Expectations theory
- Attraction of short term and long term assets varies with expectation of future interest rates
- Expected fall in interest rates make short term assets less attractive and long term assets more attractive
- Short term yields will increase, long term will decrease
- Expected rise in interest will have converse effect
What is Liquidity Preference in regard to the term structure of interest rates?
Liquidity Preference
- Long dated bonds more sensitive to interest rate changes
- Risk averse investors require greater compensation, via higher yield for greater risk of loss for longer term assets
What is Market Segmentation in regard to the term structure of interest rates?
Market Segmentation
- Bonds of different terms attractive to different investors
- Investors choose assets that match their liabilities
- Supply of bonds may vary as issuer strategies may not tie in with investors.
What is Interest rate risk?
The risk that assets and liabilities don’t
move by the same amount when rates change
Why is Impact of changes in interest rates important to actuaries?
Actuaries value assets and liabilities over long time
periods
We need to consider the vulnerability of assets and
liabilities to changes in interest rates.
State the formula we would use to measure interest rate sensitivity via the effective duration (volatility) test
If we define PV of a series of cash flows as
A =∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
Differentiate A with respect to i
ν(i) = -1/A.d/diA = -A’/A
ν(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘+1,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
Bottom line is familiar, this is the PV of a series of cash
flows payable annually in arrears.
State the formula we would use to measure interest rate sensitivity for a series of regular cashflows via the duration (Discounted Mean Term) test
DMT is an estimate of the average duration of the cash
flows
τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
Top line looks familiar, this is an increasing annuity
DMT Similar to volatility actually equal to (1+i) ν(i)
This is DMT for regular cash flows what about a bond?
Duration (Discounted Mean Term)
Consider the DMT for an eight year bond that pays
coupons annually in arrears at a rate of 6% pa and
redeemable at par. Assume i = 4% pa.
τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘 τ = ∑8,t=1 6x𝑡xv^t + 100x8xv^8/ (∑8,t=1 6v^t + 100v^8) τ = 6(Ia)8¬ + 100x8v^8 / (6a8¬ + 100v^8 ) τ = 6x28.9133 + 800x1.04^-8 / (6x6.7327 + 100x1.04^-8) τ = 758.03/113.465 = 6.68 years
What do actuaries use DMT for?
We can use DMT to estimate the impact of a change in interest rates
Consider the DMT for an eight year bond that pays
coupons annually in arrears at a rate of 6% pa and
redeemable at par. Assume i = 4% pa.
τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
τ = ∑8,t=1 6x𝑡xv^t + 100x8xv^8/ (∑n,𝑘=1 6v^t + 100v^8)
τ = 6(Ia)8¬ + 100x8v^8 / (6a8¬ + 100v^8 )
τ = 6x28.9133 + 800x1.04^-8 / (6x6.7327 + 100x1.04^-8)
τ = 758.03/113.465 = 6.68 years
We can use DMT to estimate the impact of a change in
interest rates
DMT of previous example was 6.68 years
Initial price of bond was 6a8 + 100v8 = £113.47 @ 4%pa
What is the price if interest rates increased to 5% pa?
6a8¬ + 100v^8 = 6x6.4632+100x1.05^-8 = £106.46
113.47(1.04/1.05)^6.68 = £106.44
Duration (Discounted Mean Term)
Question
Calculate the DMT for a 12 year bond that pays
coupons annually in arrears at a rate of 8% pa and
redeemable at par. Assume i = 5% pa. Calculate the
change in the price of the bond if i increases to 6% pa.
Solution: Calculate the DMT for an 12 year bond that
pays coupons annually in arrears at a rate of 8% pa and
redeemable at par. Assume i = 5% pa.
τ = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
τ = ∑12,t=1 8x𝑡xv^t + 100x12xv^12/ (∑12,t=1 8v^t +100v^12)
τ = 8(Ia)12¬ + 100x12v^12 / (8a12¬ + 100v^12 )
τ = 8x52.4873 +1200x1.05^-12 / (8x8.8633 + 100x1.05^-12)
τ = 1088.10/126.59 = 8.595 years
Solution: Calculate the change in the price of the bond
if i increases to 6% pa.
Price at 5% = 8x8.8633 + 100x1.05^-12 = £126.59
DMT 8.595 years
Expected price at 6% pa = 126.59x(1.05/1.06)^8.595 =
£116.69
P@6% = 8x8.3838 + 100x1.06^-12 = £116.77v
What is convexity?
Convexity gives a measure of the change in the duration of the bond when interest rates change
What does positive convexity imply?
Positive convexity implies that DMT is an increasing
function of i
So value will increase more when i falls than it will
decrease when i increases.
