Personal Finance 4: interest rates

by George Hatjoullis

So far the personal finance series has had a UK bias. Unfortunately, personal finance is jurisdiction specific and mine is the UK. Unlike economics it is hard to generalise it and yet keep it relevant. The target is not the high net worth individual that can afford advice but rather the ordinary Jane that might search the net for some information in a D.I.Y finance exercise. I hope this effort on interest rates is of more general value.

First some definition work is needed. The concept of interest comes from debt. If you borrow £x for a period t, the lender might charge £c, an amount of cash that must be paid as well as the original loan. This amount of cash,£c, is the interest. In order to compare the cost of alternative loans, it is common to express the interest as a ‘percentage rate’. In this case the rate of interest is (c/x) times 100. So a loan of £100 charging £6 for the period t is a rate of 6%. However this is a rate for the period, which is unspecified. Another standard practice is to express the rate of interest as a percentage per annum. This creates complications unless the loan is for only one year and the interest is paid at the end of the year along with repayment of the loan. Regulators often require financial institutions to express their interest rates using a standard calculation method. This way at least you can be reasonably certain that the comparisons are meaningful even if the method is not quite what it seems. However, standardisation is not always the case so the best place to start is with the actual cash you will be handing over and when. It is may then be possible to get a better assessment of the actual interest rate.

Loans are normally expressed as an Annual Percentage Rate (APR). What does this mean? Once again it rather depends on jurisdiction. The EU requires all APR calculations to conform to a specific formula, faithfully reproduced by Wikipedia at ( It is clear that this can take into account costs such as fees as well as the timing of repayments (recall time value of money from PF1). The APR is derived by solving the equation, which makes it a yield (more on this later). Of course, lenders may not include all costs as some are contingent. APR does however allow for the timing of payments which is very important (re-read PF1 if in doubt). You might think that paying £6 of interest on a one year loan amounts to a 6% rate of interest. It does if and only if you repay the loan and the interest at the end of the year. However, if you pay interest in equal monthly instalments of 50p and repay the loan in equal monthly instalments of £8.33 (ignore rounding) it is actually a 11.4% APR (

A simple version of the EU APR formula used is P= ∑A/(1+r/100)^t. P= the loan amount including any up-front fees, A is the periodic repayment (interest, principal and any other fees) , r is the APR , t is the time since the loan began in years and fractions of a year. The ∑ just means add up all the terms that follow and the time since the loan began is derived by dividing the period by the number of periods. To check the calculation just expand the equation with A=8.83, r/100=0.114 and for t=1/12, 2/12…12/12. The first term is 8.83/(1.114)^1/12=8.7509 and the last and 12th term is 8.83/(1.114)^12/12=7.9264. Note the average value using just these two terms is 8.3387 which times 12 gives 100.064. If you expand all terms and sum you will get exactly 100, the amount of the loan. This confirms 11.4% is the APR. Formally it is necessary to solve for r to get the APR but there are many websites that offer calculators such as the one indicated above. However, be careful to choose one relevant to your jurisdiction.

If applied consistently APR is a useful way to compare loans. However, it may not be applied consistently. In the UK a representative APR is quoted based on various assumptions. These assumptions may not apply. In the case of mortgages, for example, the representative APR may be calculated over the whole term (say 30 years) with a two-year fixed rate and then assuming a standard variable rate. The borrower may never pay the standard variable rate and could switch mortgage providers at the end of the 2 year term. A better way to compare 2 year fixed options is to calculate the APR assuming a switch will take place. The cash flows must include all costs including any switching penalties. No one knows what the standard variable rate of any provider will be in two years so there is no point in including it in the calculation when choosing mortgage provider. It is however worth considering how interest rates might vary by the end of the period and what variation might cause financial difficulty. Such sensitivity analysis is essential when judging if borrowing is an affordable option. Job security, unexpected pregnancies, death, illness and interest rate variations are all sources of financial stress. Proper planning is essential and should be included in the cost of the loan (life insurance health insurance, critical illness insurance, unemployment insurance). The cost of financial distress can be far greater than people realise until it is too late. However, more when we cover borrowing.

