Question about Duration

Hi friends, I came up with a question about Duration calculation:

Banks can purchase bonds for investment, and I know how to calculate the duration for bonds; however banks also have loans as assets, deposits as liabilities. In the IPS for banks for example, bank’s ALM needs to calculate Duration of Assets and Duration of Liabilities, and I’m confused how to calculate duration for something like a fixed rate loan or deposit?

Throughout Level 1 to Level 3, (modified) duration has been defined as the sensitivity of bond price wrt change in interest rate, i.e change in price given 1% change in interest rate. I don’t know how this concept could be extended to loan or deposit?

For example: if a 10-year loan with principal of $1,000,000, interest rate 5%, and analogous to bond duration calculation, I first calculated Macaulay Duration, and then Modified Duration, below is the Excel screenshot:

So, is my calculation process for duration of a loan correct?

If so, Modified Duration is 4.86, how to explain it? Again analogous to bond, PV is bond price, so here should the principle of $1M be the “price” of the loan? And approximate percentage change in bond price = -ModDur x change in YTM, so that means a 1% increase in the interest rate, the principle of the loan should fall by 4.86% or $48,600, so $1,000,000 loan now is as if it’s a $951,400 loan??

To calculate duration for Assets, duration of single asset must be calculated and then weighted average duration is the duration of the portfolio, and if for example the bank’s assets consists of one bond and one loan, in order to calculate the Duration of Assets, I must figure out how to calculate the duration of loan.

Any help is greatly appreciated!

Your calculation’s correct. (By the way, are you aware that Excel has built-in functions for Macaulay duration and modified duration?)

If you issue a bond (at par, say, to keep the discussion simple), that bond will have some Macaulay duration and some modified duration, even though you show it as a liability at par. That’s essentially what the bank is doing with its liabilities. The idea is that the market value of that liability will change, and that the bank (perhaps through an agent) might be able to settle the liability at the market value, much as a company that issues bonds can buy them back on the open market.

A par bond is essentially the same as a fixed-rate loan with bullet repayment in terms of cash flows (even though in one case the company issues the instrument and the “investors” buy it, i.e. lend the money, and in the other the bank, as the investor, directly lends the money), so there is no reason for duration to be different.

Keep in mind though that a very high % of bank loans are with floating rates (e.g. 3M Libor + 3% margin), thus hedging this MV risk naturally.