# Spotlight Quiz on Inflation, Index-Linking and Compounding

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1 Spotlight Quiz on Inflation, Index-Linking and Compounding Frequency of payment A major UK bank has recently written to its customers along the following lines: Through talking to customers we have found that the majority would prefer to receive their bank charges monthly rather than quarterly. Therefore we have re-designed our systems to move to monthly billing. Please be assured that we are only changing when we bill you, not what you pay. This change should make it easier to keep a close eye on your account. Question 1 The average amount of your bank charges is 1200 per year, spread evenly over each of 12 months, and your bank currently charges 0.5% interest per month. What is the effect of this change? (a) A welcome helping hand from your bank (b) A price reduction of 0.5% per annum (c) A price increase of 0.5% per annum (d) A price increase of 1.0% per annum (e) A price increase of 1.5% per annum (f) Don t know The right answer is (c) a price increase of 0.5% per annum Previously the cash flows involved in paying the charges would have been 300 per quarter. If rates are 0.5% per month then the present value of four quarterly payments of 300 would be 1, If the payment schedule changes to 100 per month the present value increase to 1, This represents an increase of 0.50%. PV of 300 in month 3 = 300 (1+0.05) -3 = PV of 300 in month 6 = 300 (1+0.05) -6 = PV of 300 in month 9 = 300 (1+0.05) -9 = PV of 300 in month 12 = 300 (1+0.05) -12 = Sum of PVs = 1, Rather than 12 calculations for the revised monthly charging, we can use an annuity formula: PV = 100 (1/0.05) (1-(1+0.05) -12 ) = 1, This is an increase in present value of 5.79, or 0.05% Compounding It is well known that interest rates vary in the way in which they are quoted. Some rates are quoted annually, some semi-annually and others such as credit cards are quoted monthly. Because of this, it is sometimes not immediately apparent how to compare different rates. An annualised equivalent rate is often quoted so that comparisons can be made, or a present value calculation might be used to value future cashflows. Question 2

2 You have 1,000 to invest for a full year, which of the following would you prefer, assuming equivalent credit and counterparty risks? (a) A bank deposit offering a simple interest rate of 6% p.a. (b) A security offering a compound annual rate of 6% p.a. (c) A security offering a semi-annual rate of 5.96% p.a. (d) A security offering a monthly compounding rate of 5.8% p.a. (e) A security offering a continuously compounded rate of 5.85% p.a. (f) Don t know The right answer is (c) A security offering a semi-annual rate of 5.96% p.a. The result for (a) and (b) is the same at 1, For (c) the result is given by 1000 (1+2.98%) 2 = 1, For (d) the result is given by 1,000 (1+(5.8%/12)) 12 = 1, For (e) the result is given by 1,000 e 5.85% = 1, (c) therefore generates the highest return. Inflation Inflation has been an ongoing issue for the British for many years. The first attempt to formally measure household inflation was just after the Second World War based on a basket of goods derived from the household expenditure survey of Over several years the contents of the basket of goods was refined and the Retail Price Index was first published in 1956, by the Cost of Living Advisory Committee. This committee instigated the regular Family Expenditure Survey partly so that the contents of the basket of goods could be continually revised to keep pace with changing purchasing trends. Among many technical changes from the index inception, from today s perspective it is interesting to note the following: meals out were included from 1968, Housing and mortgage costs were revised in 1975 The Tax and Price Index was introduced in 1979 Mortgage payments were excluded to give RPIX in 1992 The Consumer Price Index (CPI) has a much shorter history. It started life in 1996 as the Harmonised Index of Consumer Prices (HICP). Similar indices were developed across the EU as a means of determining the inflation convergence criteria for joining the Euro. As such the construction and coverage of the HICP are defined legally in European regulations. In 2003 the name was changed to the Consumer Prices Index (CPI) and it was designated as the official target rate for UK inflation by the then Chancellor. Question 3 There are differences between RPI and CPI. Which of the following lists what is included in RPI but excluded from CPI? (a) All housing costs (b) Mortgage costs and estate agent s fees (c) Mortgage costs, housing transaction fees and buildings insurance