State the formula we would use to measure the convexity for an asset with a series of regular cashflows
c(i) = -1/A.d^2/di^(2)A = -A’’/A c(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x (𝑡𝑘+1)𝑣^(𝑡,𝑘+2,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
Calculate the convexity for an asset with the following
cash flows at i = 7% pa:
£9,000 at t = 7
£12,000 at t = 13
PV = 9000x1.07^-7 +12000x1.07^-13 = £10,584.32
c(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x (𝑡𝑘+1)𝑣^(𝑡,𝑘+2,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘
c(i) = 9000x7x8x1.07^-9 +12000x13x14x1.07^-15/10584.32 c(i) = 1065724.71/10584.32 = 100.689
Question
An asset provides the following cash flows:
£8,000 at t =4
£19,000 at t = 15
Calculate the PV, volatility, DMT and convexity at i =7% pa.
PV = 8000x1.07^-4 +19000x1.07^-15 = £12,989.64
ν(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x 𝑣^(𝑡,𝑘+1,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘 ν(i) = 8000x4x1.07^-5 +19000x15x1.07^-16/ 12989.64 ν(i) = 119354.91/12989.64 = 9.189
τ = (1+i)ν(i) = 1.07x9.189 = 9.832 years
c(i) = ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑡,𝑘 x (𝑡𝑘+1)𝑣^(𝑡,𝑘+2,)/ ∑n,𝑘=1 𝐶,𝑡𝑘 x 𝑣^𝑡,𝑘 c(i) = 8000x4x5x1.07^-6 +19000x15x16x1.07^-17/ 12989.64 c(i) = 1550193.98/12989.64 = 119.34
What is the ideal scenario for an actuary when producing a portfolio of assets?
Ideal scenario is to select portfolio of assets that protect against any changes in interest rates
The ideal scenario for an actuary is to select portfolio of assets that protect against any changes in interest rates. However, what is the reality regarding this task?
- Almost impossible to select portfolio that generate
identical cash flows to that of the liabilities - Possible to select portfolio that offers some protection
How do we define a portfolio surplus?
S(i) = V,A(i) - V,L(i)
What is immunisation?
Three conditions that a portfolio must satisfy in order to protect S(i)
What are the three conditions of immunisation?
Condition 1
V,A(i,0) = V,L(i,0)
- PV of assets at i,0 = PV of liabilities at i,0
Condition 2
v,A(i,0) = v,L(i,0) or V’,A(i0) = V’,L(i0)
- Volatility of assets at i,0 = Volatility of liabilities at i,0
Condition 3
c,A(i,0) > c,L(i,0) or V’’,A(i,0) = V’’,L(i,0)
- Convexity of assets at i,0 > Convexity of liabilities at i,0
A fund must make the following payments at i = 6% pa
£40,000 at t = 5
£40,000 at t = 7
Show that immunisation can be achieved by purchasing
the following portfolio
A nominal amount Y of a 4 year zero coupon bond
A nominal amount Z of a 9 year zero coupon bond
Condition 1 V,A(i,0) = V,L(i,0) PV liabilities @ 6% pa 40000v^5 + 40000v^7 = £56,492.61 PV assets @ 6% pa Yv^4 + Zv^9 = £56,492.61 .........(1)
Condition 2
v,A(i,0) = v,L(i,0) or V’,A(i,0) = V’,L(i,0)
Volatility liabilities @ 6% pa
(40000x5v^6 + 40000x7v^8)/ V,L(i,0) = 316667.57/ V,L(i,0)
Volatility assets @ 6% pa
(Yx4v^5 + Zx9v^10)/ V,A(i,0) = 316667.57/ V,A(i,0)
Yx4v^5 + Zx9v^10 = 316667.57 …………………(2)
Eqn 1 = Yv^4 + Zv^9 = £56,492.61 Eqn 2 = Yx4v^5 + Zx9v^10 = 316667.57 Solve 1 & 2 simultaneously Multiply (1) by 4v and rearrange (1) becomes Yx4v^5 = -Zx4v^10 + £56,492.61x4v 56492.61x4v + Zx5v^10 = 316667.57 Z = £37,066.21 Y = £43,622.59
Condition 3 c,A(i,0) > c,L(i,0) or V’’,A(i,0) = V’’,L(i,0) Convexity liabilities @ 6% pa (40000x5x6v^7 + 40000x7x8v^9)/V,L(i,0) = 2123921/56,492.61 = 37.5964 Convexity assets @ 6% pa (Yx4x5v^6 + Zx9x10v^11)/ V,A(i,0) = 2372386/56,492.61 = 41.9946
All three conditions hold for this portfolio at i = 6% pa