In the UK savings accounts always quote an Annual Equivalent Rate alongside the gross rates. If interest is paid annually in arrears then the AER is the same as the annual gross rate quoted. However, if interest is paid more frequently the AER would be higher than the quoted gross rate because of compounding (interest on interest). In practice savings institutions offer lower gross rates on monthly interest paying accounts to ensure that the AER is the same whether you choose to receive interest annually or monthly. One can get a sense of the influence of compounding from here ( Note that Wikipedia refers to the effective interest rate but this is a misleading term as it has other meanings.

For most people the APR and the AER will suffice. However, some might wish to venture into the bond markets and all manner of ‘rates’ become apparent. For the purpose of this discussion we will restrict our attention to fixed coupon bonds that pay a fixed payment (hence the name) at regular intervals and repay the nominal value of the bond (usually 100) at maturity. Much UK government debt (Gilts) falls into this category. Take the UK Treasury Gilt 2.75% maturing 7/09/2024 as an example. At close of business 7/03/2014 it was quoted at a yield of 2.930587%. The clean price was 98.38. The redemption value of each bond is £100 and the coupon is paid every 6 months. So you could have a bought a bond paying £1.375 every 6 months for 98.38.  A quick look at the Debt Management Office (DMO) website ( will reveal a whole range of bonds with different coupons and prices and redemption dates. One will also see that there is something called a dirty price which includes any accrued interest that comes with the bond. How does one compare? Normal practice is to compare bonds according to the yield to maturity (YTM). This is the yield mentioned above for the UKT 2.75% 2024. Think of buying the bond as making a loan of £98.38 to the state in return for 6 monthly interest of £1.375 and a lump sum payment of £100 on 7/09/2024. Remind you of anything? The YTM is our friend the APR applied to the bond. However unlike a loan, where the amount is fixed at the start, the bond price changes continuously and so does the YTM or APR. As with APR the yield to maturity provides a way of comparing bonds of different coupons and maturing at different times. From the same logic used in PF1, if the price rises the yield will fall and if the price falls the yield will rise.

It is worth adding a few observations on Gilts. First, whatever you pay for a bond the redemption value is going to be £100. So if you pay more than £100 you have locked in a capital loss. There are plenty of bonds presently trading above £100 as one can see from the DMO website. This is because yields have fallen below where the were when the respective bonds were issued. By the same token our UKT 2.75% 2024 locks in a capital gain of £1.62 per bond. The gains and losses on coupon paying gilts are free of UK capital gain tax. This makes coupon Gilts trading below £100 very attractive to higher rate taxpayers. Note also that if you hold Gilts via a bond fund this tax efficiency may be lost depending upon the tax status of the fund. Gilts generally are available in a range of maturities and the longer maturity usually, but not always, comes with the higher a yield. This term structure of interest rates is familiar to most savers because longer fixed savings deposits usually offer a higher AER. A bond is like a longer maturity fixed deposit (in fact some such deposits are actually called bonds). The reason offered by economists for the upward sloping term structure of interest rates (higher longer maturity YTM and AER) is the uncertainty surrounding future rates. It is a little more complex than this but it is a good way to think about the issue when choosing at what maturity to fix your savings deposit or whether to buy a bond.

One could also ask what is the implied forward interest rate?Assume a deposit of £100 for one year pays interest of 5% at the end of the year. Assume also deposit of £100 fixed for 2 years pays 6% at the end of each year. The implied one year forward interest rate beginning at the start of the second year is 7.01%. The two-year can be thought of as a one year fix at 5% plus another one year fix beginning in one year. The total pay out from the two year fix at 6% is £12.36 at the end of the deposit period. The total pay out from the one year fix is £5. If we think of the two-year fix as the one year fix at 5% plus another one year fix starting after one year at r/100% we can work out the implied r. It needs to be a number that makes £105 turn into 112.36 in one year. Well £105 invested for one year at 7.01% results in £112.36. The implied forward rate is 7.01%. This is an interesting exercise. If you think the one year interest rate will be less 7.01% next year then the two year fix is a good idea. It is only a bad idea if you think the one year interest rate will rise above 7.01%. Assume the interest rate rises to 6.5%. If you invested in a one year fix then next year you can invest in another one year fix at 6.5%. The two investments result in £111.825 after two years , which is less than fixing for two years at 6% in the first place. How many people do this forward rate analysis before choosing how long to fix their savings for I wonder? Some very nice person has put a forward rate calculator onto the web at ( Try the example I have given and you will also see all the formulae pop up as well as the answer. Worth using before you fix.