3 (d) Mortgage costs, housing transaction fees, buildings insurance and council tax (e) Mortgage costs, housing transaction fees, buildings insurance, council tax and house depreciation (f) Mortgage costs, housing transaction fees, buildings insurance, council tax, house depreciation and housing maintenance costs (g) Don t know The right answer is (e) Mortgage costs, housing transaction fees, buildings insurance, council tax and house depreciation. The logic of excluding much of housing cost is due to the international nature of the index there is little that is comparable about the way that different European countries provide for their housing. Council tax is excluded because it is a tax rather than household consumption. This hits at the heart of the difference: the CPI is not intended to be a cost of living index. For the same reason television licences and road fund licences are excluded from CPI but included in RPI. ONS website: Question 4 Potential Impact of a Change to CPI The change in the relevant inflation index to determine annual pension increases may have a profound effect on the present value of future pension liabilities. If there is a long term expectation that the new measure (CPI) will be 0.5% per annum lower than the old measure (RPI) then the amount required to fund future pensions will be reduced. If: Expectation of remaining life after retirement is 20 years The first payment is 100, next year Current discount rates are, and will remain 6% for the period RPI is expected to average 3.5% and CPI is expected to average 3% over the pension period What is the change in the present value of the pension provision if annual pension payments are increased in line with CPI instead of RPI? (a) Reduce by 0.5% (b) Reduce by 2.5% (c) Reduce by 4.1% (d) Reduce by 5.2% (e) Reduce by 6.3% (f) Don t know The right answer is (c) reduce by 4.1% The present value of a twenty year growing annuity (the pension payment) can be calculated by a formula, but conceptually it is easiest to see as the difference between a perpetuity starting next year and a perpetuity starting in year 21 if we deduct the far perpetuity we are left with just the payments for the first 20 years. A perpetuity with the first payment next year has value = cashflow/(discount rate growth rate). RPI

4 We know the components so, if RPI is the inflator we can calculate Value perpetuity starting next year = 100/(6% - 3.5%) = 4000 The far perpetuity has a larger first payment, having grown at 3.5% for 20 years: So value perpetuity starting year 21 = ( 100 *(1+3.5%) 20 )/(6%-3.5%) = This has a present value of (1+6%) -20 = The difference between the perpetuities is 1, and this is the present value of the pension liability. We can repeat the same calculation using the CPI value of 3% for the inflator and the results are: PV of perpetuity starting next year = 3, PV of perpetuity starting year 21 = 1, Difference = value of pension liability = 1, So the change in use of inflator causes a reduction in PV from 1, to 1,456.15, a reduction of 4.09%, rounded to 4.1%. Question 5 Index linked bonds can have their coupons or their principal amount, or both, linked to inflation. Here we take an example where both coupons and principal are linked. Inflation is expected to fall slightly over the next few years, returning to the long term target rate of 2%. But this year it is expected to be 4.2%, next year the expectation is for 3.7% and the year after that inflation is expected to be down to 3.2%. You have decided to invest in a three-year index-linked bond with annual 2.5% coupons, the coupon and principal being index-linked. If you invest now in the bond being issued at par, and your expectations prove to be accurate, what nominal yield to maturity will you achieve if you hold the bond to maturity? (a) 5.8% p.a. (b) 6.3% p.a. (c) 6.7% p.a. (d) 7.0% p.a. (e) Don t know The right answer is (b) 6.3% p.a. The calculations lend themselves to being tabulated as below. Time Real Cashflow Inflation adjustment Actual cashflow (1+4.2%) = (1+3.7%) = (1+3.2%) = Once the cash flows are established the yield can be found, again using the table format: Time Cashflow PV at 6% PV at 7% 0 (100) (100) (100)

5 NPV 0.83 (1.90) From this it is clear that the yield is between 6% and 7%, because the Net Present Value changes from positive to negative between 6% and 7%. Using the proportions; Yield = 6% +[(0.83/( )) 1%] = 6.305% There is a short cut method which works when the bond is issued and redeemed at par and when the inflation rate remains constant for all periods. Then it is possible to compound together the coupon rate and the inflation rate to get nominal yield = (1+coupon rate) (1+inflation rate) -1. We can use an approximation of that in this instance by estimating the average inflation rate for the period as 3.7%: nominal yield = (1+2.5%) (1+3.7%) 1 = 6.029%, a pretty close approximation! Volume of Index-Linked Gilts and Method of Indexation At the end of March % of the total UK Gilt portfolio were index-linked, or billion. They are typically semi-annual coupon bonds. Coupon payments and principal are adjusted in line with RPI to take account of accrued inflation since the gilt was issued. Despite the Government s announced intention to move to CPI as its main inflation indicator, so far there have been no issues of index-linked gilts with the inflation link being to CPI rather than RPI. One reason for this may be the desire of purchasers to use the RPI link as a hedge for their corporate pension liabilities, many of which are still linked to RPI rather than CPI. Question 6 For practical reasons the indexation is lagged, otherwise the coupon might be due before the RPI has been calculated, or confirmed. What is the indexation lag for all UK Government Gilts issued since 2005? (a) One month (b) Three months (c) Six months (d) Eight months (e) Don t know The right answer is (b) three months. When index-linked gilts were first issued the indexation lag was eight months. However, this was revised for new issues in 2005 and since then all issues have used the three month indexation lag. This is regarded as international best practice. The reference rate for, say June 2011 is then the RPI for March In trading, the bonds are priced relative to their 100 real principal value. On settlement the actual price is then the price agreed in the trade times the indexation ratio published by the Debt Management Office for the day in question.

6 Debt Management Office Publication A guide to Gilts &page=investor_guide/guide

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